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MathCAD

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MathCAD Boundary value problem Second order differential equation have two initial values. They can be placed in different points. Boundary value problem Other type ... – PowerPoint PPT presentation

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Title: MathCAD


1
MathCAD
2
Boundary value problem
  • Second order differential equation
  • have two initial values. They can be placed in
    different points.

for
for
3
Boundary value problem
  • Other type of boundary conditions

for
for
b
A
4
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)

5
Boundary value problem
  • Initial conditions required to start the
    integrating procedure

for
for
a
6
Boundary value problem
  • We have to guess missing initial condition at the
    point we start the calculations

Conditions given Condition to guess
yA, yB yA or yB
yA, yB yA or yB
yA, yB yA or yB
7
Boundary value problem
  • In the chemical and process engineering
  • Displaced parameters heat and mass transfer
  • Countercurrent heat exchangers
  • Mass transfer with accompanying chemical reaction

8
Boundary value problem
  • HOW TO GUESS??!!
  • Assume missing initial value(s) at start point
  • Make the calculation to the endpoint of
    independent variable range.
  • Check the difference between boundary condition
    calculated and given on the endpoint
  • If the difference (error) is too large change the
    assumed values and go back to point 2.

9
Boundary value problem
  • Example
  • Given initial conditions of system of two
    differential equations
  • (range lta,bgt) y1a, y1b
  • To start calculations the value of y2a is
    required
  • Assume y2a
  • Calculate values of y1, y2 until the point b is
    reached
  • Calculate the difference (error) e
    y1b(calculated)-y1b,(given)
  • If egtemax change y2aand go to p. 2

10
Boundary value problem
  • What is necessary to solve the boundary values
    problem?
  • System of equations
  • Endpoints of the range of independent variable
    (range boundaries)
  • Known starting point values
  • Starting point values to be guessed
  • Calculation of error of functions values on the
    opposite (to starting point) side of the range

11
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. 34)
  • B function (could be vector type) to calculate
    error on the endpoint (b) (p. 5)
  • Result vector of searched initial conditions.

12
Boundary value problem
13
Boundary value problem
14
Odesolve
  • Overall ODE solving procedure

15
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

16
Odesolve
  • Syntax
  • Keyword Given
  • Differential equation(s) using Boolean equal(s)
    (bold ). Derivative symbols by pressing
    ctrlF7 or constructions like from calculus
    toolbar.
  • Initial/boundary condition(s) (for derivatives
    only symbols). Boolean equal.
  • function_nameOdesolve(v,x,b,initvls)

17
Odesolve
  • Additional 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.

18
Odesolve
  • One second order ODE

19
Odesolve
  • System of two first order ODE

20
Odesolve
  • Numerical 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

21
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.

22
MathCAD symbolic operations
  • Symbolic operation are accessible from the
    Symbolic Toolbar or by the keystrokes
  • ctrl. simple operations
  • shiftctrl. operations with keywords
  • To get the symbolic result NO VALUE can be
    assigned to the variables used in expressions!!

23
MathCAD symbolic operations
  • simple operations
  • Symbolic integration
  • Indefinite integration operator (symbol),
    expression, ctrl.
  • Symbolic derivation
  • Derivative operator, expression, ctrl.
  • Calculation of limits, sums

24
MathCAD symbolic operations
  • Substitute - replace all occurrences of a
    variable with another variable, an expression or
    a number
  • expression ctrlshift. substitute,
    substitution equation (use bold symbol)
  • expand - expands all powers and products of sums
    in the selected expression
  • expression ctrlshift. expand, variable
  • Simplify - carry out basic algebraic
    simplification, canceling common factors and
    apply trigonometric and inverse function
    identities
  • expression ctrlshift. simplify

25
MathCAD symbolic operations
  • Factor transforms an expression (or number)
    into a product (of prime numbers)
  • expression ctrlshift. factor
  • if the entire expression can be written as a
    product
  • To convert an equation to a partial fraction,
    type
  • expression, ctrlshift. convert,parfrac,
    variable
  • series keyword finds Taylor series
  • expression, ctrlshift. series, variable
    central point of expansion, order of
    approximation
  • To solve single equation
  • expression ctrlshift. solve, variable
  • Assumes expression equal 0

26
MathCAD symbolic operations
  • To solve system of equation
  • Type Given
  • Type equations (using ctrl)
  • find(var1, var2,..) ctrl.

27
Units in MathCAD
28
  • 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)

29
  • 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_unitmultiplierfundamental_unit,
  • e.g. kPa1000Pa

30
  • 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

31
Calculations with units.
Calculate volume of rectangular prism of size
ft
32
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
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