Estevez Mansﬁeld Clarkson utyyy + βuyuyt + βuyyut + utt = 0

2.15.13 Estevez Mansﬁeld Clarkson $u_{t y y y} + β u_{y} u_{y t} + β u_{y y} u_{t} + u_{t t} = 0$

problem number 122

Added December 27, 2018.

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

Estevez Mansﬁeld Clarkson equation. Solve for $u (x, y, t)$ $u_{t y y y} + β u_{y} u_{y t} + β u_{y y} u_{t} + u_{t t} = 0$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[D[u[x, y, t], t], {y, 3}] + beta*D[u[x, y, t], y]*D[D[u[x, y, t], y], t] + beta*D[u[x, y, t], {y, 2}]*D[u[x, y, t], t] + D[u[x, y, t], {t, 2}] == 0; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, u[x, y, t], {x, y, t}], 60*10]];

${{u (x, y, t) \to \frac{6 c_{1} (x) \tanh (- 4 t (c_{1} (x))^{3} + y c_{1} (x) + c_{3} (x))}{β} + c_{4} (x)}}$

Maple ✓

restart; 
beta='beta'; 
pde := diff(u(x,y,t),t,y,y,y)+ beta*diff(u(x,y,t),y)*diff(u(x,y,t),y,t) +  beta*diff(u(x,y,t),y$2)*diff(u(x,y,t),t) + diff(u(x,y,t),t$2)=0; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,u(x,y,t))),output='realtime'));

$u (x, y, t) = 6 \frac{\tanh (- 4 {_C 3}^{3} t +_C 2 x +_C 3 y +_C 1)_C 3}{β} +_C 5$

____________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]

2.15.13 Estevez Mansﬁeld Clarkson utyyy+βuyuyt+βuyyut+utt=0

2.15.13 Estevez Mansﬁeld Clarkson $u_{t y y y} + β u_{y} u_{y t} + β u_{y y} u_{t} + u_{t t} = 0$