I have error when I applied trapezoidal rule

This is my code, I could not figure out the error.

clear all;
xl=0; xr=1;            %domain[xl,xr]                                                                                                                         
J=100;                  % J: number of dividion for x
dx=(xr-xl)/J;          % dx: mesh size
tf=1000;                % final simulation time
Nt=100000;                 %Nt:number of time steps
dt=tf/Nt;                                                                                     
c=50;     
s=600;   % paremeter for the solution
lambdau1=0.0004;          %turning rate of susp(right)
lambdau2=0.0002;          %turning rate of susp(left)
lambdav1=0.0003;           %turning rate of infect(right)
lambdav2=0.0001;           %turning rate of infect(left)
beta=0.1;               % transimion rate
alpha=0.0;               %recovery/death rate
b=0.1;
d=0.5;
bV=0.3;
dV=0.1;
au=0.0003;                   % advection speed - susceptible
av=0.0001;                   % advection speed - infected
A=0.01;
mu1=au*dt/dx; 
mu2=av*dt/dx;
w = linspace(xl,xr,J);   
if mu1>1.0               %make sure dt satisfy stability condition
    error('mu1 should<1.0!')
end
if mu2>1.0               %make sure dt satisfy stability condition
    error('mu2 should<1.0!')
end
%Evaluate the initial conditions 
x=xl:dx:xr;               %generate the grid point
%fr=0*exp(-c*(x-0.5).^2);     %dimension f(1:J+1)initial conditions of right moving suscp
%fl=0*exp(-c*(x-0.5).^2);    %initial conditions of left moving suscep
%gr=0.5*exp(-s*(x-0.5).^2);  %initial conditions of infected
%gl=0.5*exp(-s*(x-0.5).^2);  %initial conditions of infected
k=6*pi/(xr-xl);
fr=0+0.01*(1+sin(k*x));     %dimension f(1:J+1)initial conditions of right moving suscp
fl=0+0.01*(1+sin(k*x));    %initial conditions of left moving suscep
gr=0.25+0.01*(1+sin(k*x));  %initial conditions of infected
gl=0.75+0.01*(1+sin(k*x));
g=gr+gl;
%store the solution at all grid points for all time steps
ur=zeros(J+1,Nt);
ul=zeros(J+1,Nt);
vr=zeros(J+1,Nt);
vl=zeros(J+1,Nt);
v=zeros(J+1,Nt);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tt=0:tf/(Nt-1):tf;
[T,X]=meshgrid(tt,x);
%%%%%%%%%%%%%%%%%%%%%%%%
KU = @(x) 1/(0.05*sqrt(2*pi))*exp(-0.5*(x/0.05).^2 ); 
y1 = @(x) -b*KU(x)*(gr+gl)+d*KU(x)*fr;
y2 = @(x) -b*KU(x)*(gr+gl)+d*KU(x)*fl;
KV = @(x) 1/(0.01*sqrt(2*pi))*exp(-0.5*(x/0.01).^2 ); 
F1 = @(x) -bV*KV(x)*(gr+gl)+dV*KV(x)*fr;
F2 = @(x) -bV*KV(x)*(gr+gl)+dV*KV(x)*fl;
for i=1:J+1
    x=xr+(i-1)*dx;
end
ynum1 = y1(x);
ynum2 = y2(x);
I = trapz(x,ynum1);
W = trapz(x,ynum2);
Fnum1 = F1(x); 
Fnum2 = F2(x); 
Z = trapz(x,Fnum1);
U = trapz(x,Fnum2);
%find the approximate solution at each time step 
for  n=1:Nt
    t=n*dt;               % current time
    if n==1                         % first time step
        for j=2:J                      % interior nodes
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ur(j,n)=fr(j)-0.5*mu1*(fr(j)-fr(j-1))+dt*lambdau2*(0.5+0.5*tanh(1+I(j)))*fl(j)-dt*lambdau1*(0.5+0.5*tanh(1+W(j)))*fr(j)-dt*beta*fr(j)*(gr(j)+gl(j))+dt*0.5*alpha*(gr(j)+gl(j));
            %%%%%%%%%%%%%%%%%%%upwind of left-moving of susceptible
            ul(j,n)=fl(j)+0.5*mu1*(fl(j+1)-fl(j))+dt*lambdau1*(0.5+0.5*tanh(1+W(j)))*fr(j)-dt*lambdau2*(0.5+0.5*tanh(1+I(j)))*fl(j)-dt*beta*fl(j)*(gr(j)+gl(j))+dt*0.5*alpha*(gr(j)+gl(j));
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of infected
            vr(j,n)=gr(j)-0.5*mu2*(gr(j)-gr(j-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z(j)))*gl(j)-dt*lambdav1*(0.5+0.5*tanh(1+U(j)))*gr(j)+dt*beta*fr(j)*(gr(j)+gl(j))-dt*alpha*gr(j);
            %%%%%%%%%%%%%%%%%%%lupwind of left-moving of infected
            vl(j,n)=gl(j)+0.5*mu2*(gl(j+1)-gl(j))+dt*lambdav1*(0.5+0.5*tanh(1+U(j)))*gr(j)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(j)))*gl(j)+dt*beta*fl(j)*(gr(j)+gl(j))-dt*alpha*gl(j);
            
