2.15.7 Buckmaster \( u_t = (u^4)_{xx} + (u^3)_x\)

problem number 116

Added December 27, 2018.

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

Buckmaster. Solve for \(u(x,t)\)

\[ u_t = (u^4)_{xx} + (u^3)_x \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] == D[u[x, t]^4, {x, 2}] + D[u[x, t]^3, x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, t], {x, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(u(x,t),t)= diff(u(x,t)^4,x$2)+diff(u(x,t)^3,x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = \operatorname {RootOf}\left (c_1 x +c_2 t +c_{3} +4 c_1^{2} \int _{}^{\textit {\_Z}}\frac {\textit {\_f}^{3}}{c_1 \,\textit {\_f}^{3}+4 c_{3} c_1^{2}-c_2 \textit {\_f}}d \textit {\_f} +c_{4} \right )\]

Answer in terms of RootOf.

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