Partial Differential Equations

Dr. Q. M. Zaigham Zia

Assistant ProfessorDepartment of Mathematics

COMSATS Institute of Information Technology Islamabad

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagation

Fluid Flow (air or liquid)- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings,

helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade,

atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere

- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media

- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water

- Weather: large system of coupled partial differential equations formomentum, pressure, moisture, heat,

VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat,

VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, Vibration

Mechanics of solids- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Motivation

Partial differential equations describe the behavior of many engineeringphenomena:

Heat flow and distribution

Electric fields and potentials

Diffusion of chemicals in air or water

Electromagnetism and quantum mechanics

Wave propagationFluid Flow (air or liquid)

- Air around wings, helicopter blade, atmosphere- Water in pipes or porous media- Material transport and diffusion in air or water- Weather: large system of coupled partial differential equations for

momentum, pressure, moisture, heat, VibrationMechanics of solids

- Stress-strain in material, machine part, structure

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first order

Linear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equations

Applications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equations

Classification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equations

Boundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problems

Reduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equations

Sturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville system

Technique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Course Contents

Introduction of partial differential equations

Partial Differential equations of first orderLinear and non-linear partial differential equationsApplications of first order partial differential equations

Partial Differential equations of second order

Mathematical Modeling of heat, Laplace and wave equationsClassification of second order partial differential equationsBoundary and Initial value problemsReduction to canonical form and the solution of second order partialdifferential equationsSturm-Liouville systemTechnique of Separation of Variable for the solution of partialdifferential equations

Laplace, Fourier and Hankel transforms for the solution of partialdifferential equations and their application to boundary valueproblems.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Recommended Books

Richard Haberman, Elementary partial differential equations,Prentice-Hall, INC., Englewood Cliffs, New Jersy

K. Sankara Rao, Introduction to partial differential equations,Prentice-Hall of India New Delhi.

M. Humi, W. B. Miller, Boundary value problems and partialdifferential equations, PWS-KENT publishing company, Boston.

T. Myint-U, L. Debnath, Linear partial differential equations forscientists and engineers, Fourth Edition, Birkhauser, Berlin

C. Constanda, Solution Techniques for Elementary Partial DifferentialEquations, Chapman & Hall/CRC, Washington DC.

Grading Scheme

Quiz 10%

Assignments 10%

Graded discussion 5%

Midterm 25%

Final 50%

Grading Scheme

Quiz 10%

Assignments 10%

Graded discussion 5%

Midterm 25%

Final 50%

Grading Scheme

Quiz 10%

Assignments 10%

Graded discussion 5%

Midterm 25%

Final 50%

Grading Scheme

Quiz 10%

Assignments 10%

Graded discussion 5%

Midterm 25%

Final 50%

Grading Scheme

Quiz 10%

Assignments 10%

Graded discussion 5%

Midterm 25%

Final 50%

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Differential Equation

Definition

A differential equation is an equation that relates the derivatives of afunction depending on one or more variables.

For exampled2u

dx2+

du

dx= cos x

is a differential equation involving an unknown function u(x) depending onone variable and

2u

x2+2u

2y=u

t

is a differential equation involving an unknown function u(t, x , y)depending on three variables.

Differential Equation

Definition

A differential equation is an equation that relates the derivatives of afunction depending on one or more variables.

For exampled2u

dx2+

du

dx= cos x

is a differential equation involving an unknown function u(x) depending onone variable and

2u

x2+2u

2y=u

t

is a differential equation involving an unknown function u(t, x , y)depending on three variables.

Differential Equation

Definition

A differential equation is an equation that relates the derivatives of afunction depending on one or more variables.

For exampled2u

dx2+

du

dx= cos x

is a differential equation involving an unknown function u(x) depending onone variable and

2u

x2+2u

2y=u

t

is a differential equation involving an unknown function u(t, x , y)depending on three variables.

Differential Equation

Definition

A differential equation is an equation that relates the derivatives of afunction depending on one or more variables.

For exampled2u

dx2+

du

dx= cos x

is a differential equation involving an unknown function u(x) depending onone variable and

2u

x2+2u

2y=u

t

is a differential equation involving an unknown function u(t, x , y)depending on three variables.

