Question

Use the Backward Difference Method to solve the problems of Computer Problem 1. Make a table of the exact value, the approximate value, and error at (x, t) = (0.5,1) for step sizes h = 0.02 and k = 0.02,0.01,0.005. REFERENCE: CP1: 1. Solve the equation u; = 2uxx for 0 < x <1,0<< 1, with the initial and boundary conditions that follow, using the Forward Difference Method with step sizes h = 0.1 and k=0.002. Plot the approximate solution, using the Matlab mesh command. What happens if k> 0.003 is used? Compare with the exact solutions from Exercise 1. (a) u(x,0) = 2 coshx for 0<x<1 (0,t) = 2e2t for 0 <t<1 u(1,t) = (e2 + 1)e2-1 for 0 <t<1 (b) u(x,0) = e for 0<x<1 u(0,t)=et for 0 <t<1 u(1,t) = (2++1 for 0 <i<1

MATLAB Code. Try not to use SYMS package as it does not load on Octave.

Answer 1

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(«,,x,]= [0,1] and [yo,y;] = [0,1]. Let h=0.02 and k = 0.02. Therefore, h=*, -X M M=*, -X, h 1-0 0.02 = 50 Similarly, k =); N N = ; -90 k 1-0 0.02 = 50 Use the code - Forward difference method for heat equation - in MATLAB. Write 'f = @(x) = 2*cosh(x)'. Also put 'xl = 0, xr = 1, yb = 0, yt = 1, M = 50, and N = 50'. Similarly, write l=@(t) 2*e^(2*t); r=@(t) (e.^2+1)*(e.^(2*t - 1)); The MATLAB provides the following graph:

Use the code - Forward difference method for heat equation – in MATLAB. Write ‘f = @(x) = 2*cosh(x)'. Also put xl = 0, xr = 1, yb = 0, yt = 1, M = 50, and N = 50'. Similarly, write i=@(t) 2*e.^(2*t); r=@(t) (e.^2+1)*(e.^(2*t - 1)); The MATLAB provides the following graph: Drug Detop Window Help Leth is as fol 06 09 06 Property torre

Let h=0.02 and k =0.01. Therefore, N = 100. Make this change and obtain the graph. It is as follows: e to view the took De Window OBS092603 DBS 0 Leth is as fol 0 02 Prepenter Figure 2

Prepare the following table, which gives the exact values, approximate values and the error at (x,t)=(0.5,1). h k Exact Approximate Error 0.02 0.02 16.6642 16.7023 0.0381 0.02 0.01 16.6642 16.6834 0.0192 0.02 0.005 16.6642 16.6738 0.0097

(b) Consider the following partial differential equation: u, = 24xx, where 0<x51, 05151. Let u(x,0)=e" for 0<x51 u(0,1)= 2e' for 05151 u(1,t) = e2++) for Ostsi Therefore, [x,,x,] =[0,1] and [yo, y ]= [0,1]. Let h=0.02 and k = 0.02. Therefore, h= * , - x M M=X-X, h 1-0 0.02 = 50

Similarly, k=”; Yo N N = y; -Yo k 1-0 0.02 = 50

Use the code - Forward difference method for heat equation – in MATLAB. Write 'f = @(x) = e.*x'. Also put 'XI = 0, xr = 1, yb = 0, yt = 1, M = 50, and N = 50'. Similarly, write i=@(t) e.^2*t); r=@(t) 2.^(2*t+1); The MATLAB provides the following graph: figures figure to view hot To Do Desti 17 SXSLOSE Lidory Merah WOSESLOSISI 0 www.empre. Make OB 02