            
        end
        %peridic BC for right moving of suscep(j=J+1,j+2=1)     
        ur(J+1,1)= fr(J+1)-0.5*mu1*(fr(J+1)-fr(J))+dt*lambdau2*(0.5+0.5*tanh(1+I(J+1)))*fl(J+1)-dt*lambdau1*(0.5+0.5*tanh(1+W(J+1)))*fr(J+1)-dt*beta*fr(J+1)*(gr(J+1)+gl(J+1))+dt*0.5*alpha*(gr(J+1)+gl(J+1)); 
        %peridic BC for right moving of suscep(j=1,0=J)
        ur(1,1)=fr(1)-0.5*mu1*(fr(1)-fr(J))+dt*lambdau2*(0.5+0.5*tanh(1+I(1)))*fl(1)-dt*lambdau1*(0.5+0.5*tanh(1+W(1)))*fr(1)-dt*beta*fr(1)*(gr(1)+gl(1))+dt*0.5*alpha*(gr(1)+gl(1)); %peridic BC for right moving
        %peridic BC for left moving of suscep(j=1)
        ul(1,1)=fl(1)+0.5*mu1*(fl(2)-fl(1))+dt*lambdau1*(0.5+0.5*tanh(1+W(1)))*fr(1)-dt*lambdau2*(0.5*+0.5*tanh(1+I(1)))*fl(1)-dt*beta*fl(1)*(gr(1)+gl(1))+dt*0.5*alpha*(gr(1)+gl(1)); %peridic BC for left moving;  %peridic BC for left moving
        %peridic BC for left moving of suscep(j=J+1)
        ul(J+1,1)=fl(J+1)+0.5*mu1*(fl(1)-fl(J+1))+dt*lambdau1*(0.5+0.5*tanh(1+W(J+1)))*fr(J+1)-dt*lambdau2*(0.5*+0.5*tanh(1+I(J+1)))*fl(J+1)-dt*beta*fl(J+1)*(gr(J+1)+gl(J+1))+dt*0.5*alpha*(gr(J+1)+gl(J+1));  %peridic BC for right moving(j=J+1);  %peridic BC for left moving
        %peridic BC for right moving of infected(j=J+1)
        vr(J+1,n)=gr(J+1)-0.5*mu2*(gr(J+1)-gr(J))+dt*lambdav2*(0.5+0.5*tanh(1+Z(J+1)))*gl(J+1)-dt*lambdav1*(0.5+0.5*tanh(1+U(J+1)))*gr(J+1)+dt*beta*fr(J+1)*(gr(J+1)+gl(J+1))-dt*alpha*gr(J+1);
        %peridic BC for right moving of infected(j=1)
        vr(1,n)=gr(1)-0.5*mu2*(gr(1)-gr(J))+dt*lambdav2*(0.5+0.5*tanh(1+Z(1)))*gl(1)-dt*lambdav1*(0.5+0.5*tanh(1+U(1)))*gr(1)+dt*beta*gr(1)*(gr(1)+gl(1))-dt*alpha*gr(1);
        %peridic BC for left moving of infected(j=1)
        vl(1,n)=gl(1)+0.5*mu2*(gl(2)-gl(1))+dt*lambdav1*(0.5+0.5*tanh(1+U(1)))*gr(1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(1)))*gl(1)+dt*beta*fl(J+1)*(gr(1)+gl(1))-dt*alpha*gl(1);
        %peridic BC for left moving of infected(j=J+1)
        vl(J+1,n)=gl(J+1)+0.5*mu2*(gl(1)-gl(J+1))+dt*lambdav1*(0.5+0.5*tanh(1+U(J+1)))*gr(J+1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(J+1)))*gl(J+1)+dt*beta*fl(J+1)*(gr(J+1)+gl(J+1))-dt*alpha*gl(J+1);
    else
        KU = @(x) 1/(0.05*sqrt(2*pi))*exp(-0.5*(x/0.05).^2 ); 
        y3 = @(x) -b*KU(x)*(vr+vl)+d*KU(x)*ur;
        y4 = @(x) -b*KU(x)*(vr+vl)+d*KU(x)*ul;
        KV = @(x) 1/(0.01*sqrt(2*pi))*exp(-0.5*(x/0.01).^2 ); 
        F3 = @(x) -bV*KV(x)*(vr+vl)+dV*KV(x)*ur;
        F4 = @(x) -bV*KV(x)*(vr+vl)+dV*KV(x)*ul;
        for i=1:J+1
            x=xr+(i-1)*dx;
        end
        ynum3 = y3(x);
        ynum4 = y4(x);
        I1 = trapz(x,ynum3);
        W1 = trapz(x,ynum4);
        Fnum3 = F3(x); 
        Fnum4 = F4(x); 
        Z1 = trapz(x,Fnum3);
        U1 = trapz(x,Fnum4);
        for j=2:J                % interior nodes
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ur(j,n)=ur(j,n-1)-0.5*mu1*(ur(j,n-1)-ur(j-1,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(j,n-1)))*ul(j,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(j,n-1)))*ur(j,n-1)-dt*beta*ur(j,n-1)*(vr(j,n-1)+vl(j,n-1))+dt*0.5*alpha*(vr(j,n-1)+vl(j,n-1));
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ul(j,n)=ul(j,n-1)+0.5*mu1*(ul(j+1,n-1)-ul(j,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(j,n-1)))*ur(j,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(j,n-1)))*ul(j,n-1)-dt*beta*ul(j,n-1)*(vr(j,n-1)+vl(j,n-1))+dt*0.5*alpha*(vr(j,n-1)+vl(j,n-1));    
            %%%%%%%%%%%%%%%%%%%upwindof right-moving of infected
            vr(j,n)=vr(j,n-1)-0.5*mu2*(vr(j,n-1)-vr(j-1,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(j,n-1)))*vl(j,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(j,n-1)))*vr(j,n-1)+dt*beta*ur(j,n-1)*(vr(j,n-1)+vl(j,n-1))-dt*alpha*vr(j,n-1);
            %%%%%%%%%%%%%%%%%%%upwind of left-moving of infected
            vl(j,n)=vl(j,n-1)+0.5*mu2*(vl(j+1,n-1)-vl(j,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(j,n-1)))*vr(j,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(j,n-1)))*vl(j,n-1)+dt*beta*ul(j,n-1)*(vr(j,n-1)+vl(j,n-1))-dt*alpha*vl(j,n-1);
            
