Research Article
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Year 2018, Volume: 31 Issue: 3, 879 - 888, 01.09.2018

Abstract

References

  • 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).

On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels

Year 2018, Volume: 31 Issue: 3, 879 - 888, 01.09.2018

Abstract

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.

References

  • 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).
There are 19 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Amjad Alı This is me

Zeyad Min Ullah This is me

Marjan Uddın This is me

Publication Date September 1, 2018
Published in Issue Year 2018 Volume: 31 Issue: 3

Cite

APA Alı, A., Ullah, Z. M., & Uddın, M. (2018). On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels. Gazi University Journal of Science, 31(3), 879-888.
AMA Alı A, Ullah ZM, Uddın M. On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels. Gazi University Journal of Science. September 2018;31(3):879-888.
Chicago Alı, Amjad, Zeyad Min Ullah, and Marjan Uddın. “On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels”. Gazi University Journal of Science 31, no. 3 (September 2018): 879-88.
EndNote Alı A, Ullah ZM, Uddın M (September 1, 2018) On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels. Gazi University Journal of Science 31 3 879–888.
IEEE A. Alı, Z. M. Ullah, and M. Uddın, “On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels”, Gazi University Journal of Science, vol. 31, no. 3, pp. 879–888, 2018.
ISNAD Alı, Amjad et al. “On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels”. Gazi University Journal of Science 31/3 (September 2018), 879-888.
JAMA Alı A, Ullah ZM, Uddın M. On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels. Gazi University Journal of Science. 2018;31:879–888.
MLA Alı, Amjad et al. “On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels”. Gazi University Journal of Science, vol. 31, no. 3, 2018, pp. 879-88.
Vancouver Alı A, Ullah ZM, Uddın M. On the Approximation of Highly Oscillatory Integral Equations Via Radial Kernels. Gazi University Journal of Science. 2018;31(3):879-88.