Reaction-Diffusion Equations and Matlab

[GeoTux]

I want to solve the reaction-diffusion problem, in 2D, with Matlab:

Sum_{j=1}^{4} D_{1j} * (drond^{2}z_{j} / drond x^{2} + drond^{2}z_{j} / drond y^{2}) + R_{1} = 0
Sum_{j=1}^{4} D_{2j} * (drond^{2}z_{j} / drond x^{2} + drond^{2}z_{j} / drond y^{2}) + R_{2} = 0
Sum_{j=1}^{4} D_{3j} * (drond^{2}z_{j} / drond x^{2} + drond^{2}z_{j} / drond y^{2}) + R_{3} = 0
Sum_{j=1}^{4} D_{4j} * (drond^{2}z_{j} / drond x^{2} + drond^{2}z_{j} / drond y^{2}) + R_{4} = 0
Boundary conditions: x = 0,1 z_{1} = 1 ; x = 0,1 z_{2,3,4} = 0

So, I have a system with four partial diffrential equations. and I want to plot the solution for every equations.

What it's wrong in my code ? The plots of solutions must be a surface (almost like a bell), but is not.

clc;
clear;
close all;

   a=0;b=1;n=5;
   h = (b-a)/(n+1); % step

   I=eye((n-1)*(n-1));
   A11=zeros((n-1)*(n-1));A12=zeros((n-1)*(n-1));A13=zeros((n-1)*(n-1));   A14=zeros((n-1)*(n-1));
   A21=zeros((n-1)*(n-1));A22=zeros((n-1)*(n-1));A23=zeros((n-1)*(n-1));A24=zeros((n-1)*(n-1));
   A31=zeros((n-1)*(n-1));A32=zeros((n-1)*(n-1));A33=zeros((n-1)*(n-1));A34=zeros((n-1)*(n-1));
   A41=zeros((n-1)*(n-1));A42=zeros((n-1)*(n-1));A43=zeros((n-1)*(n-1));A44=zeros((n-1)*(n-1));
   b1=zeros((n-1)*(n-1),1);b2=zeros((n-1)*(n-1),1);b3=zeros((n-1)*(n-1),1);b4=zeros((n-1)*(n-1),1);
   k1=1;k2=0.4;k3=0.5;k4=0.2;k5=0.1;
   % diffusion matrix
   D11=1; D12=-0.43; D13=-0.19; D14=-0.265;
   D21=-0.14; D22=1.475; D23=-0.025; D24=0.235;
   D31=0.105; D32=0.07; D33=1.475; D34=-0.19;
   D41=-0.085; D42=-0.245; D43=0.27; D44=1.17;
   [x,y] = meshgrid(a:h:b);      % uniform mesh

   % Matrix A
   K1D = spdiags(ones(n,1)*[-1 2 -1],-1:1,n-1,n-1);
   I1D = speye(size(K1D));                       
   Delta = kron(K1D,I1D)+kron(I1D,K1D);  
   % A1i
   A11=D11*Delta-(k1+k3)*I*h^2;
   A12=D12*Delta;
   A13=D13*Delta;
   A14=D14*Delta-k5*I*h^2;
   %A2i
   A21=D21*Delta-k1*I*h^2;
   A22=D22*Delta+k2*I*h^2;
   A23=D23*Delta-k4*I*h^2;
   A24=D24*Delta;
   %A3i
   A31=D31*Delta;
   A32=D32*Delta-k2*I*h^2;
   A33=D33*Delta+k4*I*h^2;
   A34=D34*Delta;
   %A4i
   A41=D41*Delta-k3*I*h^2;
   A42=D42*Delta;
   A43=D43*Delta;
   A44=D44*Delta+k5*I*h^2;
   % b
   b1(1)=(-1/h^2)*D11;
   b2(1)=(-1/h^2)*D21;
   b3(1)=(-1/h^2)*D31;
   b4(1)=(-1/h^2)*D41;
   b1(n-1)=(-1/h^2)*D11;
   b2(n-1)=(-1/h^2)*D21;
   b3(n-1)=(-1/h^2)*D31;
   b4(n-1)=(-1/h^2)*D41;