        end 
        %peridic BC for right moving of suscep(j=J+1,j+2=1)
        ur(J+1,n)=ur(J+1,n-1)-0.5*mu1*(ur(J+1,n-1)-ur(J,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(J+1,n-1)))*ul(J+1,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(J+1,n-1)))*ur(J+1,n-1)-dt*beta*ur(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))+dt*0.5*alpha*(vr(J+1,n-1)+vl(J+1,n-1));
        %peridic BC for right moving of suscep(j=1,0=J)
        ur(1,n)=ur(1,n-1)-0.5*mu1*(ur(1,n-1)-ur(J,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(1,n-1)))*ul(1,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(1,n-1)))*ur(1,n-1)-dt*beta*ur(1,n-1)*(vr(1,n-1)+vl(1,n-1))+dt*0.5*alpha*(vr(1,n-1)+vl(1,n-1));
        %peridic BC for left moving of suscep (j=1)
        ul(1,n)=ul(1,n-1)+0.5*mu1*(ul(2,n-1)-ul(1,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(1,n-1)))*ur(1,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(1,n-1)))*ul(1,n-1)-dt*beta*ul(1,n-1)*(vr(1,n-1)+vl(1,n-1))+dt*0.5*alpha*(vr(1,n-1)+vl(1,n-1));    
        %peridic BC for left moving of suscep(j=J+1)
        ul(J+1,n)=ul(J+1,n-1)+0.5*mu1*(ul(1,n-1)-ul(J+1,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(J+1,n-1)))*ur(J+1,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(J+1,n-1)))*ul(J+1,n-1)-dt*beta*ul(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))+dt*0.5*alpha*(vr(J+1,n-1)+vl(J+1,n-1));    
        %peridic BC for right moving of infected(j=J+1)
        vr(J+1,n)=vr(J+1,n-1)-0.5*mu2*(vr(J+1,n-1)-vr(J,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(J+1,n-1)))*vl(J+1,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(J+1,n-1)))*vr(J+1,n-1)+dt*beta*ur(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))-dt*alpha*vr(J+1,n-1);
        %peridic BC for right moving of infected(j=1)
        vr(1,n)=vr(1,n-1)-0.5*mu2*(vr(1,n-1)-vr(J,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(1,n-1)))*vl(1,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(1,n-1)))*vr(1,n-1)+dt*beta*ur(1,n-1)*(vr(1,n-1)+vl(1,n-1))-dt*alpha*vr(1,n-1);
        %peridic BC for left moving of infected(j=1)
        vl(1,n)=vl(1,n-1)+0.5*mu2*(vl(2,n-1)-vl(1,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(1,n-1)))*vr(1,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(1,n-1)))*vl(1,n-1)+dt*beta*ul(1,n-1)*(vr(1,n-1)+vl(1,n-1))-dt*alpha*vl(1,n-1);
        %peridic BC for left moving of infected(j=J+1)
        vl(J+1,n)=vl(J+1,n-1)+0.5*mu2*(vl(1,n-1)-vl(J+1,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(J+1,n-1)))*vr(J+1,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(J+1,n-1)))*vl(J+1,n-1)+dt*beta*ul(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))-dt*alpha*vl(J+1,n-1);
    end
    
