depends on the nature of the problem. "Exact" solution is only possible analytically. Numerical solutions are only approximate!
You can try Legendre multiwavelet to get the approximate solution which is same as the exact solution for some problem.. I also get the exact solution of singular integral equation, you can read my paper .
http://www.sciencedirect.com/science/article/pii/S0377042715006342
Vikash: I know that all this method gives the approximate solution, but some of these methods gives the exact solution after some steps
dear Dr Swaraj please send me your paper
http://www.sciencedirect.com/science/article/pii/S0377042715006342
The answer depending on the kind of integral equation. If the integral equation of the second kind and a linear case , we can use Picard method or Banach theorem. Picard method fails in the IE of the first kind. In the nonlinear case we can use (Picard method, Banach theorem and Schouderd theorem. Then , there exist many analytic and numerical method can be used to obtain the unique solution or at least one solution.
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