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

problem number 143

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]];
 

\[\left \{\left \{u(x,t)\to \frac {\int _{-\infty }^{\infty }e^{i K[1] \left (t K[1]^2+x\right )} \int _{-\infty }^{\infty } e^{-i x K[1]} f(x) \, dxdK[1]}{2 \pi }\right \}\right \}\]

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'));
 

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

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