2.16.5 \(u_{xy} = \sin (x) \sin (y) \)

problem number 144

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

PDE solved by Laplace transform. Solve for \(u(x,y)\)

\[ u_{xy} = \sin (x) \sin (y) \]

With boundary conditions

\begin{align*} u(x,0)&=1+\cos (x) \\ \frac {\partial u}{\partial y}(0,y) &= -2 \sin y \end{align*}

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[u[x, y], y, x] == Sin[x]*Sin[y]; 
bc  = {u[x, 0] == 1 + Cos[x], Derivative[0, 1][u][0, y] == -2*Sin[y]}; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, bc}, u[x, y], x, y], 60*10]];
 

\[\{\{u(x,y)\to (\cos (x)+1) \cos (y)\}\}\]

Maple ✓

restart; 
pde := diff(u(x, y), y,x)=sin(x)*sin(y); 
bc  := u(x,0)=1+cos(x),eval( diff(u(x,y),y),x=0)=-2*sin(y); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, bc],u(x,y))),output='realtime'));
 

\[u \left (x , y\right ) = \cos \left (y \right ) \left (1+\cos \left (x \right )\right )\]

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