Transcript Document
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