MathCAD

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MathCAD
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Boundary value problem

• Second order differential equation
• have two initial values. They can be placed in
  different points.

for
for
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Boundary value problem

• Other type of boundary conditions

for
for
b
A
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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)

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Boundary value problem

• Initial conditions required to start the
  integrating procedure

for
for
a
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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
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Boundary value problem

• In the chemical and process engineering
• Displaced parameters heat and mass transfer
• Countercurrent heat exchangers
• Mass transfer with accompanying chemical reaction

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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.

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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

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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

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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.

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Boundary value problem
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Boundary value problem
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Odesolve

• Overall ODE solving procedure

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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

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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)

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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.

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Odesolve

• One second order ODE

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Odesolve

• System of two first order ODE

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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

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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.

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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!!

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MathCAD symbolic operations

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

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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

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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

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MathCAD symbolic operations

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

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Units in MathCAD
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• 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)

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• 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

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• 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

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