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# Class Note for COSC 3361 at UH

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This 12 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15

COSC 3361 Numerical Analysis Ordinary Differential Equations VI Boundary value problem Edgar Gabriel Fall 2005 0080 3361 Numerical Analysis Edgar Gabriel C S Initial value vs boundary value problems Initial value problem y fty aStSb ya yo Boundary value problem y ftyy aStSb ya ya W yb 0080 3361 Numerical Analysis Edgar Gabriel 2 C J Existence and uniqueness of the solution of initial value Problems Existence theorem for initialvalue Problems If fis continuous in a rectangle R centered around then the initial value problem lt tol s a y yol s y f Ly ya yo a solution yt for lt tol S mina M With M maxlfU yl on R Uniqueness theorem for initialvalue problems a If f and i are continuous on the rectangle R defined previously then the initial value problem has a unique solution in the interval lttol SminW M 0080 3361 Numerical Analysis Edgar Gabriel 3 C J Existence Theorem for boundary value problems The boundary value problem y fty y0 O y1 o a has a unique solution If IS continuous nonnegative and bounded in the infinite strip defined by OStSl ooSySoo 0080 3361 Numerical Analysis Edgar Gabriel 4 C J Example Check whether the following two points boundaryvalue has a unique solution y 5ySin3ye y0 0 N 0 Solution 31 53cos 3ye 7 afiscontinuousinthestrip 031 SyS a Bf 37 IS bounded above by 8e gt Problem has a unique solution 0080 3361 Numerical Analysis Edgar Gabriel 5 C J Theorems on boundary value problems ll One can show that the problem y fty ya a W can be transformed into 5 gtz 210 a z1 gtz b a2fa b atz through a change of variables yt Zt a b 61 or Zt ya b at 0080 3361 Numerical Analysis Edgar Gabriel 6 C J Theorems on boundary value problems Ill One can show that the problem y ay y0 a y1 can be transformed into Z 80 z0 0 z1 0 gtzftza at through a change of variables yo zt a at or zt ya a at 0080 3361 Numerical Analysis Edgar Gabriel 7 C J Shooting Methods I Given the boundary value problem y ftay aStSb 81 yaa W Solution of 81 can be obtained by guessing an appropriate second initial valuey a z and solve the initial value problem Resulting value 3219 Objective select y a z such that 3 0080 3361 Numerical Analysis Edgar Gabriel 8 C J Shooting Methods ll For that we define the nonlinear function 19z z and find the zero of that function Any of the methods discussed in the section Solution of nonlinear equations can be used to solve the problem Bisection Newton Secant Note calculating 3z is very expensive since it involves the solution of an initial value problem eg using a RungeKutta or a multistep method 0080 3361 Numerical Analysis Edgar Gabriel 9 C J Shooting Methods Ill Secant Method X X xn1xnfxn n n1 fxnfxn1 Applied to the boundary value problem leads to Zn Zn l 6 Zn1 Zn Zn 6Zn0znl 0080 3361 Numerical Analysis Edgar Gabriel 10 C J Algorithm input f PhiO myrungeatb f a b alpha zO beta Phil myrungeatb f a b alpha zl beta for k 1 MAX 2 21 Phil zl zOPhil Phi0 PhiO Phil Phil myrungeatb f a b alpha 2 beta if Phi lt epsilon break 20 21 21 2 end a b 20 f 2nd order ODE ab interval alpha beta boundary value of the ODE at a resp b 2021 two initial guesses for the second initial value epsilong desired accuracy 21 alpha beta epsilon myrungegatgb solves f in the interval ab and returns the function value at b Since f describes a 2nd order ODE we need two initial values Interpolation Since calculating 62 is very expensive polynomial interpolation is often used to estimate 2 such that 6z0 Note the inverse notation 0a 9a lac Z0 Z1 Zn p9ZnZn Calculating pO 0080 3361 Numerical Analysis Edgar Gabriel 12 C J

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