# Is analytical solution available for this PDE?

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?

• Are $m$, $n$, and $p$ numbers or functions? Jul 7, 2020 at 17:43
• They are constants Jul 7, 2020 at 17:43
• This is the one-dimensional heat equation with Dirichlet conditions, right? Jul 7, 2020 at 17:54
• @SimpleProgrammer, in principle Yes! Jul 7, 2020 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 $$u'(x)$$.