A nonlinear integro-differential equation in 2D
I hope to find a closed-form solution for the following equation:
(I believe it's properly classified as 'homogenous nonlinear Fredholm integro-differential equation in two dimentions')
where is a known piecewise polinomial function (spline) of degree 3, and is a known even function. I know it might sound an ill-posed problem, but it would suffice if there was some sufficiently broad class of functions for for which the thing could be solvable. Cubic splines, again, would be ideal! I tried to substitute in a parametrized polinomial and into the equation and got a sum of sums of integrals of powers of ,x and . But I don't know what to do next...
Some books on Integral equations treat this sort of problems, but I've got the nasty partial derivative on the left side. I could find some papers on integro-differential equations of this kind, but they seem to deal with some special cases and in 1 dimension, and frankly too obsucre for my level of expertise (recent graduate in applied maths, rudimentary knowledge of mathematical physics)
I have reasons to prefer analytical solution, but, of course, numeric methods would be my last resort.
I would appreciate ANY form of help.
Thank you for your attention!
In case you care where does this problem originate from:
I came up with it doing research in image processing (to keep up with the field's notorious habbit of applying mathematical apparatus to it's problems in precarious ways). Basically, it is sort of a continuous analog of the n-body problem of physics, except that I unceremoniously replaced acceleration with velocity in the left part. It can be interpreted as equation describing translocation ( - is position of 'point' that was at postion at the moment ) of continuous matter on a line, governed by gravitation-like mutual repulsion of it's particles ( is the 'law', would be in Newton's gravitation), where is the initial density distribution. The equation was transformed from it's original formulation using the interrelation between and , to eliminate the last one.
That all might sound lunatic, but I actually constructed a programm to solve it through naive simulation, and it prooved to yield the result I expected.