MathCAD
Boundary value problem
 Second
order differential equation
y  f x, y, y
have two initial values. They can be
placed in different points.
y  A for x  a
y  B for x  b
B
A
a
b
Boundary value problem
Other type of boundary conditions
y  A for x  a
y  tg for x  b

A
a
b
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)
Boundary value problem
Initial conditions required to start the
integrating procedure
y  A for x  a
y  tg for x  a

a
b
Boundary value problem
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

Boundary value problem
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
 dy1
equations
 dx  f  x, y1 , y2 

 dy2  f  x, y , y 
1
2
 dx
(range <a,b>): y , y
1a
1b
To start calculations the value of y2a is required
1.
2.
3.
4.
Assume y2a
Calculate values of y1, y2 until the point b is reached
Calculate the difference (error)
e = |y1b(calculated)-y1b,(given)|
If e>emax change y2aand go to p. 2
Boundary value problem
What is necessary to solve the boundary
values problem?
1. System of equations
2. Endpoints of the range of independent
variable (range boundaries)
3. Known starting point values
4. Starting point values to be guessed
5. Calculation of error of functions values
on the opposite (to starting point) side of
the range
Boundary value problem

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
type! (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.
Boundary value problem
Boundary value problem
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
Odesolve
 Syntax
 Keyword
Given
 Differential equation(s) using Boolean
equal(s) (bold =). Derivative symbols ` by
pressing [ctrl][F7] or constructions like d n
from calculus toolbar.
dx n
 Initial/boundary condition(s) (for derivatives
only ` symbols). Boolean equal.
 function_name:=Odesolve([v],x,b,[initvls])
Odesolve
 Additional
information:
– 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.
v
Odesolve
 One
second
order ODE
Odesolve
 System
of
two first
order ODE
Odesolve
 Numerical
methods:
 Adams/BDF
calls:
Adams-Bashford method for non-stiff
systems of ODE
 BDF method for stiff systems of ODE

– calls rkfixed
 Adaptive – calls Rkadapt
 Radau – calls Radau method – used
with algebraic constraints
 Fixed
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.
MathCAD symbolic operations

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
VALUE can be assigned to the
variables used in expressions!!
MathCAD symbolic operations

simple operations

Symbolic integration


Symbolic derivation


Indefinite integration operator (symbol),
expression, [ctrl]+[.]
Derivative operator, expression, [ctrl]+[.]
Calculation of limits, sums
MathCAD symbolic operations

Substitute - replace all occurrences of a
variable with another variable, an expression or
a number


expand - expands all powers and products of
sums in the selected expression


expression [ctrl][shift][.] substitute, substitution
equation (use bold = symbol)
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
MathCAD symbolic operations

Factor – transforms an expression (or number) into a
product (of prime numbers)

expression [ctrl][shift][.] factor


To convert an equation to a partial fraction, type:


expression, [ctrl][shift][.] convert,parfrac, variable
series keyword finds Taylor series


if the entire expression can be written as a product
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
MathCAD symbolic operations
 To
solve system of equation
 Type
Given
 Type equations (using [ctrl]+[=])
 find(var1, var2,..) [ctrl][.]
Units in MathCAD
 System
 SI
of units available in MathCAD:
- 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 problem
 Parameters
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