Differential Equation

Definition

A differential equation is an equation that relates the derivatives of afunction depending on one or more variables.

For exampled2u

dx2+

du

dx= cos x

is a differential equation involving an unknown function u(x) depending onone variable and

2u

x2+2u

2y=u

t

is a differential equation involving an unknown function u(t, x , y)depending on three variables.

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Partial Differential Equation

Definition

A partial differential equation (PDE) is an equation that contains, inaddition to the dependent and independent variables, one or more partialderivatives of the dependent variable.

Suppose that our unknown function is u and it depends on the twoindependent variables then the general form of the PDE is

F (x , y , , u, ux , uy , uxx , uxy , uyy , ) = 0

Here subscripts denotes the partial derivatives, for example

ux =u

x, uy =

u

y, uxx =

2u

x2, uxy =

2u

xy, uyy =

2u

y2

Partial Differential Equation

Definition

A partial differential equation (PDE) is an equation that contains, inaddition to the dependent and independent variables, one or more partialderivatives of the dependent variable.

Suppose that our unknown function is u and it depends on the twoindependent variables then the general form of the PDE is

F (x , y , , u, ux , uy , uxx , uxy , uyy , ) = 0

Here subscripts denotes the partial derivatives, for example

ux =u

x, uy =

u

y, uxx =

2u

x2, uxy =

2u

xy, uyy =

2u

y2

Partial Differential Equation

Definition

A partial differential equation (PDE) is an equation that contains, inaddition to the dependent and independent variables, one or more partialderivatives of the dependent variable.

Suppose that our unknown function is u and it depends on the twoindependent variables then the general form of the PDE is

F (x , y , , u, ux , uy , uxx , uxy , uyy , ) = 0

Here subscripts denotes the partial derivatives, for example

ux =u

x, uy =

u

y, uxx =

2u

x2, uxy =

2u

xy, uyy =

2u

y2

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ;

Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ;

Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0;

Order is Three

ut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Three

ut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ;

Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ;

Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0;

Order is One

Order of Partial Differential Equation

Definition

The order of a partial differential equation is the order of the highestordered partial derivative appearing in the equation

Examples: In the following examples, our unknown function is u and itdepends on three variables t, x and y .

uxx + 2xuxy + uyy = ey ; Order is Two

uxxy + xuyy + 8u = 7y ; Order is Three

ut 6uux + uxxx = 0; Order is Threeut + uux = uxx ; Order is Two

uxxx + xuxy + yu2 = x + y ; Order is Three

ux + uy = 0; Order is One

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ;

Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ;

Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0;

Degree is Three

ut + uu3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Three

ut + uu3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ;

Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ;

Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0;

Degree is One

Degree of Partial Differential Equation

Definition

The degree of a partial differential equation is the degree of the highestorder partial derivative occurring in the equation.

Examples: In the following examples, our unknown function is u and itdepends on two variables t, x and y .

uxx + 2xuxy + uyy = ey ; Degree is One

uxxy + xu2yy + 8u = 7y ; Degree is One

ut 6uux + u3xxx = 0; Degree is Threeut + uu

3x = uxx ; Degree is One

u2xxx + xu3xy + yu

2 = x + y ; Degree is Two

ux + uy = 0; Degree is One

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants.

/t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.

However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

A linear operator L by definition satisfies

L(c1u1 + c2u2) = c1L(u1) + c2L(u2) (1.0)

for any two functions u1 and u2, where c1 and c2 are arbitrary constants./t and 2/x2 are the examples of linear operators since these twosatisfy equation (17):

t(c1u1 + c2u2) = c1

u1t

+ c2u2t

2

x2(c1u1 + c2u2) = c1

2u1x2

+ c22u2x2

Note: Any linear combination of linear operators is a linear operator.However (/t)2 is not a linear operator since it does not satisfy equation(17).

