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Original Articles

A perturbation-based approach for solving fractional-order Volterra–Fredholm integro differential equations and its convergence analysis

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Pages 1994-2014 | Received 04 Jun 2019, Accepted 23 Sep 2019, Published online: 09 Oct 2019

References

  • M.A. AL-Jawary, M.M. Azeez and G.H. Radhi, Analytical and numerical solutions for the nonlinear Burgers and advection–diffusion equations by using a semi-analytical iterative method, Comput. Math. Appl. 76 (2018), pp. 155–171. doi: 10.1016/j.camwa.2018.04.010
  • A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Soliton Fract. 40 (2009), pp. 521–529. doi: 10.1016/j.chaos.2007.08.001
  • K. Assaleh and W.M. Ahmad, Modeling of speech signals using fractional calculus, 9th International Symposium on Signal Processing and Its Applications, 2007.
  • R.L. Bagley and P.J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol. 27 (1983), pp. 201–210. doi: 10.1122/1.549724
  • D. Baleanu, K. Diethelm, E. Scalas and J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, New Jersey, 2016.
  • M. Chandru, P. Das and H. Ramos, Numerical treatment of two-parameter singularly perturbed parabolic convection diffusion problems with non-smooth data, Math. Methods Appl. Sci. 41 (2018), pp. 5359–5387. doi: 10.1002/mma.5067
  • M. Chandru, T. Prabha, P. Das and V. Shanthi, A numerical method for solving boundary and interior layers dominated parabolic problems with discontinuous convection coefficient and source terms, Differ. Equ. Dyn. Syst. 27(1) (2019), pp. 91–112. doi: 10.1007/s12591-017-0385-3
  • P. Das, Comparison of a priori and a posteriori meshes for singularly perturbed nonlinear parameterized problems, J. Comput. Appl. Math. 290 (2015), pp. 16–25. doi: 10.1016/j.cam.2015.04.034
  • P. Das, A higher order difference method for singularly perturbed parabolic partial differential equations, J. Differ. Equ. Appl. 24(3) (2018), pp. 452–477. doi: 10.1080/10236198.2017.1420792
  • P. Das, An a posteriori based convergence analysis for a nonlinear singularly perturbed system of delay differential equations on an adaptive mesh, Numer. Algorithms 81 (2019), pp. 465–487. doi: 10.1007/s11075-018-0557-4
  • P. Das and V. Mehrmann, Numerical solution of singularly perturbed convection–diffusion–reaction problems with two small parameters, BIT 56 (2016), pp. 51–76. doi: 10.1007/s10543-015-0559-8
  • P. Das and J. Vigo-Aguiar, Parameter uniform optimal order numerical approximation of a class of singularly perturbed system of reaction diffusion problems involving a small perturbation parameter, J. Comput. Appl. Math. 354 (2019), pp. 533–544. doi: 10.1016/j.cam.2017.11.026
  • C. Dong, Z. Chen and W. Jiang, A modified homotopy perturbation method for solving the non-linear mixed Volterra–Fredholm integral equation, J. Comput. Appl. Math. 239 (2013), pp. 359–366. doi: 10.1016/j.cam.2012.09.003
  • A. Ghorbani, Beyond adomian polynomials: He polynomials, Chaos Soliton Fract. 39 (2009), pp. 1486–1492. doi: 10.1016/j.chaos.2007.06.034
  • R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Springer-Verlag, Berlin, 1991.
  • A.A. Hamoud, K.P. Ghadle, M.Sh. Bani Issa and G. Giniswamy, Existence and uniqueness theorems for fractional Volterra–Fredholm integro-differential equations, Int. J. Appl. Math. 31 (2018), pp. 333–348. doi: 10.12732/ijam.v31i3.3
  • S. Kheybari, M.T. Darvishi and A.M. Wazwaz, A semi-analytical approach to solve integro-differential equations, J. Comput. Appl. Math. 317 (2017), pp. 17–30. doi: 10.1016/j.cam.2016.11.011
  • S.A. Khuri and A. Sayfy, Generalizing the variational iteration method for BVPs: Proper setting of the correction functional, Appl. Math. Lett. 68 (2017), pp. 68–75. doi: 10.1016/j.aml.2016.11.018
  • A.A. Kilbas, H.M. Srivastav and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier B.V., Amsterdam, 2006.
  • K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley-Interscience, New York, 1993.
  • D. Nazari and S. Shahmorad, Application of the fractional differential transform method to fractional-order integro-differential equations with nonlocal boundary conditions, J. Comput. Appl. Math. 234 (2010), pp. 883–891. doi: 10.1016/j.cam.2010.01.053
  • A. Prakash, M. Kumar and K.K. Sharma, Numerical method for solving fractional coupled Burgers equations, Appl. Math. Comput. 260 (2015), pp. 314–220.
  • H. Ramos and J. Vigo-Aguiar, A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems, Int. J. Comput. Math. 85 (2008), pp. 603–611. doi: 10.1080/00207160701199773
  • P.K. Sahu and S. Saha Ray, A novel Legendre wavelet Petrov Galerkin method for fractional Volterra integro-differential equations, Comput. Math. Appl. 290 (2016), pp. 1–5.
  • F.M. Scudo, Vito Volterra and theoretical ecology, Theor. Popul. Biol. 2 (1971), pp. 1–23. doi: 10.1016/0040-5809(71)90002-5
  • R.D. Small, Population growth in a closed system, SIAM Rev. 25 (1983), pp. 93–95. doi: 10.1137/1025005

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