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'));
\[u \left ( x,t \right ) ={\frac {{\it \_C3}}{{{\rm e}^{{\it \_c}_{{1}}t}}}\sqrt {{{\rm e}^{{\frac {x\beta }{d}}}}} \left ( {{\rm e}^{{\frac {x}{2\,d}\sqrt {{\beta }^{2}-4\,d{\it \_c}_{{1}}}}}}{\it \_C1}+{{\rm e}^{-{\frac {x}{2\,d}\sqrt {{\beta }^{2}-4\,d{\it \_c}_{{1}}}}}}{\it \_C2} \right ) }\]
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