32 Tricomi PDE

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32.1 Boundary value problem for the Tricomi equation

problem number 182

From Mathematica DSolve helps pages.

Solve for \(u(x,y)\) \[ \frac {\partial ^2 u}{\partial x^2} + y \frac {\partial ^2 u}{\partial y^2} =0 \]

With boundary conditions \begin {align*} u(x,0)&=0 \\ \frac {\partial u}{\partial y}(x,0) &= x^2 \end {align*}

Mathematica ✓

ClearAll[u, x, y]; 
 pde = D[u[x, y], {x, 2}] + y*D[u[x, y], {y, 2}] == 0; 
 bc = {u[x, 0] == 0, Derivative[0, 1][u][x, 0] == x^2}; 
 sol = AbsoluteTiming[TimeConstrained[DSolve[{pde, bc}, u[x, y], {x, y}], 60*10]];
 

\[ \left \{\left \{u(x,y)\to -y \left (y-x^2\right )\right \}\right \} \]

Maple ✓

 
x:='x'; y:='y';u:='u'; 
pde:= diff(u(x,y),x$2)+ y*diff(u(x,y),y$2)=0; 
bc:=u(x,0)=0,(D[2](u))(x,0)=x^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde,bc],u(x,y))),output='realtime'));
 

\[ u \left ( x,y \right ) =y \left ( {x}^{2}-y \right ) \]