Question

Write a MATLAB code to solve below 2nd order linear ordinary differential equation by finite difference method: y"-y'-0 in domain (-1, 1) with boundary condition y(x-1)--1 and y(x-1)-1. with boundary condition y an Use 2nd order approximation, i.e. O(dx2), and dx-0.05 to obtain numerical solution. Then plot the numerical solution as scattered markers together wi exp(2)-explx+1) as a continuous curve. Please add legend in your plot th the analytical solution y-1+

Answer 1

Please refer to the matlab script bvpSolver.m at the end. Run the script to generate the required plot, the screen shot of the code and the plot are also attached at the end.

Such that y'( 1)--1 and y (x -1) - 1 The second order approximation to y' and y' respectively are 2y; +yi- and (Ax)2 2(4x) Substitute the approximations in the given diffe rential equation to obtain the discretized difference equation at the ith hode Y,+, - 2y, yi- Note ťİ; is an internal node i.e. the number of nodes are given as 1-1 Ther fore, the above difference equation holds fOr 0.05

I=¼ 2, 3, , 40 except the end nodes i=1 and 41. Now, use boundary conditions associated with nodes i and 41, 2 4x ~-3 yi + 4yi+,-yi+2 İs a second order accurate forward finite diffevence scheme so, the system of difference equations are for nodes ronging From i 1, 2, 40 are as under Whee 2, 3, , 40 and atrix to 2 For i=3

Likewise, for i=40 24 Δχ 4 D Rewrite in matrix form the system of equations 4 (2+A2) -4 (2+5%) (2-AX) 4- 24 BX 40X 40 520% y2 (2-Ax) 40 40X1 40XI

so, the matrix equation is Ay=b and Later calculations and plots are shown in the Matlab.

bvpSolver.m

clear, close all; clc;

a = -1 ;
b = 1 ;
dx = 0.05 ;
numIntervals = ( b - a ) / dx ;
numNodes = numIntervals + 1 ;

e0 = (2 + dx) * ones( numNodes-1, 1 ) ;
e1 = -4 * ones( numNodes-1, 1 ) ;
e2 = (2 - dx) * ones( numNodes-1, 1 ) ;
A = spdiags( [e0, e1, e2], -1:1, numNodes-1, numNodes-1 ) ;
A(1, 1) = 3; A(1, 2) = -4; A(1, 3) = 1;

b = zeros( 1, numNodes-1 );
b(1) = 2 * dx ;
b(end) = -(2 - dx) ;

y = A\b' ;

yAll = [ y; 1 ];
xAll = a + (0 : numNodes-1) * dx ;
yTrue = 1 + exp(2) - exp(xAll + 1);

plot( xAll, yAll, 'o', xAll, yTrue, '-' )
xlabel( '\bf x' )
ylabel( '\bf y' )
legend( 'numerical', 'analytical' )
grid on
grid minor

O numerical analytical 4 0.5 0.5

Name 1-clear, close all; clc; 40x40 sparse do... 1x40 double 0.0500 40x1 double 40x1 double 40x1 double 3- a=-1; 4-b=1; 5- dx = 0.05 ; 6-numIntervals (b - a/ dx; dx e0 numintervals 40 XAll yAll 7-numNodesnumIntervals1; Ix4l double 40x1 double 41x1 double Ix41 double 8 9- e0 (2 + dx) * ones ( numNodes-1, 1 ) ; 10- e1 = -4 * ones ( numNodes-1, 1 ) 1e2(2 - dx) *ones( numNodes-1, 1; 12-A spdiags[e0, e, e2], 1:1, numNodes-1, numNodes-1; yTrue 14 15-b-zeros( 1, numNodes-1 16-b (1) 2 * dx; 17-b (end) - (2 - dx); 18 20 22- XA11 = a + (0 : numNodes-1) * dx ; 23 yTrue-1 exp (2) exp(xAl 1): 24 25-plot( xALl, yAll, o xAl, yTrue, '-') 26-xlabel'bf x') 27-ylabel'lbf y' 28-egend 'numerical, 'analytical') 29grid on 30grid minor Command Window New to MATLAB? See resources for Getting Started