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. =
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