Linear Partial Differential EquationLinear operator

To prove the non-linearity of the operator (/t)2, we do the followingcalculations. If our L = (/t)2 then Lu = (u/t)2

u derivative

u

tSquare

(u

t

)2

Now

((c1u1 + c2u2)

t

)2=

(c1u1t

+ c2u2t

)2=

(c1u1t

)2+

(c2u2t

)2+ 2c1c2

u1t

u2t

= c21

(u1t

)2+ c22

(u2t

)2+ 2c1c2

u1t

u2t

6= c1(u2t

)2+ c2

(u2t

)2

Linear Partial Differential EquationLinear operator

To prove the non-linearity of the operator (/t)2, we do the followingcalculations. If our L = (/t)2 then Lu = (u/t)2

u derivative

u

tSquare

(u

t

)2

Now

((c1u1 + c2u2)

t

)2=

(c1u1t

+ c2u2t

)2

=

(c1u1t

)2+

(c2u2t

)2+ 2c1c2

u1t

u2t

= c21

(u1t

)2+ c22

(u2t

)2+ 2c1c2

u1t

u2t

6= c1(u2t

)2+ c2

(u2t

)2

Linear Partial Differential EquationLinear operator

To prove the non-linearity of the operator (/t)2, we do the followingcalculations. If our L = (/t)2 then Lu = (u/t)2

u derivative

u

tSquare

(u

t

)2

Now

((c1u1 + c2u2)

t

)2=

(c1u1t

+ c2u2t

)2=

(c1u1t

)2+

(c2u2t

)2+ 2c1c2

u1t

u2t

= c21

(u1t

)2+ c22

(u2t

)2+ 2c1c2

u1t

u2t

6= c1(u2t

)2+ c2

(u2t

)2

Linear Partial Differential EquationLinear operator

To prove the non-linearity of the operator (/t)2, we do the followingcalculations. If our L = (/t)2 then Lu = (u/t)2

u derivative

u

tSquare

(u

t

)2

Now

((c1u1 + c2u2)

t

)2=

(c1u1t

+ c2u2t

)2=

(c1u1t

)2+

(c2u2t

)2+ 2c1c2

u1t

u2t

= c21

(u1t

)2+ c22

(u2t

)2+ 2c1c2

u1t

u2t

6= c1(u2t

)2+ c2

(u2t

)2

Linear Partial Differential EquationLinear operator

To prove the non-linearity of the operator (/t)2, we do the followingcalculations. If our L = (/t)2 then Lu = (u/t)2

u derivative

u

tSquare

(u

t

)2

Now

((c1u1 + c2u2)

t

)2=

(c1u1t

+ c2u2t

)2=

(c1u1t

)2+

(c2u2t

)2+ 2c1c2

u1t

u2t

= c21

(u1t

)2+ c22

(u2t

)2+ 2c1c2

u1t

u2t

6= c1(u2t

)2+ c2

(u2t

)2

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = u

First-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0

Schrodingers equation: iut + u = 0

Linear Partial Differential Equation

Definition

A partial differential equation is said to be a linear if

i) it is linear in the unknown function and

ii) all the derivatives of the unknown functions with constant coefficientsor the coefficients depends on the independent variables.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are linear partialdifferential equations

Laplaces equations: u = 0, where = uxx + uyy

Helmholtzs equation: = uFirst-order linear transport equation: ut + cux = 0

Heat or diffusion equation: ut u = 0Schrodingers equation: iut + u = 0

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Quasi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a quasi-linear if the derivativesof unknown function are linear, while the coefficients depends on theindependent variables as well as the dependent variables.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are quasi-linear partialdifferential equations

uux + uy = 0

yuxx + 2xyuuyy + u = 1

uxxy + xuuyy + 8u = 7y

Quasi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a quasi-linear if the derivativesof unknown function are linear, while the coefficients depends on theindependent variables as well as the dependent variables.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are quasi-linear partialdifferential equations

uux + uy = 0

yuxx + 2xyuuyy + u = 1

uxxy + xuuyy + 8u = 7y

Quasi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a quasi-linear if the derivativesof unknown function are linear, while the coefficients depends on theindependent variables as well as the dependent variables.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are quasi-linear partialdifferential equations

uux + uy = 0

yuxx + 2xyuuyy + u = 1

uxxy + xuuyy + 8u = 7y

Quasi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a quasi-linear if the derivativesof unknown function are linear, while the coefficients depends on theindependent variables as well as the dependent variables.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are quasi-linear partialdifferential equations

uux + uy = 0

yuxx + 2xyuuyy + u = 1

uxxy + xuuyy + 8u = 7y

Quasi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a quasi-linear if the derivativesof unknown function are linear, while the coefficients depends on theindependent variables as well as the dependent variables.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are quasi-linear partialdifferential equations

