Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?

Answer

Integral equations are often harder to solve by nature.. Every differential equation can be recast as an integral equation (or iterated integral equation) but the converse does not hold. Unfortunately, techniques for solving integral equations are somewhat few and far between with no real overarching paradigm. I am aware of texts (e.g. Handbook of Integral Equations) that attempt such things but the approaches are much less clean, much more restrictive and often on a case-by-case basis. Really the issue is kernels can be as wild as we want them and for that reason, integral equations can be very difficult to deal with. The Fourier transform and similar integral transforms are nice because their kernels have some very desirable properties but once you leave the realm of linear or multiplicative kernels, quite frankly, all hell breaks loose.

I should add that this lack of a cohesive paradigm for analyzing integral transforms and equations has attracted my interest and I’ve done quite a bit of study on them for research purposes.

Edit:

For example, how would you go about finding solutions to

\lambda f(t) = \int_0^{\infty} e^{-st}f(s)ds

in a straightforward manner? You would first need to examine which values of \lambda are even eigenvalues (somehow!) and then find solutions from there. One solution (and the only one I am aware of) is f(t) = \frac{1}{\sqrt{t}}.

With differential equations, we have the full brunt of Sturm-Liouville theory at our disposal but this in no way can apply to integral transforms. Sturm-Liouville operators are compact (if I recall correctly) and there is quite a bit of nice theory about compact operators on Hilbert space. However general integral operators are not necessarily compact so we can’t use the machinery from compact operators. An example that is not is the Fourier transform (and a broader class of integral operators I am working on). The nature of your integral equation is extremely dependent upon the nature of your kernel and boundary conditions and there is no one technique that works for the broad spectrum (hah) of integral equations since the operators are of extremely different forms. It would be like trying to use compactness arguments to sets that aren’t compact! Hell, one of the more popular results (well, amongst those of us who study integral equations anyway..) is Schur’s test but even that only works under very restrictive conditions. The moral of the story is that integral operators have extremely varying behavior so there can’t be an overarching theory that gives meaningful results for the whole class of integral equations. I hope this answers your question.

Attribution
Source : Link , Question Author : Victor , Answer Author : Cameron Williams

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