Module 4 Linear Boundary value problems

Transcription

1 Module 4 Linear Boundary value problems Lecture Content Hours 1 Finite Difference Metods: Diriclet type boundary 1 condition Finite Difference Metods: Mixed boundary condition 1 3 Sooting Metod 1 4 Sooting Metod contd 1

2 Module 4 Lecture 1 Finite Difference Metods: Diriclet type boundary condition keywords: Diriclet type boundary conditions, discretize, finite differences

3 Linear Boundary Value Problems An initial value problem consists of a differential equation and associated initial conditions for finding te unique solution of te problem. All te initial conditions are specified at one specified point called as initial point. Typical initial conditions for nt order differential equation involve specification of y, y,,y (n-1) at some initial point, say at t=0. For a boundary value problem boundary conditions are associated wit te differential equation. Te boundary conditions are specified at different points of te domain of independent variable t known as boundary points. For example, consider a second order ordinary differential equation y f(y,y,x);a x b 1(x)y(x) 1(x)y (x) 1atx a (4.1) (x)y(x) (x)y (x) atx b Te Boundary value is called two-point linear boundary value problem wen te arbitrary function f is given as a linear combination of dependent variable and its derivative as f(y,y,x) c(x)y (x) d(x)y(x) e(x) (4.) Te boundary conditions are specified at te boundary points x=a and x=b as linear combination of y and its derivative y. Tis is Robin mixed boundary conditions. In particular wen i 0, i 0;i 1, ten te boundary conditions are known as Diriclet boundary conditions. Te conditions are Neumann s boundary conditions wen i 0, i 0;i 1, Finite difference metod for two point linear Boundary Value problem wit Diriclet type conditions y c(x)y (x) d(x)y(x) e(x);a x b (4.3) y(a) 1; y(b) To apply finite difference metod first discretize te domain a x binto N-1 computational grid points x i;i 1,...N 1and two boundary points x 0 and x N as a x0 x1 x... xn1 xn b

4 Te grid points are equi-spaced and computed as b a xi x0 i; N Te step size is a critical parameter for stability and convergence of te numerical sceme. Te differential equation is now written at eac internal grid point x i;i 1,...N 1. For tis, te derivatives are replaced by corresponding finite differences: yi 1 yi y y i 1 x i o( ) yi 1 yi 1 y x i o( ) tat is or yi1yiyi1 yi 1 y c(x i 1 i) d(x i)yi e(x i) ( 1 c i)y i1( d)y i i ( 1 c i)yi1 e;i i 1,,N 1 (4.4) Te unknown yi s are on te left side and known quantities are on rigt side of te equation for i=,3, N-. Using boundary conditions for i=1and i=n-1 give ( d 1)y 1( 1 c 1)y e 1(1 c 1) 1 (4.6a) ( 1 c N1)y N( d N1)y N1 [ e N1(1 c N1) ] (4.6b) Tis reduces te system of differential equations to linear system of N-1 algebraic equations wic can be written in te matrix form as AX=B were u 1 1 u u N N N N1 N1 (N1)X(N1)

5 i i i d i 1 ci u 1 c i x1 x X. xn x N1 e 1 (1 ) 1 e B... e e (1 ) N N 1 (4.7) Te system of equations must admit unique solution for wic te sufficient condition is te diagonal dominance of te matrix A. Suppose d(x) as positive values in te domain and c(x) is continuous. Let L be upper bound of te function c(x) over te domain ten te step size smaller tan /L guarantees te uniqueness of te solution. Example 4.1: Solve te boundary value problem using N=4 y 1y16;y(0) y() 5 Solution: It is observed tat c(x)=0, d(x)=1>0, e(x)=-16. for N=4, te Bvp will reduce to system of tree algebraic equations wit step size =/4=0.5 Te equivalent system given below: 5y1y y15y y3 1 4 y 5y 1 3 Te system of algebraic equations corresponding to te BVP as unique solution irrespective of step size. Te solution of te system is y 1 1,y 18 1,y Example 4.1: Solve te boundary value problem using N=4 3 xy (x 1)y (x 1)y; y(1) e, y(3) 10e Solution: For te given boundary value problem a=1, b=3, N=4 =0.5 x 1 x 1 c(x),d(x),e(x) 0 x x

6 It may be noted tat d(x) is positive and c(x) is a decreasing function in (1, 3). Terefore, L=C(1)=3. Accordingly, te condition for unique solution ( </L) is satisfied for =0.5 Te grid points are x0 1.0,x11.5,x.0,x3.5,x4 3.0 Te coefficients of te matrix are computed from te expressions given below xi 1 xi 1 xi 1 di, i 1 ( ),ui 1 ( ), xi xi xi Applying finite differences gives te following system of equations:.41667y y = y y y 3=0-1.6 y +.35 y = Te coefficient matrix is diagonally dominant. Te system of equations can be solved using Gauss-Seidel iterative sceme wit initial guess as (0,0,0). For numeric computations refer to NPTEL-II\BVP-I.xls Te final solution is obtained as y 1= , y = , y 3=

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