Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the nonavailability of methods to find the exact solutions and hence lot of approximation methods were referred. I wanted to know that how effective these methods are while resolving problems and is it really any requirement that exact solution methods will be of real need?
Kendall Atkinson has done a big work in numerical solutions of integral equations. As with almost all other kinds of equations, the explicit solution of the integral equation is rarely known. On the other hand, you shall ask yourself what is meant by the explicit solution. Usually, it means that the solution function $f$ of some integral equation $\mathcal I[f] = 0$ can be expressed via the finite number of algebraic operations involving some "known" functions, e.g. polynomials, exponentials and trigonometric functions. So that if you know that the solution is $f(x) = \sin x$, would you declare that it is explicit? You can barely compute the explicit value of $\sin x$ even if $x = 1$. Moreover, if you allow for a more general class of functions to appear in the expression of $f$, such as $\Gamma$function or Bessel functions, perhaps you can solve explicitly a bigger class of equations. Finally, if you know that for some integral operator $\mathcal I$ it holds that $$ \mathcal I[f] = 0 \tag{1} $$ has a unique solution $\hat f$, the equation $(1)$ by itself serves as an unambiguous definition of the function $\hat f$. You can see now, that an explicit solvability of $(1)$ is quite a relative notion and it depends, in which class of functions are you looking for the expression of solution. In the end, to answer your question about the real need for exact solutions, let me mention that it hinges upon the problem in hand.
I hope, it helps  though I am not sure whether this is the best place to ask such questions, as it may be not considered as a researchlevel question. Better try MSE 

