2.16.4 \(u_t + u_{xxx} = 0\)

problem number 144

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Third oder PDE. Solve for \(u(x,y)\) \[ u_t + u_{xxx} = 0 \] With initial conditions \begin {align*} u(x,0)&=f(x) \end {align*}

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[u[x, t], t] == -D[u[x, t], {x, 3}]; 
ic  = u[x, 0] == f[x]; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, u[x, t], {x, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde := diff(u(x, t), t)=- diff(u(x, t), x$3); 
ic  := u(x,0)=f(x); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic],u(x,t))),output='realtime'));
 

\[u \left ( x,t \right ) =1/4\,{\frac {1}{{\pi }^{2}}\int _{-\infty }^{\infty }\!4/3\,{\frac {\pi \,f \left ( -\zeta \right ) }{\sqrt [3]{-t}}\sqrt {-{\frac {x+\zeta }{\sqrt [3]{-t}}}}\BesselK \left ( 1/3,-2/9\,{\frac {\sqrt {3} \left ( x+\zeta \right ) }{\sqrt [3]{-t}}\sqrt {-{\frac {x+\zeta }{\sqrt [3]{-t}}}}} \right ) }\,{\rm d}\zeta }\]

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