I been trying to solve the below PDE analytically for awhile now but with no success, perhaps someone here can help me with it. $$ \frac{\partial w(x,t)}{\partial t}= \frac{\partial}{\partial x} \bigg(\beta \exp (-g(x)) \frac{\partial}{\partial x} \big\{ w(x,t) \exp(g(x)) \big\} \bigg) $$

And the boundary and initial conditions: (1) $$ w(x,0)=m $$ and (2) $$w(0,t)=n, w(L,t)=p$$.

Is analytical solution possible here?

  • $\begingroup$ Are $m$, $n$, and $p$ numbers or functions? $\endgroup$ – Josh B. Jul 7 '20 at 17:43
  • $\begingroup$ They are constants $\endgroup$ – Lucas Derck Jul 7 '20 at 17:43
  • $\begingroup$ This is the one-dimensional heat equation with Dirichlet conditions, right? $\endgroup$ – SimpleProgrammer Jul 7 '20 at 17:54
  • $\begingroup$ @SimpleProgrammer, in principle Yes! $\endgroup$ – Lucas Derck Jul 7 '20 at 17:57

If you write $w(x,t) = v(x,t) \exp(-g(t))$, the PDE becomes

$$ \dfrac{\partial v}{\partial t} = \beta\dfrac{\partial^2 v}{\partial x^2} - \beta g'(x) \dfrac{\partial v}{\partial x} $$ This has separation-of-variables solutions $v(x,t) = X(x) T(t)$ where $c$ is an arbitrary constant and $$ T'(t) = c \beta T(t), \ X''(x) = c X(x) + g'(x) X'(x) $$ In general I think the equation for $X$ can't be solved in closed form, but can be reduced to a first-order nonlinear equation: if $X(x) = \exp(u(x))$, $u'' + (u')^2 = g'(x) u'(x) - c$ which is a first-order nonlinear equation for $u'(x)$.


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