    %calculate the analytical solution 
    for j=1:J+1
        xj=xl+(j-1)*dx;
        u_ex(j,n)=exp(-c*(xj-au*t-0.5)^2);
    end
end 

Index exceeds the number of array elements. Index must not exceed 1.

figure
ss=surf(T,X,ur+ul);
set(ss,'edgecolor','none');
xlabel('t','FontSize',28);
ylabel('x','FontSize',28);
title('u^++u^-','FontSize',28);
set(gca,'Fontsize',28);
view(2)
%%%%%%%%%%%%

Answer 1

[Walter Roberson]

I = trapz(x,ynum1);

That is going to return a scalar.

ur(j,n)=fr(j)-0.5*mu1*(fr(j)-fr(j-1))+dt*lambdau2*(0.5+0.5*tanh(1+I(j)))*fl(j)-dt*lambdau1*(0.5+0.5*tanh(1+W(j)))*fr(j)-dt*beta*fr(j)*(gr(j)+gl(j))+dt*0.5*alpha*(gr(j)+gl(j));

That requires I(j) which is indexing the scalar I.

Perhaps you wanted cumtrapz() instead of trapz() ?

[Walter Roberson]

for i=1:J+1
    x=xr+(i-1)*dx;
end

Before that loop, x had already been defined as

x=xl:dx:xr;               %generate the grid point

but in that loop, each cycle you overwrite all of the variable x . Perhaps you want to write to x(i) ?

You already created x as a grid from xl to xr. Is there a reason to overwrite it with values from xr to something in increments of dx ? If you need a vector then surely an xr:dx:SOMETHING would make more sense?

[lulu]

I appreciate your responce. Actaully I used this loop

for i=1:J+1

x=xr+(i-1)*dx;

end

As we know trapezoidal needs f(x0), f(x1)....When I delete this loop and I consider x=xl:dx:xr; I got error.

[Walter Roberson]

You are not computing f at each of those locations.

