Please someone should help me debug my error. The approximate solution is supposed to converge to the exact solution. I think i have problem with the code.

i want to plot a graph of the numerical and exact solution. however, the errore is too much. can someone help me to trace my error. the solution are supposed to be ALMOST THE SAME. Here is my code:

alpha=0.25;
r=5;
t=0.002;
x=[-5:0.2:5];
k = reshape(0:100, 1, 1, []);
A=exp(x);
B = (t.^(k.*alpha))./factorial(k).*alpha.^k;
C = sum(B,3);
Numerical=A.*C;
Exact=A.*exp((r-4).*(t.^alpha)./(alpha));
Z1=Numerical;
Z2=Exact;
hZ1 = plot(x,Z1,'-r');
hold on
hZ2 = plot(x,Z2,'-g');
hold off
grid on
xlabel('x')
ylabel('u')
legend([hZ1(1),hZ2(1)], 'CFRDTM','EXACT', 'Location','NE')

Answer 1

[Star Strider]

There are several typographical errors, at lelast with respect to the code representing the symbolic equations in the image.

With those corrections, ‘Numerical’ and ‘Exact’ are the same (within floating-point approximation error), at least for the values provided.

alpha=0.25; 
r=5; 
t=0.002; 
x=[-5:0.2:5]; 
k = reshape(0:100, 1, 1, []); 
A=exp(x); 
% B =(t.^(k.*alpha))./(factorial(k).*alpha.^k); 
B = (t.^(k.*alpha)).*(r-4).^k ./ (factorial(k).*alpha.^k); 
C = sum(B,3); 
Numerical=A.*C; 
% Exact=A.*exp((r-4).*(t.^alpha)./(alpha)); 
Exact = exp(x+((r-4).*(t.^alpha)./(alpha))); 
Z1=Numerical 

Z1 = 1×51

0.0157 0.0192 0.0234 0.0286 0.0349 0.0427 0.0521 0.0637 0.0778 0.0950 0.1160 0.1417 0.1731 0.2114 0.2582 0.3153 0.3852 0.4704 0.5746 0.7018 0.8572 1.0470 1.2788 1.5619 1.9077 2.3301 2.8460 3.4761 4.2457 5.1857

Z2=Exact 

Z2 = 1×51

0.0157 0.0192 0.0234 0.0286 0.0349 0.0427 0.0521 0.0637 0.0778 0.0950 0.1160 0.1417 0.1731 0.2114 0.2582 0.3153 0.3852 0.4704 0.5746 0.7018 0.8572 1.0470 1.2788 1.5619 1.9077 2.3301 2.8460 3.4761 4.2457 5.1857

Error = Numerical - Exact;
ErrorRMS = sqrt(mean(Error.^2))

ErrorRMS = 1.1945e-14

figure
hZ1 = plot(x,Z1,'-r'); 
hold on 
hZ2 = plot(x,Z2,'--g');                                     % Change To Dashed Line
hold off 
grid on 
xlabel('x')  
ylabel('u') 
legend([hZ1(1),hZ2(1)], 'CFRDTM','EXACT', 'Location','NE')          

It would also help if the code was parsed into different lines (as I had to do here), in order for it to run.

.

[Omorodion Solomon]

Thank you so much. It works perfectly. Now, suppose I want to compute a table that will display the following, x=-4,-3,-2,-1,2,3,4; t=0.002,0.004,0.006,0.008;EXACT; numerical; Absolute Error. How do I do that? Thanks you in advance.

[Star Strider]

As always, my pleasure!

‘Now, suppose I want to compute a table that will display the following,’

I am not absolutely certain what you want, however there are 7 values of ‘x’ and 4 values of ‘t’, so there appears to be only one way to approach this:

x = [-4,-3,-2,-1,2,3,4]; 
t = [0.002,0.004,0.006,0.008];
[Xm,Tm] = ndgrid(x,t);

[Exact,Numerical] = runCode(Xm(:),Tm(:));
absErr = abs(Exact-Numerical);

Results = table(Xm(:), Tm(:), Exact, Numerical, absErr, 'VariableNames',{'x','t','Exact','Numerical','AbsoluteError'})

Results = 28×5 table
    x       t       Exact      Numerical    AbsoluteError
    __    _____    ________    _________    _____________

