Approximate Numerical Solution of Singular Integrals and Singular Initial Value Problems

Numerical integration is one of the important branch of mathematics. Singular integrals arises in different applications in applied and engineering mathematics. The evaluation of singular integrals is one of the most challenging jobs. Earlier different techniques were developed for evaluating such integrals, but these were not straightforward. Recently various order straightforward formulae have been developed for evaluating such integrals but; all these integral formulae depend on Romberg technique for more accurate results. Based on these integral formulae, different order (up to fifth) implicit methods have been developed for solving singular initial value problems. These implicit methods give better results than those obtained by implicit Runge-Kutta methods but; the derivation of such higher order formulae are not so easy. In this article, a new third order straightforward integral formula has been proposed for evaluating singular integrals. This new formula is able to evaluate more efficiently than others existing formulae, moreover it has the independent ability to calculate very near accurate result to the exact value of the numerical integrals. Based on this new integral formula a new third order implicit method has been proposed for solving singular initial value problems. The new method provides significantly better results than other existing methods.