Amjad ALI, Zeyad Min ULLAH, Marjan UDDIN

On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels

In this work we used radial kernels for computing more generalized fast oscillatory integral equations. The proposed method is based on radial kernels. The present method is efficient for computing oscillatory integral equations with large oscillation parameters. The proposed method is very robust and capable of handling fast oscillatory integral equations.

Keywords:

Oscillatory integral equations, Interpolation Scheme, Radial kernels,

PDF

___

Brunner. H., “Collocation Methods for Volterra Integral and Related Functional Equations”, Cambridge University Press, Cambridge, (2004).
Davies. P. J., Duncan. D. B., “Stability and convergence of collocation schemes for retarded potential integral equations”, SIAM journal on numerical analysis, 42(3): 1167-1188, (2004).
De Hoog. F., Weiss. R., “On the solution of Volterra integral equations of the first kind”, Numerische Mathematik, 21(1): 22-32, (1973).
Brunner. H., Iserles. A., Norsett. S., “Open problems in the computational solution of Volterra functional equations with highly oscillatory kernels”, Isaac Newton Institute, HOP 2007, (2007).
Iserles. A., Norsett. S. P, “On quadrature methods for highly oscillatory integrals and their implementation”, BIT Numerical Mathematics, 44(4): 755-772, (2004).
Linz. P, “Product integration methods for Volterra integral equations of the first kind”, BIT Numerical Mathematics, 11(4): 413-421, (1971).
Levin. D, “Fast integration of rapidly oscillatory functions”, Journal of Computational and Applied Mathematics, 67(1): 95-101, (1996).
Levin. D, “Procedures for computing one- and two-dimensional integrals of functions with rapid irregular oscillations”, Math. Comput., 38: 531-538, (1982).
Longman. I. M., “A method for numerical evaluation of finite integrals of oscillatory functions”, Mathematics of Computation, 14(69): 53-59, (1960).
Polyanin.A. D., Manzhirov. A. V., “Handbook of Integral Equations”, CRC Press, (2008).
Chen. R., “Numerical approximation for highly oscillatory {Bessel} transform and applications”, Journal of Mathematical Analysis and Applications, 421(2): 1635-1650, (2015).
Fasshauer. G., Mc Court. M., “Kernel-based approximation methods using Matlab”, World Scientific Publishing Co Inc, (2015).
Sarler. B., Vertnik. R., “Meshfree explicit local radial basis function collocation method for diffusion problems”, Computers and mathematics with applications, 51(8): 1269-1282, (2006).
Uddin. M, Minullah. Z., Ali. A, “On the local kernel based approximation of highly oscillatory integrals”, Miskolc Mathematical Notes, 16(2): 1253-1264, (2015).
Wang. H., Xiang. S, “Asymptotic expansion and Filon-type methods for a Volterra integral equation with a highly oscillatory kernel”, IMA Journal of Numerical Analysis, 31(2): 469-490, (2010).
Piessens. R., Branders. M., “Modified Clenshaw-Curtis method for the computation of {Bessel} functions integralsl”, BIT Numerical Mathematics, 23(3): 370-381, (1983).
Piessens. R., Branders. M., “A survey on numerical methods for the computation of Bessel functions integrals”, Rend. Sem. Mat. Univ. Politec. Torino Fascicolo Speciale. Special Functions: Theory and Computation, 249: 265, (1985).
Puoskari. M., “A method for computing {Bessel} functions integrals”, Journal of Computational Physics, 75(2): 334-344, (1988).
Uddin. M., Shah. I. A., Ali. H., “On the numerical solution of evolution equation via soliton kernels”, Gazi University Journel of Science, 28(4): 631-637, (2015).