   % aprox solution
   A=[A11 A12 A13 A14;A21 A22 A23 A24;A31 A32 A33 A34;A41 A42 A43 A44];
   bn=[b1;b2;b3;b4];
   ub=reshape(bn,(n-1)*(n-1)*4,1);
   v=inv(A)*ub;
   % split the solution vector
   pas=size(A11);
   v1=v(1:n+2);
   v2=v(pas+1:2*pas);
   v3=v(2*pas+1:3*pas);
   v4=v(3*pas+1:4*pas);
   vv1=zeros(n+2,n+2);
   vv2=zeros(n+2,n+2);
   vv3=zeros(n+2,n+2);
   vv4=zeros(n+2,n+2);
   % boundary conditions
   vv1=[ones(n+2,1) v1 ones(n+2,1) ones(n+2,1) ones(n+2,1) ones(n+2,1) ones(n+2,1)];
   vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];
   vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];
   vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];
   % plot
   figure
   set(gcf,'DefaultAxesFontSize',8,'PaperPosition', [0 0 3.5 3.5])
   subplot(2,2,1);
   surf(x,y,vv1);
   title('Numerical solution for Ecuation1');
   subplot(2,2,2);
   surf(x,y,vv2);
   title('Numerical solution for Ecuation2');
   subplot(2,2,3);
   surf(x,y,vv3)
   title('Numerical solution for Ecuation3');
   subplot(2,2,4);
   surf(x,y,vv4);
   title('Numerical solution for Ecuation4');

 

[GregLocock]

% boundary conditions
vv1=[ones(n+2,1) v1 ones(n+2,1) ones(n+2,1) ones(n+2,1) ones(n+2,1) ones(n+2,1)];
vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];
vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];
vv1=[zeros(n+2,1) v1 zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1) zeros(n+2,1)];

makes no sense to me, and you don't seem to do anything much with vv2 etc

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[GeoTux]

Hi Greg,

Thanks for your reply.
I think my issues start when I try to split the big vector v, into 4 small vectors (one for each equation:v1,v2,v3,v4).
After that, the boundary conditions(vv1,vv2,vv3,vv4) is not correctly implemented (it's need to be created dynamically).
In conclusion, there is a size issue between:
- [x,y] = meshgrid(a:h:b);
- split the vector v
- surf(x,y,vv1);
But, I don't have any idea to resolv the issue.

Cheers,
George Sarbu

 

[GregLocock]

Frankly I'm amazed it runs at all

v2=v(pas+1:2*pas);

pas is a 1x2 matrix, what do you think that line is supposed to accomplish? I suspect that you really want

v2=v(pas(1)+1:2*pas(1));

doubtless there are similar problems with array sizes later on

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[GeoTux]

I make some changes:

% split the solution vector
pas=size(A11);
v1=v(1:pas);
v2=v(pas(1)+1:2*pas(1));
v3=v(2*pas(1)+1:3*pas(1));
v4=v(3*pas(1)+1:4*pas(1));

% boundary conditions
for i=1:pas
  vv1(i,1)=1;
  vv2(i,1)=0;
  vv3(i,1)=0;
  vv4(i,1)=0;
end
vv1(:,2)=v1;
vv2(:,2)=v2;
vv3(:,2)=v3;
vv4(:,2)=v4;

for i=3:pas
  for j=1:pas
    vv1(j,i)=1;
    vv2(j,i)=0;
    vv3(j,i)=0;
    vv4(j,i)=0;
  end
end

Now, if I take h very small: h = (b-a)/(n+10), the surf function run well. But, of course the plot is not like a bell

 

[GregLocock]

Sadly I don't have the time to figure out what you are actually tying to accomplish (the first 4 lines in your first post don't actually mean anything to me) so all I can suggest is to break the problem down into small parts, check that each step works, including the sizes of the arrays, at each step.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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