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Which method to solve

  1. Sep 19, 2012 #1
    Hi, I have the attached diffusion based equations I am looking to solve, however I am unsure about how to implement. I have solved similar equations before, with a simplified r(cG,cO) using the explicit finite difference method and it worked fine. However, I am wondering if that would be OK for this set of equations. I read that people used shooting method to solve these, but I am unfamiliar with this method. Can I also use the differential equation solvers in MATLAB to do this? Thanks.
     

    Attached Files:

  2. jcsd
  3. Sep 24, 2012 #2
    Hi, I have developed a program in matlab to solve these equations, however I was wondering if someone can take a look for me. My solutions are all coming out to be "NaN", which is "not-a-number" in Matlab. I am using the explicit finite difference scheme to solve. Any suggestions would be great. My code below will run in matlab from just a copy/paste from here. My boundary conditions are the following:

    Code (Text):

    x = 0: dcG/dx=0, dcO/dx=0 and cH = 0  
    x = d: cG = cG0, cO = O20, cH = 0
     
     Initial conditions are
     
     cH(t = 0) = 0
     cG(t = 0) =cG0
     cO(t = 0) = O20
     

    Code (Text):


    clear all;

    numx =50;                       %number of grid points in space
    numt =82000;                    %number of time steps to be iterated over
    tmax = .06;

    Length =50E-7;                  %length of grid
    DG = 4E-10;                     %requirement DG(dt)/dx^2 < .5, cm^2/sec
    DO = 5E-9;                      %requirement DO(dt)/dx^2 < .5, cm^2/sec
    DH = 5E-9;                      %requirement DH(dt)/dx^2 < .5, cm^2/sec            

    Vmax = 60E-6;                   %A/cm^2
    KG = 33E-6;                     %mol/cm^3
    KO = .2E-6;
    G0 =2E-6;                       %mol/cm^3 initial substrate concentration
    O20 = .12E-6;                   %mol/cm^3 inital oxygen concentration

    cG = zeros(numx,1);              %initialize everything to zero
    cH = zeros(numx,1);              %initialize everything to zero
    cO = zeros(numx,1);

    zcG = linspace(0,Length,numx)';   %vector of x values, to be used for plotting
    tG = linspace(0,tmax,numt)';      %vector of t values, to be used for plotting

    zcO = linspace(0,Length,numx)';   %vector of x values, to be used for plotting
    tO = linspace(0,tmax,numt)';      %vector of t values, to be used for plotting

    zcH = linspace(0,Length,numx)';   %vector of x values, to be used for plotting
    tH = linspace(0,tmax,numt)';      %vector of t values, to be used for plotting

    dxG = zcG(2)-zcG(1);              %Define grid spacing in time
    dtG = tG(2)-tG(1);                %Define grid spacing in time

    dxO = zcO(2)-zcO(1);              %Define grid spacing in time
    dtO = tO(2)-tO(1);                %Define grid spacing in time

    dxH = zcH(2)-zcH(1);              %Define grid spacing in time
    dtH = tH(2)-tH(1);                %Define grid spacing in time

    S_Stability1 = (DG*dtG)/((dxG)^2) %check requirement DG(dtG)/dxG^2 < .5, cm^2/sec
    O_Stability1 = (DO*dtO)/((dxO)^2) %check requirement DO(dtO)/dxO^2 < .5, cm^2/sec
    H_Stability1 = (DH*dtH)/((dxH)^2) %check requirement DH(dtH)/dxH^2 < .5
           
    %{
    Dirchelet Boundary Conditions, where Neumann BC dcG/dx=0 and dcO/dx=0
    is in body of for loop below, IC is given by matrix initialization (zeros) and BC
    %}

    cG(numx) = G0;                 %cG(x=d,t)=G0
    cO(numx) = O20;                %cO(x=d,t)=O20
    cH(1) = 0;                     %cH(x=0,t)=0
    cH(numx) = 0;                  %cH(x=d,t)=0

    %iterate central difference equation%

    for j=1:numt %time loop  
       
               cGlast = cG;
               cOlast = cO;
               cHlast = cH;
               
         
        for i=2:numx-1 %space loop
                 
          cG(i) = cGlast(i) + (dtG/dxG^2)*DG*(cGlast(i+1) - 2*cGlast(i) + cGlast(i-1))-((Vmax*dtG*cGlast(i)*cOlast(i))/(cGlast(i)*cOlast(i)+(KG*cOlast(i))+KO*cGlast(i)));
          cO(i) = cOlast(i) + (dtO/dxO^2)*DO*(cOlast(i+1) - 2*cOlast(i) + cOlast(i-1))-((Vmax*dtO*cGlast(i)*cOlast(i))/(cGlast(i)*cOlast(i)+(KG*cOlast(i))+KO*cGlast(i)));
          cH(i) = cHlast(i) + (dtH/dxH^2)*DH*(cHlast(i+1) - 2*cHlast(i) + cHlast(i-1))+((Vmax*dtH*cGlast(i)*cOlast(i))/(cGlast(i)*cOlast(i)+(KG*cOlast(i))+KO*cGlast(i)));
         
          y(j,1) = cH(2);  %store data in vector for current plotting
         
        end
       
        cG(1)= cGlast(1)+dtG*DG*2*(cGlast(2)-cGlast(1))./dxG.^2; %Neumann Boundary Condition
        cO(1)= cOlast(1)+dtO*DO*2*(cOlast(2)-cOlast(1))./dxO.^2; %Neumann Boundary Condition
       
    end

    figure(1);
    plot(zcH,cH);
    xlabel('Length (cm)');
    ylabel('[H2O2] (M)');

    figure(2);
    plot(zcG,cG);
    xlabel('Length (cm)');
    ylabel('[Glucose] (M)');


    figure(3);
    plot(zcO,cO);
    xlabel('Length (cm)');
    ylabel('[O2] (M)');

    figure(4);  
    i = 2*(96500)*DH*(y/dxH);
    %i = 2*(96500)*DH*(y/dxH);
    plot(tH,i);
    xlabel('Time (s)');
    ylabel('Current (A)');
     
     
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