How to make graph that plot between Analytical Solution (Exact Solution) and Numerical Method Solution?

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Yusuf Arya Yudanto 2022-3-5

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评论： Yusuf Arya Yudanto 2022-3-6

Greetings!

Can someone help to guide me how to make graph that plot between Analytical Solution (Exact Solution) and Adams-Moulton Solution? Here I attached my script.

%% 2nd-order Adams-Moulton Solution
fun = @(t,y) ((1+4*t)*((y)^1/2));
y0 = 1;
tspan = [0,1];
N = 4; 
%% Initial Values
h = (tspan(2) - tspan(1))/N;
exactY = @(t) (((3*t)/2)+3*t.^2).^(2/3);
t1 = tspan(1) + h; y1 = exactY(t1);
t2 = tspan(1) + 2*h; y2 = exactY(t2);
[t2,Y2] = AM2(fun,tspan,y0,y1,N);
%% Display Solution
Y = exactY(t2);
disp('-----------------------------'); 
disp('t_i   y(t_i)   AM2   Error')
disp('-----------------------------'); 
formatSpec = '%.2f   %.5f   %.5f   %.5f\n';
fprintf(formatSpec,[t2';Y';Y2';abs(Y'-Y2')])

I really appreciate any help you can provide. Thank you.

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Torsten 2022-3-5

在 MATLAB Online 中打开

fun = @(t,y) (1+4*t)*sqrt(y);
y0 = 1;
tspan = [0,1];
N = 4; 
%% Initial Values
h = (tspan(2) - tspan(1))/N;
exactY = @(t)(t/2 + t.^2 + 1).^2;
t1 = tspan(1) + h; y1 = exactY(t1);
[T,Ynum] = AM2(fun,tspan,y0,y1,N);
%% Display Solution
Yexact = exactY(T);
plot(T,[Ynum,Yexact])
function [t,y] = AM2(fun,tspan,y0,y1,N)
a = tspan(1);
b = tspan(2);
h = (b-a)/N;
t = zeros(N+1,1);
y = zeros(N+1,1);
t(1) = a; y(1) = y0;
t(2) = a+h; y(2) = y1;
for i = 2:N
    t(i+1) = t(i) + h;
    Fi = fun(t(i),y(i));
    Fi1 = fun(t(i-1),y(i-1));
    funh = @(yh) yh - y(i) - h/12*(5*fun(t(i+1),yh) + 8*Fi - Fi1);
    y(i+1) = fsolve(funh,y(i));
end
end

Yusuf Arya Yudanto 2022-3-5

Many thanks! I really appreciate your help.

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Answer 1

Alan Stevens 2022-3-5

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此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/1664029-how-to-make-graph-that-plot-between-analytical-solution-exact-solution-and-numerical-method-soluti#answer_910384

在 MATLAB Online 中打开

After Y = exactY(t2) you probably need something like:

plot(t2,Y2,t2,Y)
legend('AM','Exact')

However, you haven't supplied the code for your AM function, so I can't check!

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Alan Stevens 2022-3-5

在 MATLAB Online 中打开

Like so:

%% 2nd-order Adams-Moulton Solution

fun = @(t,y) ((1+4*t)*((y)^1/2));

y0 = 1;

tspan = [0,1];

N = 4;

%% Initial Values

h = (tspan(2) - tspan(1))/N;

exactY = @(t) (((3*t)/2)+3*t.^2).^(2/3);

t1 = tspan(1) + h; y1 = exactY(t1);

t2 = tspan(1) + 2*h; y2 = exactY(t2);

[t2,Y2] = AM2(fun,tspan,y0,y1,N);

%% Display Solution

Y = exactY(t2);

plot(t2,Y2,t2,Y),grid

xlabel('t2'),ylabel('y2 and Y')

legend('AM','Exact')

disp('-----------------------------');

-----------------------------

disp('t_i y(t_i) AM2 Error')

t_i y(t_i) AM2 Error

disp('-----------------------------');

-----------------------------

formatSpec = '%.2f %.5f %.5f %.5f\n';

fprintf(formatSpec,[t2';Y';Y2';abs(Y'-Y2')])

0.00 0.00000 1.00000 1.00000 0.25 0.68142 0.68142 0.00000 0.50 1.31037 0.78457 0.52580 0.75 1.99248 0.96652 1.02596 1.00 2.72568 1.26418 1.46150

function [t,y] = AM2(fun,tspan,y0,y1,N)

a = tspan(1);

b = tspan(2);

h = (b-a)/N;

t = zeros(N+1,1);

y = zeros(N+1,1);

t(1) = a; y(1) = y0;

t(2) = a+h; y(2) = y1;

for i = 2:N

t(i+1) = t(i) + h;

Fi = fun(t(i),y(i));

Fi1 = fun(t(i-1),y(i-1));

Fiplus1 = fun(t(i+1),y(i+1));

y(i+1) = y(i) + (h/12)*(5*Fiplus1 + 8*Fi - Fi1);

end

Yusuf Arya Yudanto 2022-3-5

Oh. Thank you for the correction. I really appreciate it!

Yusuf Arya Yudanto 2022-3-6

Hello, Sir Alan. Can you help me make a loglog scale plot of the multiple solutions? I've posted the question, please kindly check it. How to make convergence plot (error VS time step) in a log-log scale among four numerical methods and exact solution?

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How to make graph that plot between Analytical Solution (Exact Solution) and Numerical Method Solution?

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How to make graph that plot between Analytical Solution (Exact Solution) and Numerical Method Solution?

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