How to solve coupled partial differential equation with an integral term?

Question

MANOJ KUMAR 2020-11-15

0
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https://ww2.mathworks.cn/matlabcentral/answers/648423-how-to-solve-coupled-partial-differential-equation-with-an-integral-term

回答： Hewa selman 2021-12-23

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I want to solve the below given PDE system in MATLAB,

I have written a code but I am not able to solve integro differential equation in this system.

%% discretizing a 
dx=0.25;
x=0:dx:20;
m=length(x);
% discretizing time
dt=0.01;
%Tp=(10)^(-1);
% Tf=input('enter the max time');
Tf=20;
t=0:dt:Tf;
n=length(t);
%% parameter values
a3=0; a4=1000;  rho = 0.2; 
c=1; alpha = 0.40; mu = 0.30; k=0.5; lambda=50;
%% size of the matrix
A=zeros(m,n);
B=zeros(m,n);
S=zeros(1,n);
R=zeros(1,n);
sum=zeros(1,n);
%% initial data 
S(1) = a4; % S_s
R(1) = a3; % R_s
%% defining functions
 for i=1:m
       a1(i) = 0;      %l_s(a)
       a2(i) = 0;      %i_s(a)
      p(i) = 1700*(1-exp(-x(i)));       %p(a)
%        p(i) = i;       %p(a)
       delta1(i) = 0.01; % delta_l(a)
       delta2(i) = 0.4; % delta_i(a)
 end
%% initial conditions
 for i=1:m
     A(i,1) = a1(i);
     B(i,1) = a2(i);
     sum(1)=trapz(p(i).* B(i,1))*dx;
 end
 % %trapz(p(i)* B(:,i))*dx%
%% discretizing age and time derivative
  
  for j=1:n-1
       
      for i=2:m-1
       A(i,j+1)=A(i,j)+dt*(-(A(i+1,j) - A(i-1,j))/(2*dx) - alpha*A(i,j)-delta1(i)*A(i,j));
       B(i,j+1)=B(i,j)+dt*(-(B(i+1,j) - B(i-1,j))/(2*dx)  +alpha*A(i,j)-delta2(i)*B(i,j));
       
       sum(j+1)=trapz(p(i).* B(i,j))*dx;
       S(j+1) = S(j) + dt*(lambda - k*R(j).*S(j) - mu*S(j));
       R(j+1) = R(j)+ dt*sum(j+1) - dt*c*R(j);
      end
      % boundary conditions
      A(1,j+1)= rho*k*R(j)*S(j); 
      B(1,j+1)= (1-rho)*k*R(j)*S(j);   
 end   
plot(t,S,'b-', t,R,'r-');

I have plotted the solution but the plot of R(t) is not correct, I feel that I am not able to solve integro differential correctly. Please help me to solve integro differential equation by some method.