TITLE: Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?
QUESTION [50 upvotes]: Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?

REPLY [18 votes]: 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.