    -4    0.002    0.042677    0.042677      1.3878e-17  
    -3    0.002     0.11601     0.11601      2.7756e-17  
    -2    0.002     0.31534     0.31534      5.5511e-17  
    -1    0.002     0.85718     0.85718      1.1102e-16  
     2    0.002      17.217      17.217      3.5527e-15  
     3    0.002      46.801      46.801      1.4211e-14  
     4    0.002      127.22      127.22      4.2633e-14  
    -4    0.004    0.050084    0.050084      1.3878e-17  
    -3    0.004     0.13614     0.13614      2.7756e-17  
    -2    0.004     0.37007     0.37007               0  
    -1    0.004       1.006       1.006               0  
     2    0.004      20.205      20.205               0  
     3    0.004      54.924      54.924      1.4211e-14  
     4    0.004       149.3       149.3      2.8422e-14  
    -4    0.006    0.055758    0.055758               0  
    -3    0.006     0.15157     0.15157               0  

figure
scatter3(Results.x, Results.t, Results.Exact, 'xb')
hold on
scatter3(Results.x, Results.t, Results.Numerical, '+r')
hold off
grid on
xlabel('x')
ylabel('t')
set(gca, 'ZScale','log')
legend('Exact', 'Numerical', 'Location','bestoutside')

function [Exact,Numerical] = runCode(x,t)
alpha=0.25; 
r=5; 
% t=0.002; 
% x=[-5:0.2:5]; 
k = reshape(0:100, 1, 1, []); 
A=exp(x); 
% B =(t.^(k.*alpha))./(factorial(k).*alpha.^k); 
B = (t.^(k.*alpha)).*(r-4).^k ./ (factorial(k).*alpha.^k); 
C = sum(B,3); 
Numerical=A.*C; 
% Exact=A.*exp((r-4).*(t.^alpha)./(alpha)); 
Exact = exp(x+((r-4).*(t.^alpha)./(alpha))); 
end

That would appear to cover all the possibilities.

.

[Omorodion Solomon]

</matlabcentral/answers/uploaded_files/718029/IMG_20210821_223615_715.jpg> I'm getting this error while running it

[Omorodion Solomon]

[Star Strider]

The function is part of the code I posted.

Please copy the entire code, then run it, as I do here —

x = [-4,-3,-2,-1,2,3,4]; 
t = [0.002,0.004,0.006,0.008];
[Xm,Tm] = ndgrid(x,t);

[Exact,Numerical] = runCode(Xm(:),Tm(:));
absErr = abs(Exact-Numerical);

Results = table(Xm(:), Tm(:), Exact, Numerical, absErr, 'VariableNames',{'x','t','Exact','Numerical','AbsoluteError'})

Results = 28×5 table
    x       t       Exact      Numerical    AbsoluteError
    __    _____    ________    _________    _____________

    -4    0.002    0.042677    0.042677      1.3878e-17  
    -3    0.002     0.11601     0.11601      2.7756e-17  
    -2    0.002     0.31534     0.31534      5.5511e-17  
    -1    0.002     0.85718     0.85718      1.1102e-16  
     2    0.002      17.217      17.217      3.5527e-15  
     3    0.002      46.801      46.801      1.4211e-14  
     4    0.002      127.22      127.22      4.2633e-14  
    -4    0.004    0.050084    0.050084      1.3878e-17  
    -3    0.004     0.13614     0.13614      2.7756e-17  
    -2    0.004     0.37007     0.37007               0  
    -1    0.004       1.006       1.006               0  
     2    0.004      20.205      20.205               0  
     3    0.004      54.924      54.924      1.4211e-14  
     4    0.004       149.3       149.3      2.8422e-14  
    -4    0.006    0.055758    0.055758               0  
    -3    0.006     0.15157     0.15157               0  

figure
scatter3(Results.x, Results.t, Results.Exact, 'xb')
hold on
scatter3(Results.x, Results.t, Results.Numerical, '+r')
hold off
grid on
xlabel('x')
ylabel('t')
set(gca, 'ZScale','log')
legend('Exact', 'Numerical', 'Location','bestoutside')

function [Exact,Numerical] = runCode(x,t)
alpha=0.25; 
r=5; 
% t=0.002; 
% x=[-5:0.2:5]; 
k = reshape(0:100, 1, 1, []); 
A=exp(x); 
% B =(t.^(k.*alpha))./(factorial(k).*alpha.^k); 
B = (t.^(k.*alpha)).*(r-4).^k ./ (factorial(k).*alpha.^k); 
C = sum(B,3); 
Numerical=A.*C; 
% Exact=A.*exp((r-4).*(t.^alpha)./(alpha)); 
Exact = exp(x+((r-4).*(t.^alpha)./(alpha))); 
end

.