[Walter Roberson]

The below code computes the first of the two plots. It was not obvious to me what the difference was between the two parts. You can take this code and adjust it to suit whatever your second plot does differently than your first plot.

Note that I guessed that you should use cumtrapz() instead of trapz() -- it is the only thing that made sense considering the indexing you were doing.

Note that Nt (number of time steps) was cut down by a factor of 10. This was done so that you could see some progress: with the original Nt the computation time is long enough that you would have definite grounds to wonder whether anything is happening with the code.

xl=0; xr=1;            %domain[xl,xr]                                                                                                                         
J=100;                  % J: number of dividion for x
dx=(xr-xl)/J;          % dx: mesh size
tf=1000;                % final simulation time
Nt=10000; %Nt=100000;                 %Nt:number of time steps
dt=tf/Nt;                                                                                     
c=50;     
s=600;   % paremeter for the solution
lambdau1=0.0004;          %turning rate of susp(right)
lambdau2=0.0002;          %turning rate of susp(left)
lambdav1=0.0003;           %turning rate of infect(right)
lambdav2=0.0001;           %turning rate of infect(left)
beta=0.1;               % transimion rate
alpha=0.0;               %recovery/death rate
b=0.1;
d=0.5;
bV=0.3;
dV=0.1;
au=0.0003;                   % advection speed - susceptible
av=0.0001;                   % advection speed - infected
A=0.01;
mu1=au*dt/dx; 
mu2=av*dt/dx;
w = linspace(xl,xr,J);   
if mu1>1.0               %make sure dt satisfy stability condition
    error('mu1 should<1.0!')
end
if mu2>1.0               %make sure dt satisfy stability condition
    error('mu2 should<1.0!')
end
%Evaluate the initial conditions 
x=xl:dx:xr;               %generate the grid point
%fr=0*exp(-c*(x-0.5).^2);     %dimension f(1:J+1)initial conditions of right moving suscp
%fl=0*exp(-c*(x-0.5).^2);    %initial conditions of left moving suscep
%gr=0.5*exp(-s*(x-0.5).^2);  %initial conditions of infected
%gl=0.5*exp(-s*(x-0.5).^2);  %initial conditions of infected
k=6*pi/(xr-xl);
fr=0+0.01*(1+sin(k*x));     %dimension f(1:J+1)initial conditions of right moving suscp
fl=0+0.01*(1+sin(k*x));    %initial conditions of left moving suscep
gr=0.25+0.01*(1+sin(k*x));  %initial conditions of infected
gl=0.75+0.01*(1+sin(k*x));
g=gr+gl;
%store the solution at all grid points for all time steps
ur=zeros(J+1,Nt);
ul=zeros(J+1,Nt);
vr=zeros(J+1,Nt);
vl=zeros(J+1,Nt);
v=zeros(J+1,Nt);
u_ex = zeros(J+1,Nt);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
tt=0:tf/(Nt-1):tf;
[T,X]=meshgrid(tt,x);
%%%%%%%%%%%%%%%%%%%%%%%%
KU = @(x) 1 ./ (0.05 .* sqrt(2*pi)) .* exp(-0.5*(x./0.05).^2 );
bKU = @(x) -b .* KU(x) .* g;
dKU = @(x) d .* KU(x);
y1 = @(x) bKU(x) + dKU(x) .* fr;
y2 = @(x) bKU(x) + dKU(x) .* fl;
KV = @(x) 1 ./ (0.01 .* sqrt(2*pi)) .* exp(-0.5*(x./0.01).^2 );
bKV = @(x) -bV .* KV(x) .* g;
dKV = @(x) dV .* KV(x);
F1 = @(x) bKV(x) + dKV(x) .* fr;
F2 = @(x) bKV(x) + dKV(x) .* fl;
ynum1 = y1(x);
ynum2 = y2(x);
I = cumtrapz(x,ynum1);
W = cumtrapz(x,ynum2);
Fnum1 = F1(x); 
Fnum2 = F2(x); 
Z = cumtrapz(x,Fnum1);
U = cumtrapz(x,Fnum2);
wb = waitbar(0, 'please wait (N)');
cleanMe = onCleanup(@() delete(wb));

