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

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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.

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Can't? Maybe there can be but we are too stupid at the moment. –  Jacob Wakem Mar 19 at 0:42
    
Well perhaps but you would have to build some monolith of a machine with which to analyze that it might not even be useful for practical purposes given the extremely varying behavior of integral operators. –  Cameron Williams Mar 19 at 0:43
    
@CameronWilliams just curious. How do you justify $\lambda f(t) = \int_0^{\infty} e^{-st}f(s)ds$? Why $f$ then the Laplace transform of $f$ is in the same space. Shouldn't we view one as a distribution? –  tst Mar 19 at 3:18
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Integral equations are often harder to solve by nature.

They may be harder to solve than ordinary differential equations, but the questioner asked about partial differential equations. I'm not aware of any general theory for partial differential equations. Yes, there are theories for elliptic equations, for parabolic equations, and for systems of hyperbolic conservation laws. Yes, there are also theories for many other types of partial differential equations, but the theory is there because the equations arise in certain applications, not because anybody would try to develop a general theory of partial differential equations.

What we do have for partial differential equations is a flow of information. If we take a finite volume bounded by a sufficiently smooth surface, then we can uniquely determine the solution inside the finite volume by prescribing "enough" information on the surface. Sadly, we often can't avoid prescribing "too much" information on the surface when we do this, so that this sort of procedure often would lead to ill-posed problems.


Integral equations may really be underused. We don't need to go to partial differential equations to run into troublesome differential equations. Already coupled systems of ordinary algebraic equations and ordinary differential equations (called differential algebraic equations or DAE) have their own difficulties (like nonholonomic systems) preventing a smooth theory. Some of the solution techniques like dummy variables used by the Pantelides algorithm would be much more natural for an integral equation formulation. Actually I really do think that the standard presentation of a DAE as $F(\dot{x}(t),x(t),t)=0$ is misguided, and that a presentation as $G(x(t),t)=\int_{t_0}^tds \; F(x(s),s)$ would make much more sense. An ordinary algebraic equation $g(x,t)=0$ would then naturally fit into the theory in the form $g(x,t)=\int_{t_0}^t0$, and subtle distinctions between the linear and the non-linear theory would just turn into normal regularity conditions (i.e. regular points are characterized by rank conditions for certain matrices).

However, whether integral equations are really underused is not a question of how difficult they are to solve, but how often they would arise in actual applications, if we had the courage to admit them. I don't know the answer for that question.

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