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I've searched around, but I couldn't find anything too helpful on the subject, so here goes. I am trying to find (if possible) an analytical solution to the differential equation of the form: $$ \pm\frac{\partial F}{\partial \tau} = \frac{a}{x^{2/3}}F + \frac{b}{x} \frac{\partial F}{\partial x} + c\frac{\partial^2 F}{\partial x^2} $$ where $a$, $b$, and $c$ are distinct constants, while $F = F(x,t)$. Owing to the similarity between this equation and a general diffusion equation, I have tried massaging it into various 'known' forms of such differential equations like $$ \frac{\partial F}{\partial \tau} = \frac{\partial}{\partial x}\left(g(x) \frac{\partial F}{\partial x} \right) + h(x)F + \text{constant}\times F $$ but have so far been unable to yield anything useful. I am really quite novice when it comes to solving such ODE's (so I might be missing a trick), but am not sure where to go from here. Any help would be greatly appreciated!

P.S. I might be completely over-complicating this; that's entirely possible.

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    $\begingroup$ This is not a non-linear equation. $\endgroup$
    – okrzysik
    Commented Jul 12, 2016 at 10:55
  • $\begingroup$ @okrzysik That's very true; I was getting myself confused looking in the literature! Still stuck though unfortunately. $\endgroup$
    – PhysyCola
    Commented Jul 12, 2016 at 11:02
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    $\begingroup$ @PhysyCola : You are opening a too wide subject if you don't say what are the boundary and initial conditions. $\endgroup$
    – JJacquelin
    Commented Jul 12, 2016 at 11:44
  • $\begingroup$ @JJacquelin Ok, so if you need initial conditions; the condition on x would be that $F=0$ at $x \rightarrow \infty$, and $F = 0$ at $\tau \rightarrow \infty$ for the $+$ on the LHS, and $F = F_0$ at some $\tau = \tau_{max}$ for the $-$ on the LHS, with the aim of taking $\tau_{max} \rightarrow \infty$. Does that help? $\endgroup$
    – PhysyCola
    Commented Jul 12, 2016 at 12:35
  • $\begingroup$ Please make it less ambiguous. First condition : $F(x\to \infty\:,\:\tau)=0$ but what is the value of $\tau$? . Second condition : $F(x\:,\:\tau\to\infty)=0$ but what is the value of $x$ ? Third condition : ... etc. $\endgroup$
    – JJacquelin
    Commented Jul 12, 2016 at 12:45

1 Answer 1

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$$ \pm\frac{\partial F}{\partial \tau} = \frac{a}{x^{2/3}}F + \frac{b}{x} \frac{\partial F}{\partial x} + c\frac{\partial^2 F}{\partial x^2} $$ With the method of separation of variables, one can find an infinity of particular solutions.

Let $F(x,\tau)=f(x)g(\tau) \quad\to\quad \pm\frac{g'}{g}=\frac{a}{x^{2/3}}+\frac{b}{x}\frac{f'}{f}+c\frac{f''}{f}=\lambda$ $$g(\tau)=e^{\pm\lambda\tau}$$ $f_\lambda(x)$ is solution of the ODE : $\quad cf_\lambda''(x)+\frac{b}{x}f_\lambda'(x)+\left(\frac{a}{x^{2/3}}-\lambda \right)f_\lambda(x)=0 $

The PDE is linear. So, a more general solution can be formally given in terms of infinite series (or finite series in some cases).

$$F(x,\tau)=\sum_{\text{any } \lambda}C_\lambda f_\lambda(x)e^{\pm\lambda\tau}$$ where $C_\lambda$ are constants to be determined to make $F(x,\tau)$ fit to the boundary and initial conditions.

The hitch is that the analytic solving of the ODE in order to express $f_\lambda(x)$ on closed form is certainly very arduous.

Moreover, the calculus of the coefficients $C_\lambda$ in order to satisfy the boundary and initial conditions is generally much too complicated (except in some particular and favorable cases, which is the raison of the necessity to precisely know how those conditions are).

Most likely, the analytical solving of the above PDE is not realistic. Numerical methods of solving are recommended.

Only in case of very favorable boundary and initial conditions this pessimistic opinion might be reconsidered.

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