I have a partial differential equation as follows: $$\frac{\partial p(x,t)}{\partial t}=\text{Dp} \frac{\partial ^2p(x,t)}{\partial x^2}-\frac{p(x,t)-\text{p0}}{\tau }$$ What I try to do was to get its general solution by

DSolve[D[p[x, t], t] == -((p[x, t] - p0)/τ) + Dp D[p[x, t], {x, 2}], p, {x, t}]

However, the output returns what I have inputted, which means Mathematica cannot solve this PDE.

By looking up the help document, I find that DSolve can only find the general solution for a restricted type of homogeneous linear second-order PDEs, that is, for such a linear second-order PDE as $b \frac{\partial ^2u}{\partial x\, \partial y}+a \frac{\partial ^2u}{\partial x^2}+c \frac{\partial ^2u}{\partial y^2}+d \frac{\partial u}{\partial x}+e \frac{\partial u}{\partial y}+f u=g$, only when d=0,e=0,f=0 and a,b and c are all constants can we get the general solution.

I have tried this PED in Maple, and get a solution with conditions, but I don't know whether it is correct. The help document tells that the algorithm used by DSolve is not applicable in this case.

So, I wonder whether there exist some other ways or some packages to solve this PDE. If finding the general solution is impossible, is it possible to get a special solution?

  • $\begingroup$ You can refer to this answer. $\endgroup$ – xzczd Mar 11 '14 at 2:48
  • $\begingroup$ None of us could make a living doing PDEs if it was that easy.. $\endgroup$ – Kai Sikorski Mar 11 '14 at 6:15
  • $\begingroup$ and get a solution with conditions, but I don't know whether it is correct. You can always substitute the solution back in the PDE and see if it satisfies it? Or since you used Maple, you can use pdetest maplesoft.com/support/help/Maple/view.aspx?path=pdetest $\endgroup$ – Nasser Mar 11 '14 at 6:33
  • $\begingroup$ @xzczd Thanks for this information. I'll try it again. $\endgroup$ – Z-Y.L Mar 11 '14 at 6:59
  • $\begingroup$ @Nasser Thank you! $\endgroup$ – Z-Y.L Mar 11 '14 at 7:00

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