%find the approximate solution at each time step 
for  n=1:Nt
    waitbar(n/Nt, wb);
    t=n*dt;               % current time
    if n==1                         % first time step
        for j=2:J                      % interior nodes
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ur(j,n)=fr(j)-0.5*mu1*(fr(j)-fr(j-1))+dt*lambdau2*(0.5+0.5*tanh(1+I(j)))*fl(j)-dt*lambdau1*(0.5+0.5*tanh(1+W(j)))*fr(j)-dt*beta*fr(j)*(gr(j)+gl(j))+dt*0.5*alpha*(gr(j)+gl(j));
            %%%%%%%%%%%%%%%%%%%upwind of left-moving of susceptible
            ul(j,n)=fl(j)+0.5*mu1*(fl(j+1)-fl(j))+dt*lambdau1*(0.5+0.5*tanh(1+W(j)))*fr(j)-dt*lambdau2*(0.5+0.5*tanh(1+I(j)))*fl(j)-dt*beta*fl(j)*(gr(j)+gl(j))+dt*0.5*alpha*(gr(j)+gl(j));
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of infected
            vr(j,n)=gr(j)-0.5*mu2*(gr(j)-gr(j-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z(j)))*gl(j)-dt*lambdav1*(0.5+0.5*tanh(1+U(j)))*gr(j)+dt*beta*fr(j)*(gr(j)+gl(j))-dt*alpha*gr(j);
            %%%%%%%%%%%%%%%%%%%lupwind of left-moving of infected
            vl(j,n)=gl(j)+0.5*mu2*(gl(j+1)-gl(j))+dt*lambdav1*(0.5+0.5*tanh(1+U(j)))*gr(j)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(j)))*gl(j)+dt*beta*fl(j)*(gr(j)+gl(j))-dt*alpha*gl(j);
            
