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Use the difference equation wi,j+1 = 120;+1,j + 2(1 – 12); +120;-1,j - Ui,j-1 to approximate the solution of the boundary-val

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Answer #1

MATLAB CODE FOR BVP

function [M]= BVP(problem)

if problem==1
c=1;a=1;T=1;n=4;m=20;
f=@(x)(x.*(x-1)); %initial condion problem 1
x=linspace(0,a,m );
u_t0=f(x); % using initial condition
end
if problem==2
c=1;a=2;T=1;n=5;m=20;
f=@(x)(exp(-18*(x-1)^2)); %initial condion problem 2
x=linspace(0,a,m );
u_t0=f(x); % using initial condition
end
if problem==3
c=sqrt(2);a=1;T=1;n=10;m=50;
x=linspace(0,a,m );
u_t0=problem3_ini(x); % using initial condition
end
M=zeros(m,n);
u_BC=[0,0];%Boundary condition

  
A=sparse(m);B=sparse(m,1); % for solving system
t=linspace(0,T,n); % discretising domain in given number of points
% A(1,1)=1;A(m,m)=1;B(1,1)=0;B(m,1)=0; % By using BC
dx=x(2)-x(1);dt=t(2)-t(1); lamda=(dt)^2/(dx)^2;

M(1:end,1)=u_t0
% we use initial velocity du/dx=0 at t=0
u_t1=u_t0;
M(1:end,2)=u_t0
for N=3:1:length(t)
A(1,1)=1;A(m,m)=1;B(1,1)=0;B(m,1)=0; % By using BC
for i=2:1:length(x)-1
A(i,i)=1; % by given difference equation
B(i)=lamda^2*c^2*(u_t1(i+1)-2*u_t1(i)+u_t1(i-1))+2*u_t1(i)-u_t0(i);% by given difference equation
end
u_t2=A\B;
M(1:end,N)=u_t2;
u_t0=u_t1; % updating previous time solution
u_t1=u_t2;% updating previous time solution

  
  
end
double(M)
end

function u=problem3_ini(x)
u=zeros(length(x),1);
for i=1:1:length(x)
if x(i)>0.5
u(i)=0.6;
else
u(i)=0;
end
end
end

Results  

Above function take the value problem no. 1 for (a), 2 for (b) 3 for (c). and return the solution matrix in which column denote the time step solution and row represents the solution at each space steps on same time. there is a function problem3_ini need to save in the same file where we are saving BVP since we are calling this function in BVP for calculating initial condition for problem 3.

BVP(1)

ans =

1.0e+04 *

0 0 0 0
-0.0000 -0.0000 0.0009 -1.4314
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 -1.4314
0 0 0 0

ans =

1.0e+04 *

0 0 0 0
-0.0000 -0.0000 0.0009 -1.4314
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0026
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 0.0027
-0.0000 -0.0000 0.0009 -1.4314
0 0 0 0

  

>> BVP(3)

ans =

1.0e+27 *

0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0.0002
0 0 0 0 0 0 0 0 0.0000 -0.0032
0 0 0 0 0 0 0 0.0000 -0.0000 0.0223
0 0 0 0 0 0 0.0000 -0.0000 0.0000 -0.0967
0 0 0 0 0 0.0000 -0.0000 0.0000 -0.0001 0.2899
0 0 0 0 0.0000 -0.0000 0.0000 -0.0000 0.0002 -0.6375
0 0 0 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0003 1.0621
0 0 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0004 -1.3653
0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0004 1.3653
0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0003 -1.0621
0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0002 0.6375
0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0001 -0.2899
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0967
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0223
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0032
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0002
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0001
0.0000 0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0001
0.0000 0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0001
0.0000 0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0000 0.0000 -0.0001
0.0000 0.0000 0 0 0 0 0 0 0 0

>>


  

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