# calculation – Could this particularly symmetric nonlinear PDE provide a solution?

For a physics problem that interests me, I am looking for a (partial) solution to the following particularly symmetric EDP:

$$fc {(1,0)} (t, x)} {f {{0,1}} (t, x)} = g_1 (t) h_1 (x) + g_2 (t) h_2 ( X)$$

In this PDE, all functions and variables have a real value. $$x$$ has the meaning of a spatial coordinate and $$t$$ has sense of time. Exhibitors in parentheses indicate the order of differentiation. respective arguments. While the two functions of time $$g_1 (t)$$ and $$g_2 (t)$$ can at best be considered unspecified1, the two functions of space $$h_1 (x)$$ and $$h_2 (x)$$ are given by simple algebraic expressions:

• $$h_1 (x) = + x (x ^ gamma-1)$$
• $$h_2 (x) = – x (x-1)$$,

or $$gamma$$ is a real positive number. Since the two time-dependent functions are left undetermined, some information about the form (in space) from $$f (t, x)$$. Approximations, serial solutions or simply information on certain borderline cases would already be welcome.

Normally, when I solve particular nonlinear differential equations, I am able to discover a mathematical literature that deals with them. However, in this case, I am not sure of the term to look for. Therefore, I would also be grateful for any indication of sources of information that may help me to solve this problem on my own. In case this information is omnipresent and I could have found it, I apologize.

There are some constraints that $$f (t, x)$$ should adhere to:

• The domain of $$f (t, x)$$ is $$[0infty)[0infty)[0infty)[0infty)$$, $$(0,1) and his codomaine is [0infty)[0infty)[0infty)[0infty).$$
• $$f (t, 1) = 0$$
• $$lim_ {x downarrow 0} f (t, x) = + infty$$.
• $$f (t, x)$$ is a monotonous decreasing function of $$x$$.
• $$lim_ {x downarrow 0} f ^ {(0,1)} (t, x) = – infty$$
• $$lim_ {x uparrow 1} f ^ {(0,1)} (t, x) = – infty$$

In short, the inverse function of $$f (t, x)$$ w.r.t. his second argument might look like a Gaussian or a Lorentzian. In fact, the EDP considered here is derived from a much more horrible PDE, formulated according to this inverse, as well as other considerations.

1 They each depend on their own set of non-linear evolution equations for which, I am quite sure, have no solution or approximation. I think drawing these equations in the context of this question probably would not improve it.