            
        end
        %peridic BC for right moving of suscep(j=J+1,j+2=1)     
        ur(J+1,1)= fr(J+1)-0.5*mu1*(fr(J+1)-fr(J))+dt*lambdau2*(0.5+0.5*tanh(1+I(J+1)))*fl(J+1)-dt*lambdau1*(0.5+0.5*tanh(1+W(J+1)))*fr(J+1)-dt*beta*fr(J+1)*(gr(J+1)+gl(J+1))+dt*0.5*alpha*(gr(J+1)+gl(J+1)); 
        %peridic BC for right moving of suscep(j=1,0=J)
        ur(1,1)=fr(1)-0.5*mu1*(fr(1)-fr(J))+dt*lambdau2*(0.5+0.5*tanh(1+I(1)))*fl(1)-dt*lambdau1*(0.5+0.5*tanh(1+W(1)))*fr(1)-dt*beta*fr(1)*(gr(1)+gl(1))+dt*0.5*alpha*(gr(1)+gl(1)); %peridic BC for right moving
        %peridic BC for left moving of suscep(j=1)
        ul(1,1)=fl(1)+0.5*mu1*(fl(2)-fl(1))+dt*lambdau1*(0.5+0.5*tanh(1+W(1)))*fr(1)-dt*lambdau2*(0.5*+0.5*tanh(1+I(1)))*fl(1)-dt*beta*fl(1)*(gr(1)+gl(1))+dt*0.5*alpha*(gr(1)+gl(1)); %peridic BC for left moving;  %peridic BC for left moving
        %peridic BC for left moving of suscep(j=J+1)
        ul(J+1,1)=fl(J+1)+0.5*mu1*(fl(1)-fl(J+1))+dt*lambdau1*(0.5+0.5*tanh(1+W(J+1)))*fr(J+1)-dt*lambdau2*(0.5*+0.5*tanh(1+I(J+1)))*fl(J+1)-dt*beta*fl(J+1)*(gr(J+1)+gl(J+1))+dt*0.5*alpha*(gr(J+1)+gl(J+1));  %peridic BC for right moving(j=J+1);  %peridic BC for left moving
        %peridic BC for right moving of infected(j=J+1)
        vr(J+1,n)=gr(J+1)-0.5*mu2*(gr(J+1)-gr(J))+dt*lambdav2*(0.5+0.5*tanh(1+Z(J+1)))*gl(J+1)-dt*lambdav1*(0.5+0.5*tanh(1+U(J+1)))*gr(J+1)+dt*beta*fr(J+1)*(gr(J+1)+gl(J+1))-dt*alpha*gr(J+1);
        %peridic BC for right moving of infected(j=1)
        vr(1,n)=gr(1)-0.5*mu2*(gr(1)-gr(J))+dt*lambdav2*(0.5+0.5*tanh(1+Z(1)))*gl(1)-dt*lambdav1*(0.5+0.5*tanh(1+U(1)))*gr(1)+dt*beta*gr(1)*(gr(1)+gl(1))-dt*alpha*gr(1);
        %peridic BC for left moving of infected(j=1)
        vl(1,n)=gl(1)+0.5*mu2*(gl(2)-gl(1))+dt*lambdav1*(0.5+0.5*tanh(1+U(1)))*gr(1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(1)))*gl(1)+dt*beta*fl(J+1)*(gr(1)+gl(1))-dt*alpha*gl(1);
        %peridic BC for left moving of infected(j=J+1)
        vl(J+1,n)=gl(J+1)+0.5*mu2*(gl(1)-gl(J+1))+dt*lambdav1*(0.5+0.5*tanh(1+U(J+1)))*gr(J+1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z(J+1)))*gl(J+1)+dt*beta*fl(J+1)*(gr(J+1)+gl(J+1))-dt*alpha*gl(J+1);
    else
        KU = @(x) 1 ./ (0.05 .* sqrt(2*pi)) .* exp(-0.5*(x./0.05).^2 );
        bKU2 = @(x) -b .* KU(x).' .* (vr+vl);
        dKU = @(x) d .* KU(x);
        y3 = @(x) bKU2(x) + dKU(x).' .* ur;
        y4 = @(x) bKU2(x) + dKU(x).' .* ul;
        KV = @(x) 1 ./ (0.01 .* sqrt(2*pi)) .* exp(-0.5*(x./0.01).^2 );
        bKV2 = @(x) -bV .* KV(x).' .* (vr+vl);
        dKV = @(x) dV .* KV(x);
        F3 = @(x) bKV2(x) + dKV(x).' .* ur;
        F4 = @(x) bKV2(x) + dKV(x).' .* ul;
        ynum3 = y3(x);
        ynum4 = y4(x);
        I1 = cumtrapz(x,ynum3);
        W1 = cumtrapz(x,ynum4);
        Fnum3 = F3(x); 
        Fnum4 = F4(x); 
        Z1 = cumtrapz(x,Fnum3);
        U1 = cumtrapz(x,Fnum4);
        for j=2:J                % interior nodes
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ur(j,n)=ur(j,n-1)-0.5*mu1*(ur(j,n-1)-ur(j-1,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(j,n-1)))*ul(j,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(j,n-1)))*ur(j,n-1)-dt*beta*ur(j,n-1)*(vr(j,n-1)+vl(j,n-1))+dt*0.5*alpha*(vr(j,n-1)+vl(j,n-1));
            %%%%%%%%%%%%%%%%%%%upwind of right-moving of susceptible
            ul(j,n)=ul(j,n-1)+0.5*mu1*(ul(j+1,n-1)-ul(j,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(j,n-1)))*ur(j,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(j,n-1)))*ul(j,n-1)-dt*beta*ul(j,n-1)*(vr(j,n-1)+vl(j,n-1))+dt*0.5*alpha*(vr(j,n-1)+vl(j,n-1));    
            %%%%%%%%%%%%%%%%%%%upwindof right-moving of infected
            vr(j,n)=vr(j,n-1)-0.5*mu2*(vr(j,n-1)-vr(j-1,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(j,n-1)))*vl(j,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(j,n-1)))*vr(j,n-1)+dt*beta*ur(j,n-1)*(vr(j,n-1)+vl(j,n-1))-dt*alpha*vr(j,n-1);
            %%%%%%%%%%%%%%%%%%%upwind of left-moving of infected
            vl(j,n)=vl(j,n-1)+0.5*mu2*(vl(j+1,n-1)-vl(j,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(j,n-1)))*vr(j,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(j,n-1)))*vl(j,n-1)+dt*beta*ul(j,n-1)*(vr(j,n-1)+vl(j,n-1))-dt*alpha*vl(j,n-1);
            
