Index in position 1 is invalid but I need w(0,j), w(0,0), and w(m,0)

I am following an algorithm for wave equations (shown on the pic). I get the error "Index in position 1 is invalid. Array indices must be positive integers or logical values." What do I do, if w(0,j), w(0,0), and w(m,0) are needed? Also, in this line "wij(i,1)=(1-lambda^2)*f(i*h)+(lambda^2/2)*(f((i+1)*h)+f((i-1)*h))+k*g(i*h);" i get the error "Unable to convert expression containing symbolic variables into double array." Help would be very much appreciated, thanks.

This is my program:

%STEP0

l=1; %endpoint

T=1; %maximum time

alpha = 2; %constant

m=10;

N=20;

syms f(x) g(x,t)

f(x)=sin(pi*x);

g(x)=(sin(pi*x))*(cos(2*pi*t));

%STEP1

h=l/m;

k=T/N;

lambda=k*alpha/h;

%STEP2

for j=1:N

wij(0,j)=0;

wij(m,j)=0;

end

%STEP3

wi(0,0)=f(0);

wi(m,0)=f(l);

%STEP4

for i=1:m-1

wij(i,0)=f(i*h);

wij(i,1)=(1-lambda^2)*f(i*h)+(lambda^2/2)*(f((i+1)*h)+f((i-1)*h))+k*g(i*h);

end

%STEP5

for j=1:N-1

for i=1:m-1

wij(i,j+1)=2*(1-lambda^2)*w(i,j)+lambda^2*(wij(i+1,j)+wij(i-1,j))-wij(i,j-1);

end

end

%STEP6

for j=0:N

t=j*k;

for i=0:m

x-i*h;

fprintf( '%.1f %.1f %.10f',x,t,wij(i,j))

end

end

Answer 1

[Jos Jordan]

I'm pretty sure the answer here is that unlike most other languages, MATLAB starts indexing at 1, not 0. So for example in the code you wrote:

%STEP 2
for j=1:N
    wij(1,j)=0;
    wij(m,j)=0;
end

would give the desired result - you would then have to change the others to have an index of 1 instead of 0.

See https://uk.mathworks.com/help/matlab/math/array-indexing.html for further information.

[LightFury Yeji]

Does this mean I can change, for example, the w(i,1) to w(i,2) since I will be adjusting everything? Thanks for this!

[Jos Jordan]

I think so - in the problem everything is indexed from 0, so you want to index from 1 - if you think about what you're doing carefully you should be able to see which parts you need to add 1 to.

In Step 4 you are doing:

which in your code is:

for i=1:m-1
    wij(i,1)=f(i*h);
    wij(i,2)=(1-lambda^2)*f(i*h)+(lambda^2/2)*(f((i+1)*h)+f((i-1)*h))+k*g(i*h);
end

as you starting the index of the second subscript at 1 as oppose to 2.

By the looks of it, you should also set

for i=2:m
    wij(i,1)=f(i*h);
    wij(i,2)=(1-lambda^2)*f(i*h)+(lambda^2/2)*(f((i+1)*h)+f((i-1)*h))+k*g(i*h);
end

changing the value of i you are looping over, as again you are adding one to the index there as in the problem it states:

so you want to be iterating from the second index of the i in your code. If you cange the rest of it - it should work, I believe. If not I'll test it and see.

[LightFury Yeji]

Thank you so much :)