uux + uy = 0

yuxx + 2xyuuyy + u = 1

uxxy + xuuyy + 8u = 7y

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Semi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a semi-linear if the derivativesof unknown function are linear and the coefficients of derivatives dependson the independent variables only.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are semi-linear partialdifferential equations

ux + uxy = u2

yuxx + (2x + y)uyy + u3 = 1

uxxy + xuyy + 8u = 7xyu2

Semi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a semi-linear if the derivativesof unknown function are linear and the coefficients of derivatives dependson the independent variables only.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are semi-linear partialdifferential equations

ux + uxy = u2

yuxx + (2x + y)uyy + u3 = 1

uxxy + xuyy + 8u = 7xyu2

Semi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a semi-linear if the derivativesof unknown function are linear and the coefficients of derivatives dependson the independent variables only.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are semi-linear partialdifferential equations

ux + uxy = u2

yuxx + (2x + y)uyy + u3 = 1

uxxy + xuyy + 8u = 7xyu2

Semi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a semi-linear if the derivativesof unknown function are linear and the coefficients of derivatives dependson the independent variables only.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are semi-linear partialdifferential equations

ux + uxy = u2

yuxx + (2x + y)uyy + u3 = 1

uxxy + xuyy + 8u = 7xyu2

Semi-Linear Partial Differential Equation

Definition

A partial differential equation is said to be a semi-linear if the derivativesof unknown function are linear and the coefficients of derivatives dependson the independent variables only.

Examples: Suppose u is our unknown function which depends on twoindependent variables x and y then the following are semi-linear partialdifferential equations

ux + uxy = u2

yuxx + (2x + y)uyy + u3 = 1

uxxy + xuyy + 8u = 7xyu2

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)

Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Non-Linear Partial Differential Equation

Definition

A partial differential equation which is not linear is called non-linear partialdifferential equation.

Examples: Suppose u is our unknown function which depends on threeindependent variables t, x and y then the following are non-linear partialdifferential equations:

Non linear Poisson equation: u = f (u)Burgers equation: ut + uux = 0

Korteweg-deVries equation(KdV): u + uux + uxxx = 0

Reaction-diffusion equation: ut u = f (u)

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Homogeneous Partial Differential Equation

Definition

A partial differential equation is said to be a homogeneous partialdifferential equation if its all terms contain the unknown functions or itsderivatives.

The simplest way to test whether an equation is homogeneous is tosubstitute the function u identically equal to zero. If u = 0 satisfies theequation then the equation is called homogeneous.Examples:

uxx + xuxy + yu2 = 0 is a homogeneous equation

uxx + 2xuxy + 5u = 0 is a homogeneous equation

3ux + uuy = f (x , y) is a non-homogeneous equation

Homogeneous Partial Differential Equation

Definition

A partial differential equation is said to be a homogeneous partialdifferential equation if its all terms contain the unknown functions or itsderivatives.

The simplest way to test whether an equation is homogeneous is tosubstitute the function u identically equal to zero. If u = 0 satisfies theequation then the equation is called homogeneous.

Examples:

uxx + xuxy + yu2 = 0 is a homogeneous equation

uxx + 2xuxy + 5u = 0 is a homogeneous equation

3ux + uuy = f (x , y) is a non-homogeneous equation

Homogeneous Partial Differential Equation

Definition

A partial differential equation is said to be a homogeneous partialdifferential equation if its all terms contain the unknown functions or itsderivatives.

The simplest way to test whether an equation is homogeneous is tosubstitute the function u identically equal to zero. If u = 0 satisfies theequation then the equation is called homogeneous.Examples:

uxx + xuxy + yu2 = 0 is a homogeneous equation

uxx + 2xuxy + 5u = 0 is a homogeneous equation

3ux + uuy = f (x , y) is a non-homogeneous equation

Homogeneous Partial Differential Equation

Definition

A partial differential equation is said to be a homogeneous partialdifferential equation if its all terms contain the unknown functions or itsderivatives.