        end 
        %peridic BC for right moving of suscep(j=J+1,j+2=1)
        ur(J+1,n)=ur(J+1,n-1)-0.5*mu1*(ur(J+1,n-1)-ur(J,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(J+1,n-1)))*ul(J+1,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(J+1,n-1)))*ur(J+1,n-1)-dt*beta*ur(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))+dt*0.5*alpha*(vr(J+1,n-1)+vl(J+1,n-1));
        %peridic BC for right moving of suscep(j=1,0=J)
        ur(1,n)=ur(1,n-1)-0.5*mu1*(ur(1,n-1)-ur(J,n-1))+dt*lambdau2*(0.5+0.5*tanh(1+I1(1,n-1)))*ul(1,n-1)-dt*lambdau1*(0.5+0.5*tanh(1+W1(1,n-1)))*ur(1,n-1)-dt*beta*ur(1,n-1)*(vr(1,n-1)+vl(1,n-1))+dt*0.5*alpha*(vr(1,n-1)+vl(1,n-1));
        %peridic BC for left moving of suscep (j=1)
        ul(1,n)=ul(1,n-1)+0.5*mu1*(ul(2,n-1)-ul(1,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(1,n-1)))*ur(1,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(1,n-1)))*ul(1,n-1)-dt*beta*ul(1,n-1)*(vr(1,n-1)+vl(1,n-1))+dt*0.5*alpha*(vr(1,n-1)+vl(1,n-1));    
        %peridic BC for left moving of suscep(j=J+1)
        ul(J+1,n)=ul(J+1,n-1)+0.5*mu1*(ul(1,n-1)-ul(J+1,n-1))+dt*lambdau1*(0.5+0.5*tanh(1+W1(J+1,n-1)))*ur(J+1,n-1)-dt*lambdau2*(0.5*+0.5*tanh(1+I1(J+1,n-1)))*ul(J+1,n-1)-dt*beta*ul(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))+dt*0.5*alpha*(vr(J+1,n-1)+vl(J+1,n-1));    
        %peridic BC for right moving of infected(j=J+1)
        vr(J+1,n)=vr(J+1,n-1)-0.5*mu2*(vr(J+1,n-1)-vr(J,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(J+1,n-1)))*vl(J+1,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(J+1,n-1)))*vr(J+1,n-1)+dt*beta*ur(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))-dt*alpha*vr(J+1,n-1);
        %peridic BC for right moving of infected(j=1)
        vr(1,n)=vr(1,n-1)-0.5*mu2*(vr(1,n-1)-vr(J,n-1))+dt*lambdav2*(0.5+0.5*tanh(1+Z1(1,n-1)))*vl(1,n-1)-dt*lambdav1*(0.5+0.5*tanh(1+U1(1,n-1)))*vr(1,n-1)+dt*beta*ur(1,n-1)*(vr(1,n-1)+vl(1,n-1))-dt*alpha*vr(1,n-1);
        %peridic BC for left moving of infected(j=1)
        vl(1,n)=vl(1,n-1)+0.5*mu2*(vl(2,n-1)-vl(1,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(1,n-1)))*vr(1,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(1,n-1)))*vl(1,n-1)+dt*beta*ul(1,n-1)*(vr(1,n-1)+vl(1,n-1))-dt*alpha*vl(1,n-1);
        %peridic BC for left moving of infected(j=J+1)
        vl(J+1,n)=vl(J+1,n-1)+0.5*mu2*(vl(1,n-1)-vl(J+1,n-1))+dt*lambdav1*(0.5+0.5*tanh(1+U1(J+1,n-1)))*vr(J+1,n-1)-dt*lambdav2*(0.5*+0.5*tanh(1+Z1(J+1,n-1)))*vl(J+1,n-1)+dt*beta*ul(J+1,n-1)*(vr(J+1,n-1)+vl(J+1,n-1))-dt*alpha*vl(J+1,n-1);
    end
    
    %calculate the analytical solution 
    for j=1:J+1
        xj=xl+(j-1)*dx;
        u_ex(j,n)=exp(-c*(xj-au*t-0.5)^2);
    end
end 
figure
ss=surf(T,X,ur+ul);
set(ss,'edgecolor','none');
xlabel('t','FontSize',28);
ylabel('x','FontSize',28);
title('u^++u^-','FontSize',28);
set(gca,'Fontsize',28);
view(2)