Analytically solve following PDE: $$u_{tt}+2u_{xt}+u_{xx}=u_t-u_x, \ u(x,x)=e^{-4x^2}$$ I can solve it by hand $$u(x,t)=\frac{e^{-\frac{(x+t)^2}{1+8(t-x)}}}{\sqrt{1+8(t-x)}}$$ But this boundary condition can't be recognized by my mathematica:

  pde = D[u[x, t], t, t] + 2 D[u[x, t], x, t] + D[u[x, t], x, x]
  - D[u[x, t], t] + D[u[x, t], x] == 0;
  soln = DSolve[{pde, u[x, x] == E^(-4 x^2)}, u[x, t], {x, t}]

They said it can't recognize this boundary condition.

  • $\begingroup$ In general hand is better than CAS. $\endgroup$ – Mariusz Iwaniuk Oct 20 '17 at 20:28
  • $\begingroup$ @MariuszIwaniuk I hope mathematica can do these kind of thing. I don't want to cost my time to do these kinds of computation in my research. $\endgroup$ – maplemaple Oct 20 '17 at 20:31
  • 1
    $\begingroup$ Mathematica can't solve this. Dsolve is weak, it can solve only most popular and simple equations.Try Maple :) $\endgroup$ – Mariusz Iwaniuk Oct 20 '17 at 20:36

We don't normally allow mixed time/space BCs since they are not, in general, well posed, and it is better to return no answer than the wrong answer. There is a further complication that it is trying to figure out which equations are differential equations and which are BCs (a cost of allowing you to freely specify as many of each as you want, in any order), and it's being confused here by seeing $u(x,x)$. I will file a suggestion about this, but it is not so easy to fix.

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