MathCAD

1. MathCAD

2. B A a b Boundary value problem • Second order differential equation have two initial values. They can be placed in different points. for for

3. a b Boundary value problem Other type of initial conditions for for tga=B A

4. 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)

5. a b Boundary value problem Initial conditions to start the integrating procedure for for tga=B

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

7. 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.

8. Boundary value problem Example: Given initial conditions of system of two differential equations (range <a,b>): y1a, y1b To start calculations the value of y2a is necessary • Assume y2a • Calculate values of y1, y2until the point b is reached • Calculate the difference (error) e= |y1b(calculated)-y1b,(given)| • If e>emax change y2aand go to p. 2

9. 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 • Calculationof error of functionsvalues on the opposite side of interval

10. 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)

11. Boundary value problem

12. Boundary value problem

13. 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,

14. 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!!

15. MathCAD symbolic operations • simple operations • Symbolic integration • Indefinite integration sign, expresion, [ctrl]+[.] • Symbolic derivation • Derivative sign, expression, [ctrl]+[.]

16. MathCAD symbolic operations • Substitute - replace all occurrences of a variable with another variable, an expression or a number • expression [ctrl][shift][.] substitute, substitution equation • expand - expands all powers and products of sums in the selected expression • expression [ctrl][shift][.] expand • Simplify - carry out basic algebraic simplification and apply trigonometric and inverse function identities • expression [ctrl][shift][.] simplify

17. MathCAD symbolic operations • Factor – transforms an expression into a product • expression [ctrl][shift][.] factor • If product of differences of type independent variable integer data exists • To convert an equation to a partial fraction, type: • expression, [ctrl][shift][.] convert,parfrac, variable • series keyword finds Taylor series • 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

18. MathCAD symbolic operations • To solve system of equation • Type Given • Type equations (using [ctrl]+[=]) • find(var1, var2,..) [ctrl][.]

19. Units in MathCAD

20. 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)

21. 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

22. 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

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