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 initial conditions

for
for
tgaB
A
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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)

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Boundary value problem

• Initial conditions to start the integrating
  procedure

for
for
tgaB
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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

• HOW TO GUESS??!!
• Assume missing initial value(s) at start point
• Make the calculation to the endpoint of
  independent variable
• 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
  necessary
• 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
• Known starting point values
• Starting point values to guess
• Calculation of error of functions values on the
  opposite side of interval

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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)
• 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. 34)
• B function (could be vector type) to calculate
  error on the endpoint (b) (p. 5)

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Boundary value problem
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Boundary value problem
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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,

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MathCAD symbolic operations

• Symbolic operation are accessible from the
  Symbolic Toolbar or through the keys
• ctrl. simple operations
• shiftctrl. operations with keywords
• To get the symbolic result NO VALEUE can be
  assigned to the variables used in expressions!!

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MathCAD symbolic operations

• simple operations
• Symbolic integration
• Indefinite integration sign, expresion,
  ctrl.
• Symbolic derivation
• Derivative sign, expression, ctrl.

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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
• expand - expands all powers and products of sums
  in the selected expression
• expression ctrlshift. expand
• Simplify - carry out basic algebraic
  simplification and apply trigonometric and
  inverse function identities
• expression ctrlshift. simplify

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MathCAD symbolic operations

• Factor transforms an expression into a product
• expression ctrlshift. factor
• If product of differences of type independent
  variable integer data exists
• 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.
• It 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