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.

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Omorodion Solomon 2021-8-21

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评论： Star Strider 2021-8-21

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')

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Answer 1

Star Strider 2021-8-21

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在 MATLAB Online 中打开

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.

.

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Star Strider 2021-8-21

在 MATLAB Online 中打开

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 2021-8-21

Star Strider 2021-8-21

在 MATLAB Online 中打开

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

.

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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.

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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.

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