Math CAD Boundary value problem Second order differential

Math. CAD

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

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

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)

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

Boundary value problem We have to guess missing initial condition at the point we start the calculations Conditions given Condition to guess y. A, y. B y’A or y’B y. A, y’B y’A or y. B y’A, y. B y. A or y’B

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 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 equations (range <a, b>): y 1 a, y 1 b To start calculations the value of y 2 a is necessary 1. 2. 3. 4. Assume y 2 a Calculate values of y 1, y 2 until the point b is reached Calculate the difference (error) e = |y 1 b(calculated)-y 1 b, (given)| If e>emax change y 2 aand 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 3. Known starting point values 4. Starting point values to guess 5. Calculation of error of functions values on the opposite side of interval

Boundary value problem ® To find missing initial values in the Math. CAD 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! 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) ®

Boundary value problem

Boundary value problem

Math. CAD symbolic operations ® Chosen symbolic operations accessible in Math. CAD ® ® Simple symbolic evaluation: algebraic expressions, derivating, integrating, matrix operations, calculation of limits etc. Symbolic with keyword: substitute, expand, simplify, convert, parfrac, series, solve,

Math. CAD 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!!

Math. CAD symbolic operations ® simple operations ® Symbolic integration ® ® Indefinite integration sign, expresion, [ctrl]+[. ] Symbolic derivation ® Derivative sign, expression, [ctrl]+[. ]

Math. CAD 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

Math. CAD symbolic operations ® Factor – transforms an expression ® expression [ctrl][shift][. ] factor ® into a product 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

Math. CAD symbolic operations ® To solve system of equation ® Type Given ® Type equations (using [ctrl]+[=]) ® find(var 1, var 2, . . ) [ctrl][. ]

Units in Math. CAD

® System ® SI of units available in Math. CAD: - 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 (Math. CAD will add multiplication sign between number and units) ® Math. CAD 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. : k. Pa: =1000*Pa

® 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

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