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 initial conditions
yA
y  B
tga=B
A
a
b
for
for
xa
xb
Boundary value problem
 Concerns
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 to start the integrating
procedure
yA
y  B
a
tga=B
b
for
for
xa
xa
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
HOW TO GUESS??!!
1. Assume missing initial value(s) at start point
2. Make the calculation to the endpoint of
independent variable
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 necessary
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
3. Known starting point values
4. Starting point values to guess
5. Calculation of error of functions values
on the opposite side of interval
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)





a, b – endpoints of the interval on which the differential
equation is being evaluated (p. 2)
v – vector of guesses of searched initial values in the
starting point a (p. 4)
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 vector of guesses
(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)
Boundary value problem
Boundary value problem
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,
MathCAD symbolic operations

Symbolic operation are accessible
from the Symbolic Toolbar or through
the keys:



[ctrl][.] simple operations
[shift][ctrl][.] operations with keywords
To get the symbolic result NO
VALEUE can be assigned to the
variables used in expressions!!
MathCAD symbolic operations

simple operations

Symbolic integration


Indefinite integration sign, expresion,
[ctrl]+[.]
Symbolic derivation

Derivative sign, expression, [ctrl]+[.]
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
expression [ctrl][shift][.] expand
Simplify - carry out basic algebraic
simplification and apply trigonometric and
inverse function identities

expression [ctrl][shift][.] simplify
MathCAD symbolic operations

Factor – transforms an expression into a product

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 product of differences of type independent variable integer data
exists
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.
 It
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