Question

For the following differential equation, answer the questions. -y"+y=x (0.0 SX 35.0) y(0.0)=1.0, y(5.0)= 2.0 (1) Solve the differential equation analytically. (2) Solve the differential equation using centered finite difference approximation of y" with a step size of 1.0 and check the accuracy of the solutions.

solve using matlab or by hand.

Answer 1

The problem is solved analytically by hand and then for comparison, the result is inserted into the MATLAB in the form of the equation.

While solving (2) question central finite difference is used and formed a tridiagonal matrix using the MATLAB and solve the value of y at X=0,1,2,3,4,5 to compare the same value with the analytical solution.

In MATLAB results y is the solution from the analytical equation while D is the solution from the discretize equation.

I have included the script which you can directly copy-paste in MATLAB to get the result shown in the snip of the result.

comment if any query.

y- Audeng equotion " mt particwon soutian 제 1 DAI 머 Lori - [+] = 3 lito 즐 Cs +1-8)

= کیا کی [1-o] + x[ito ass - so Solution y=ceatsex tx at x=0, y=1 Ct2=1, C = 1-2 at x=s, y=2 a=cies the +5 c estges= -3 -(a) es & hos=-3 ca estes=-3 Caleses)=-3-c 5 Ca 3tes e – e 5 a = - 3tes cs -és-3- es_cs (3+65) estes

х ч = -( 3-5) estes + (3 tes es-c-s 2) Ч = x р 6 b: w - nes 1 3 - 4 = -х Ox2 9. - 24 Ч- - Ч Чs - 2 ч. - Ч. - 3 + - с = 2 XA S= = = 3. -3-33 - - - Ч - 3ч. = 3 1 = 21 - 2 - 343 3-4 ст31 - 3 Чs - 3 че

=s, x= ч Чs - 3 1 = -Ч - 5s ч -Ч- - 3Чs 4 = -6 this form a tridiagona matrix of to SS

SCRIPT OF MATLAB TO COMPARE BOTH RESULTS

clear all
clc
% %ANALYTICAL RESULT
x=0:5;
y=((-(3+exp(-5))*exp(x)+(3+exp(5))*exp(-x))/(exp(5)+exp(-5)))+x
%MATLAB DISCRETIZE RESULT
nn=6;
nn1=nn-2;
y(1)=1;
y(6)=2;
c1=-(2+1);
c2=1;
E=[x(2):x(5)]';
time=0;
%DEFINING TRIDIAGONAL MATRIX
A=full(gallery('tridiag',4,c2,c1,c2));
B=-E;
B(1)=B(1)-y(1); B(4)=B(4)-y(6);
C=A\B;
%y value AFTER FIRST ITERATION
D=[y(1);C;y(6)]'

SNIP OF MATLAB PROGRAM

clear all clc SANALYTICAL RESULT x=0:5; y=((-(3+exp(-5)) *exp(x)+(3+exp (5)) *exp(-x))/(exp(5) +exp(-5) ) ) +x SMATLAB DISCRETIZE RESULT nn=6; nnl=nn-2; y (1)=1; y (6)=2; cl=- (2+1); c2=1; E=[x (2) :x (5) ]'; time=0; DEFINING TRIDIAGONAL MATRIX A=full (gallery('tridiag', 4, c2,cl,c2)); B=-E; B(1)=B (1)-v (1); B (4)=B (4) -Y (6); C=A\B; Sy value AFTER FIRST ITERATION D=(y(1);C;y(6)]'

SNIP OF RESULT FROM MATLAB:

y = 0.9999 1.3202 1.9884 2.6439 2.9126 2.0003 D = 1.0000 1.3273 1.9818 2.6182 2.8727 2.0000

where y is the value that is obtained from the Analytical solution while D is the value of y from the MATLAB discretization.

comment if any query.