Efficient computation of some highly oscillatory improper integrals arising in natural and applied sciences

Highly oscillatory integrals with logarithmic or algebraic singularity are required in the solutions of problems arising in important fields such as optics, electromagnetic, electrodynamics, quantum mechanics, image analysis, computed tomography, fluid mechanics. In this thesis, a Gaussian integration method has been developed based on a modification of the Steepest Descent Method, which is used to compute efficiently and economically such integrals. In this context, in this method, coefficients in the three-term recurrance relations of orthogonal polynomials with respect to some non-standard weight functions are used. The obtained method has been applied to some selected integrals. Highly oscillatory singular integrals also occur in solutions of some partial differential equations encountered in practice. For example, the solution of some boundary value problems can be expressed by oscillatory and singular integrals. In this study, these boundary value problems have been solved and highly oscillatory singular integrals in these solutions have been computed efficiently. Some examples are given to demonstrate the effectiveness of the presented methods. The results obtained from these examples show that the proposed methods are effeciently and very fast and economical in terms of computation time.