MathCAD

Boundary value problem Second order differential equation

have two initial values. They can be placed in different points.

yyxfy ,,

a b

A

B

Ay for ax By for bx

Boundary value problem

Other type of boundary conditions

Ay for ax tgy for bx

a b

A

Boundary value problem Applies to second order differential

equations or systems of first order differential equations

Initial conditions are given on opposite boundaries of solving range

Numerical methods (usually) needs initial values focused in one point (one of the boundaries)


Initial conditions required to start the integrating procedure

Ay for ax tgy for ax

a b


We have to guess missing initial condition at the point we start the calculations

Conditions given Condition to guess

yA, yB y’A or y’B

yA, y’B y’A or yB

y’A, yB yA or y’B

Boundary value problem In the chemical and process

engineering: Displaced parameters: heat and mass

transfer Countercurrent heat exchangers Mass transfer with accompanying chemical

reaction


HOW TO GUESS??!!

1. Assume missing initial value(s) at start point

2. Make the calculation to the endpoint of independent variable range.

3. Check the difference between boundary condition calculated and given on the endpoint

4. If the difference (error) is too large change the assumed values and go back to point 2.

Boundary value problem Example:

Given initial conditions of system of two differential equations

(range <a,b>): y1a, y1b

To start calculations the value of y2a is required

1. Assume y2a

2. Calculate values of y1, y2 until the point b is reached

3. Calculate the difference (error) e = |y1b(calculated)-y1b,(given)|

4. If e>emax change y2aand go to p. 2

212

211

,,

,,

yyxfdx

dy

yyxfdx

dy

Boundary value problem What is necessary to solve the boundary

values problem?1. System of equations2. Endpoints of the range of independent

variable (range boundaries)3. Known starting point values4. Starting point values to be guessed5. Calculation of error of functions values

on the opposite (to starting point) side of the range

To find missing initial values in the MathCAD the sbval procedure can be used. SYNTAX: sbval(v, a, b, D, S, B) v – vector of guesses of searched initial values in the starting

point a (p. 4)

a, b – endpoints of the range on which the differential equation is being evaluated (p. 2)

D – vector function of independent variable and dependent variable vector, consists of right hand sides of equations. Dependent variables in the equations HAVE TO BE vector HAVE TO BE vector typetype! (p. 1)

S – vector function of starting point and known and searched (v) defining initial conditions on starting point (p. 3&4)

B – function (could be vector type) to calculate error on the endpoint (b) (p. 5)

Result: vector of searched initial conditions.


Odesolve

Overall ODE solving procedure

Odesolve Returns a function(s) of independent variable

which is a solution to the single ordinary differential equation or ODE system

Solving initial condition problem as well as boundary problem

Can solve single ODE and system of ODE Result is an implicit function

OdesolveSyntax

Keyword Given Differential equation(s) using Boolean

equal(s) (bold =). Derivative symbols ` by pressing [ctrl][F7] or constructions like from calculus toolbar.

Initial/boundary condition(s) (for derivatives only ` symbols). Boolean equal.

function_name:=Odesolve([v],x,b,[initvls])

n

n

dx

d

OdesolveAdditional information:

v – vector of functions names - for ODE system only

b – terminal point of the integration Initvls – number of discretization intervals

(def. 1000) functions have to be defined explicitly (y(x)

not just y) Algebraic constraints are accepted.

OdesolveOne second

order ODE

OdesolveSystem of

two first order ODE

OdesolveNumerical methods:

Adams/BDF calls: Adams-Bashford method for non-stiff

systems of ODE BDF method for stiff systems of ODE

Fixed – calls rkfixed Adaptive – calls Rkadapt Radau – calls Radau method – used

with algebraic constraints

MathCAD symbolic operations

Chosen symbolic operations accessible in MathCAD

Simple symbolic evaluation: algebraic expressions, derivating, integrating, matrix operations, calculation of limits etc.

Symbolic with keyword: substitute, expand, simplify, convert, parfrac, series, solve,...etc.


Symbolic operation are accessible from the Symbolic Toolbar or by the keystrokes:

[ctrl][.] simple operations [shift][ctrl][.] operations with keywords

To get the symbolic result NO NO VALUEVALUE can be assigned to the variables used in expressions!!


simple operations Symbolic integration

Indefinite integration operator (symbol), expression, [ctrl]+[.]

Symbolic derivation Derivative operator, expression, [ctrl]+[.]

Calculation of limits, sums

Substitute - replace all occurrences of a variable with another variable, an expression or a number expression [ctrl][shift][.] substitute, substitution

equation (use bold = symbol) expand - expands all powers and products of

sums in the selected expression expression [ctrl][shift][.] expand, variable

Simplify - carry out basic algebraic simplification, canceling common factors and apply trigonometric and inverse function identities expression [ctrl][shift][.] simplify


Factor – transforms an expression (or number) into a product (of prime numbers) expression [ctrl][shift][.] factor

if the entire expression can be written as a product

To convert an equation to a partial fraction, type: expression, [ctrl][shift][.] convert,parfrac, variable

series keyword finds Taylor series expression, [ctrl][shift][.] series, variable = central point of

expansion, order of approximation To solve single equation

expression [ctrl][shift][.] solve, variable Assumes expression equal 0


To solve system of equation Type Given Type equations (using [ctrl]+[=]) find(var1, var2,..) [ctrl][.]


Units in MathCAD

System of units available in MathCAD: SI - fundamental units: meters (m), kilograms

(kg), seconds (s), amps (A), Kelvin (K), candella (cd), moles (mole).

MKS - fundamental units: meters (m), kilograms (kg), seconds (sec), coulombs (coul), Kelvin (K)

CGS - fundamental units: centimeters (cm), grams (gm), seconds (sec), coulombs (coul), Kelvin (K)

US - fundamental units: feet (ft), pounds (lb), seconds (sec), coulombs (coul), Kelvin (K)

To add unit: type unit after number (MathCAD will add multiplication sign between number and units)

MathCAD converts units between Units Systems and between fundamental and derived unit. User can define new derived units as fallows:

derived_unit:=multiplier*fundamental_unit, e.g.: kPa:=1000*Pa

Independently of units used in data the results are given in fundamental units of actual Units System. Result unit can be changed!!

After the result of evaluation the placeholder appears. In these placeholder type the desired unit

Calculations with units.

Calculate volume of rectangular prism of size

ft

Units problemParameters with units can not be used

in the vector function definition of system of differential equations (especially from transformation of second order ODE to the system of first order ODE)

Solution: Multiply each element of sum in vector

function definition by inversion of its unit

MathCAD

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