2.16.9 Einstein-Weiner \(u_t = -\beta u_x + D u_{xx}\)

problem number 148

Added January 2, 2018.

Einstein-Weiner PDE. Solve for \(u(x,t)\) with \(x>0,t>0\)

\[ u_t = -\beta u_x + D u_{xx} \]

Assuming \(\beta >0,D>0\)

Mathematica ✓

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

\[\{\{u(x,t)\to \cosh (c_2 (\beta t+c_2 d t+x)+c_1)+\sinh (c_2 (\beta t+c_2 d t+x)+c_1)+1\}\}\]

Maple ✓

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

\[u \left (x , t\right ) = \sqrt {{\mathrm e}^{\frac {x \beta }{d}}}\, c_{3} \left (c_{1} {\mathrm e}^{\frac {x \sqrt {\beta ^{2}-4 d \textit {\_c}_{1}}}{d}}+c_{2} \right ) {\mathrm e}^{-\frac {2 \textit {\_c}_{1} t d +x \sqrt {\beta ^{2}-4 d \textit {\_c}_{1}}}{2 d}}\]

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