The simplest way to test whether an equation is homogeneous is tosubstitute the function u identically equal to zero. If u = 0 satisfies theequation then the equation is called homogeneous.Examples:

uxx + xuxy + yu2 = 0 is a homogeneous equation

uxx + 2xuxy + 5u = 0 is a homogeneous equation

3ux + uuy = f (x , y) is a non-homogeneous equation

Homogeneous Partial Differential Equation

Definition

A partial differential equation is said to be a homogeneous partialdifferential equation if its all terms contain the unknown functions or itsderivatives.

The simplest way to test whether an equation is homogeneous is tosubstitute the function u identically equal to zero. If u = 0 satisfies theequation then the equation is called homogeneous.Examples:

uxx + xuxy + yu2 = 0 is a homogeneous equation

uxx + 2xuxy + 5u = 0 is a homogeneous equation

3ux + uuy = f (x , y) is a non-homogeneous equation

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Solution of Partial Differential Equation

Definition

Functions u = u(x , y , ) which satisfy the following partial differentialequation

F (x , y , , u, u, ux , uy , , uxx , uxy , uyy , ) = 0

identically in a suitable domain D of the ndimensional space Rn in theindependent variables x , y , , if exist are called solutions.Example: Functions u(x , y) = (x + y)3 and u(x , y) = sin(x y) aresolutions of the differential equation

uxx uyy = 0.

Solution of Partial Differential Equation

Definition

Functions u = u(x , y , ) which satisfy the following partial differentialequation

F (x , y , , u, u, ux , uy , , uxx , uxy , uyy , ) = 0

identically in a suitable domain D of the ndimensional space Rn in theindependent variables x , y , , if exist are called solutions.Example: Functions u(x , y) = (x + y)3 and u(x , y) = sin(x y) aresolutions of the differential equation

uxx uyy = 0.

Solution of Partial Differential Equation

Definition

Functions u = u(x , y , ) which satisfy the following partial differentialequation

F (x , y , , u, u, ux , uy , , uxx , uxy , uyy , ) = 0

identically in a suitable domain D of the ndimensional space Rn in theindependent variables x , y , , if exist are called solutions.

Example: Functions u(x , y) = (x + y)3 and u(x , y) = sin(x y) aresolutions of the differential equation

uxx uyy = 0.

Solution of Partial Differential Equation

Definition

Functions u = u(x , y , ) which satisfy the following partial differentialequation

F (x , y , , u, u, ux , uy , , uxx , uxy , uyy , ) = 0

identically in a suitable domain D of the ndimensional space Rn in theindependent variables x , y , , if exist are called solutions.Example: Functions u(x , y) = (x + y)3 and u(x , y) = sin(x y) aresolutions of the differential equation

uxx uyy = 0.

Outline

1 Basic DefinitionsDifferential EquationsPartial Differential Equation (PDE)Order of Partial Differential EquationDegree of Partial Differential EquationLinear Partial Differential Equation

Linear Operator

Quasi-Linear Partial Differential EquationSemi-Linear Partial Differential EquationNon-Linear Partial Differential EquationHomogeneous Partial Differential Equation

2 Solution of Partial Differential EquationPrinciple of Superposition

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),

that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,

1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition,

c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2

is again thesolution of equation (31).

Principle of Superposition

Definition

If u1 and u2 are two solutions of a linear homogeneous partial differentialequation then an arbitrary linear combination of them c1u1 + c2u2 alsosatisfies the same linear homogeneous differential equation.

Examples: Consider a one dimensional heat equation

1

k

u

t=2u

x2. (2.0)

If u1 and u2 are two solutions of equations (31) then they must satisfyequation (31),that is,

1

k

u1t

=2u1x2

,1

k

u2t

=2u2x2

.

According to the principle of superposition, c1u1 + c2u2 is again thesolution of equation (31).

Principle of Superposition

L.H.S.

=1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2)

= c11

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2)

= R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions.

Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations, then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =1

k

t(c1u1 + c2u2) = c1

1

k

u1t

+ c21

k

u2t

= c12u1x2

+ c22u2x2

=2

x2(c1u1 + c2u2) = R. H. S.

Note: Here the superposition principle is stated for two solutions only, it istrue for any finite linear combinations of solutions. Furthermore, underproper restrictions, it is also true for infinite number of solutions. Ifui , i = 1, 2, are solutions of a homogeneous linear partial differentialequations,

then

w =n

i=1

ciui

is also a solution.

Principle of Superposition

L.H.S. =