Solving Cauchy problem for first order PDE

How to solve the following Cauchy problem using MATLAB: x*U_x + y*U_y = U+1; U(x,y)= x^2 on y= x^2

I was trying using the following command: dsolve('Dx=x','Dy=y','Du=u+1');

But the problem is how to apply Cauchy data? Please advice.

[John D'Errico]

What Cauchy data? You have not shown any data.

As well, you might also explain what you intended by the expression:

U+!

[Akansha Mittal]

Sorry it is edited now

Answer 1

[Torsten]

There is no ready-to-use MATLAB program to solve this kind of problem.

I suggest you look up "method-of-characteristics". Your PDE can be solved analytically.

Best wishes

Torsten.

[Akansha Mittal]

Sir, the command which I wrote is computing characteristics only but the problem is that how to include the Cauchy data in the command? Please suggest. Thank you

[Torsten]

Don't know if it works, but you'll see how to proceed:

syms x(s) y(s) u(s) X0
[solx,soly,solu] = dsolve(diff(x,s)==x, diff(y,s)==y, diff(u,s)==u+1, x(0)==X0, y(0)==X0^2, u(0)==X0^2);
[X0sol,ssol]=solve([solx==x,soly==y],[X0,s]];
subs(solu,[X0,s],[X0sol,ssol])

Best wishes

Torsten.

[naren BORO]

programming is not correct