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

problem number 138

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'));
 

\[u \left (x , y\right ) = \frac {-\ln \left (\frac {\alpha ^{2}-2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }{\nu ^{2} \left ({\mathrm e}^{\left (x -y \right ) \sqrt {\alpha ^{2}-2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }} c_{1} -c_{2} \right )^{2}}\right )-\ln \left (\frac {\alpha ^{2}+2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }{\nu ^{2} \left ({\mathrm e}^{\left (x +y \right ) \sqrt {\alpha ^{2}+2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }} c_{3} -c_{4} \right )^{2}}\right )+\left (-x +y \right ) \sqrt {\alpha ^{2}-2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }+\left (-x -y \right ) \sqrt {\alpha ^{2}+2 \alpha \beta +\beta ^{2}-4 \textit {\_c}_{1} \nu }-2 \beta x -2 \alpha y -4 \ln \left (2\right )}{2 \nu }\]

___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________