DOI QR코드

DOI QR Code

COMBINED LAPLACE TRANSFORM WITH ANALYTICAL METHODS FOR SOLVING VOLTERRA INTEGRAL EQUATIONS WITH A CONVOLUTION KERNEL

  • AL-SAAR, FAWZIAH M. (RESEARCH SCHOLAR AT. DEPARTMENT OF MATHEMATICS, DR. BAM UNIVERSITY) ;
  • GHADLE, KIRTIWANT P. (DEPARTMENT OF MATHEMATICS, DR. BABASAHEB AMBEDKAR MARATHWADA UNIVERSITY)
  • Received : 2018.02.11
  • Accepted : 2018.06.12
  • Published : 2018.06.25

Abstract

In this article, a homotopy perturbation transform method (HPTM) and the Laplace transform combined with Taylor expansion method are presented for solving Volterra integral equations with a convolution kernel. The (HPTM) is innovative in Laplace transform algorithm and makes the calculation much simpler while in the Laplace transform and Taylor expansion method we first convert the integral equation to an algebraic equation using Laplace transform then we find its numerical inversion by power series. The numerical solution obtained by the proposed methods indicate that the approaches are easy computationally and its implementation very attractive. The methods are described and numerical examples are given to illustrate its accuracy and stability.

References

  1. J. Abdalkhani: A modied approach to the numerical solution of linear weakly singular Volterra integral equa- tions of the second kind, The Journal of Integral Equations and Applications, 2 (1993), 149-166.
  2. E. Babolian, A. Salimi Shamloo: Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions, Journal of Computational and Applied Mathematics, 214 (2008), 495-508. https://doi.org/10.1016/j.cam.2007.03.007
  3. S. Dixit, P. Singh, S. Kumar: A stable numerical inversion of generalized Abel's integral equation, Appl. Numer. Math. 62 (2012), 567-579 . https://doi.org/10.1016/j.apnum.2011.12.008
  4. A. Hamoud, K. Ghadle: The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro-differential equations, Journal of the Korean Society for Industrial and Applied Mathematics, 21 (1) (2017), 17-28.
  5. A. Hamoud, K. Ghadle: The reliable modified of Laplace Adomian decomposition method to solve nonlinear interval Volterra-Fredholm integral equations, Korean J. Math. 25 (2017), 323-334.
  6. A. Hamoud, K. Ghadle: On the numerical solution of nonlinear Volterra-Fredholm integral equations by variational iteration method, International Journal of Advanced Scientific and Technical Research, 3 (2016), 45-51.
  7. J. He: A coupling method of homotopy technique and perturbation technique for nonlinear problems, Int. J. Nonlinear Mech. 35 (2000), 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7
  8. J. He: Homotopy perturbation technique, Computer methods in applied mechanics and engineering, 178 (1999), 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3
  9. L. Huang, Y. Huang and X. Li: Approximate solution of Abel integral equation, Computers and Mathematics with Applications, 56 (7) ( 2008), 1748-1757. https://doi.org/10.1016/j.camwa.2008.04.003
  10. H. Jafari, C. Khalique, M. Nazari: Application of the Laplace decomposition method forsolving linear and nonlinear fractional diffusionwave equations, Appl. Math. Lett. 24 (2011), 1799-1805. https://doi.org/10.1016/j.aml.2011.04.037
  11. A. Kamyad, M. Mehrabinezhad, J. Saberi-Nadja: A numerical approach for solving linear and nonlinear Volterra integral equations with controlled error, International Journal of Applied Mathematics, 40 (2) (2010), 27-32.
  12. M. Khan, M. Hussain: Application of Laplace decomposition method on semiinnite domain, Numerical Algorithms, 56 (2011), 211-218. https://doi.org/10.1007/s11075-010-9382-0
  13. M. Khodabin, K. Maleknejad, F. Shekarabi: Application of triangular functions to numerical solution of stochastic Volterra integral equations, International Journal of Applied Mathematics, 43 (1) (2013), 1-9.
  14. S. Kumar: A numerical study for solution of time fractional nonlinear shallow-water equation in oceans, Zeitschrift fr Naturforschung, 68 (2013), 1-7. https://doi.org/10.5560/ZNC.2013.68c0001
  15. S. Kumar, A. Kumar: Analytical solution of Abel integral equation arising in astrophysics via Laplace trans- form, Journal of the Egyptian Mathematical Society, 23 (2015), 102-107. https://doi.org/10.1016/j.joems.2014.02.004
  16. S. Kumar, P. Singh: Numerical inversion of Abel integral equation using homotopy perturbation method, Zeitschrift fr Naturforschung, 65a (2010), 677-682.
  17. K. Maleknejad, A. Shamloo: Numerical solution of singular Volterra integral equations system of convolution type by using operational matrices, Applied Mathematics and Computation, 195 ( 2008), 500-505. https://doi.org/10.1016/j.amc.2007.05.001
  18. A. Orsi: Product integration for Volterra integral equations of the second kind with weakly singular kernels, Mathematics of Computation of the American Mathematical Society, 215 (1996), 1201-1212.
  19. H. Teriele: Collocation method for weakly singular second kind Volterra integral equations with non-smooth solution, IMA Journal of Numerical Analysis, 2 (4) (1982), 437-449. https://doi.org/10.1093/imanum/2.4.437
  20. C. Yang: Chebyshev polynomial solution of nonlinear integral equations, Journal of the Franklin Institute, 349 (3) (2012), 947-956. https://doi.org/10.1016/j.jfranklin.2011.10.023
  21. C. Yang, J. Hou: Numerical method for solving Volterra integral equations with a convolution kernel, IAENG International Journal of Applied Mathematics, 43 (4) (2013), 43-47.
  22. S. Yousefi: Numerical solution of Abel's integral equation by using Legendre wavelets, Applied Mathematics and Computation, 175 (1) (2006), 574-580. https://doi.org/10.1016/j.amc.2005.07.032