2.15.29 Thomas equation \( u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0\)

problem number 139

Added December 27, 2018.

Taken from https://en.wikipedia.org/wiki/List_of_nonlinear_partial_differential_equations

Thomas equation. Solve for \(u(x,t)\) \[ u_{xy} + \alpha u_x + \beta u_y+ \nu u_x u_y =0 \]

Mathematica

ClearAll["Global`*"]; 
pde =  D[u[x, y], x, y] + alpha*D[u[x, y], x] + beta*D[u[x, y], y] + nu*D[u[x, y], x]*D[u[x, y], y] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y], {x, y}], 60*10]];
 

Failed

Maple

restart; 
pde := diff(u(x,y),x,y)+alpha*diff(u(x,y),x)+beta*diff(u(x,y),y) 
      +nu* diff(u(x,y),x)*diff(u(x,y),y)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y),'build')),output='realtime'));
 

\[u \left ( x,y \right ) =1/2\,{\frac {1}{\nu } \left ( -\ln \left ( {\frac {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ( {\it \_C1}\,{{\rm e}^{ \left ( x-y \right ) \sqrt {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C2} \right ) ^{2}}} \right ) -\ln \left ( {\frac {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }{{\nu }^{2} \left ( {\it \_C3}\,{{\rm e}^{ \left ( x+y \right ) \sqrt {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }}}-{\it \_C4} \right ) ^{2}}} \right ) + \left ( -x+y \right ) \sqrt {{\alpha }^{2}-2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }+ \left ( -x-y \right ) \sqrt {{\alpha }^{2}+2\,\alpha \,\beta +{\beta }^{2}-4\,{\it \_c}_{{1}}\nu }-2\,\beta \,x-2\,\alpha \,y-4\,\ln \left ( 2 \right ) \right ) }\]

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