2.14.9 Chaffee Infante \(u_t = u_{xx} + \lambda (u^3 - u) = 0\)

problem number 111

Added December 27, 2018.

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

Chaffee Infante equation. Solve for \(u(x,t)\) \[ u_t = u_{xx} + \lambda (u^3 - u) = 0 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] - D[u[x, t], {x, 2}] + lambda*(u[x, t]^3 - u[x, t]) == 0; 
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),x$2)+lambda*(u(x,t)^3-u(x,t))=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,t))),output='realtime'));
 

\[u \left (x , t\right ) = \frac {\tanh \left (-\frac {3 \lambda t}{4}+\frac {\sqrt {2}\, \sqrt {\lambda }\, x}{4}+c_{1}\right )}{2}-\frac {1}{2}\]

____________________________________________________________________________________