lecture 15

lecture 15 - CE 30125 - Lecture 15 LECTURE 15 APPLICATIONS...

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CE 30125 - Lecture 15 p. 15.1 LECTURE 15 APPLICATIONS OF FD APPROXIMATIONS FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS Ordinary Differential Equations Initial Value Problems • For Initial Value problems (IVP’s), conditions are specified at only one value of the independent variable initial conditions (i.c.’s) • For example a simple harmonic oscillator is described by = location dependent variable = time independent variable A d 2 y dt 2 -------- B dy ----- Cy ++ gt  = y 0 y o = -----0 V o = y t
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CE 30125 - Lecture 15 p. 15.2 Boundary Value Problems • For Boundary Value Problems (BVP’s) conditions are specified at two values of the independent variable (which represent the actual physical boundaries) •Ex amp l e General Initial Value Problems • Any IVP can be represented as a set of one or more 1st order d.e.’s each with an i.c. l e • Let and we can develop a system of 2 first order O.D.E.’s which are coupled d 2 y dx 2 -------- D dy ----- Ey ++ hx  = y 0 y o = yL y l = A d 2 y dt 2 B C gt = y 0 y o = -----0 v o = z = z = y 0 y o = dz B A --- z C A A --------- + = z 0 v o =
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CE 30125 - Lecture 15 p. 15.3 • Therefore the general IVP can be written as: . . . • Possible solution strategies for higher order o.d.e.’s • Solutions of 1st order d.e.’s • single equation with associated i.c. • extension to coupled sets • Solution of higher order equations without reduction to a 1st order system must develop a method for a specific equation not as general may be advantageous in certain cases dy 1 dt -------- f 1 y 1 y 2 y n t   = y 1 0 y 10 = n f n y 1 y 2 y n t = y n 0 y n 0 =
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CE 30125 - Lecture 15 p. 15.4 Solution to a 1st Order Single Equation IVP with specified i.c.
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This note was uploaded on 03/02/2012 for the course CE 30125 taught by Professor Westerink,j during the Fall '08 term at Notre Dame.

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lecture 15 - CE 30125 - Lecture 15 LECTURE 15 APPLICATIONS...

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