DOI QR코드

DOI QR Code

EXISTENCE AND APPROXIMATE SOLUTION FOR THE FRACTIONAL VOLTERRA FREDHOLM INTEGRO-DIFFERENTIAL EQUATION INVOLVING ς-HILFER FRACTIONAL DERIVATIVE

  • Awad T. Alabdala (Management Department - Universite Francaise d'Egypte) ;
  • Alan jalal abdulqader (Mathematical Department, College of Education, Al-Mustansiriyah University) ;
  • Saleh S. Redhwan (School of Mathematical Sciences, Zhejiang Normal University) ;
  • Tariq A. Aljaaidi (Department of Mathematics, Hajjah University)
  • Received : 2023.03.04
  • Accepted : 2023.04.17
  • Published : 2023.12.15

Abstract

In this paper, we are motivated to evaluate and investigate the necessary conditions for the fractional Volterra Fredholm integro-differential equation involving the ς-Hilfer fractional derivative. The given problem is converted into an equivalent fixed point problem by introducing an operator whose fixed points coincide with the solutions to the problem at hand. The existence and uniqueness results for the given problem are derived by applying Krasnoselskii and Banach fixed point theorems respectively. Furthermore, we investigate the convergence of approximated solutions to the same problem using the modified Adomian decomposition method. An example is provided to illustrate our findings.

Keywords

References

  1. A. T. Alabdala, B. N. Abood, S. S. Redhwan and S. Alkhatib, Caputo delayed fractional differential equations by Sadik transform, Nonlinear Funct. Anal. Appl, 28(2) (2023), 439-448.
  2. B.N. Abood, S.S. Redhwan and M.S. Abdo, Analytical and approximate solutions for generalized fractional quadratic integral equation, Nonlinear Funct. Anal. Appl, 26(3) (2021), 497-512.
  3. B.N. Abood, S.S. Redhwan, O. Bazighifan and K. Nonlaopon, Investigating a generalized fractional quadratic integral equation, Fractal and Fractional, 6(5)(2022), 251.
  4. G. Adomian, A review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135(2) (1988), 501-544. https://doi.org/10.1016/0022-247X(88)90170-9
  5. G. Adomian and R. Rach, Inversion of nonlinear stochastic operators, J. Math. Anal. Appl., 91(1) (1983), 39-46. https://doi.org/10.1016/0022-247X(83)90090-2
  6. G. Adomian and D. Sarafyan, Numerical solution of differential equations in the deterministie limit of stochastic theory, Appl. Math. Comput., 8 (1981), 111-119. https://doi.org/10.1016/0096-3003(81)90002-3
  7. R.P. Agarwal, M. Benchohra and S.A. Hamani, A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta Appl. Math., 109(3) (2010), 973-1033. https://doi.org/10.1007/s10440-008-9356-6
  8. A.H. Ahmed, M.S. Abdo and K.B. Ghadle. Existence and uniqueness results for Caputo fractional integro-differential equations, J. Kor. Soc. Indust. Appl. Math., 22(3) (2018), 163-177.
  9. M.A. Almalahi, O. Bazighifan, S.K. Panchal, S.S. Askar and G.I. Oros, Analytical study of two nonlinear coupled hybrid systems involving generalized Hilfer fractional operators, Fractal and Fractional, 5(4) (2021), 178.
  10. M.A. Almalahi, S.K. Panchal, K. Aldwoah and M. Lotayif, On the Explicit Solution of ψ-Hilfer Integro-Differential Nonlocal Cauchy Problem, Progr. Fract. Differ. Appl., 9(1) (2023), 65-77. https://doi.org/10.18576/pfda/090104
  11. S.Y. Al-Mayyahi, M.S. Abdo, S.S. Redhwan and B.N. Abood, Boundary value problems for a coupled system of Hadamard-type fractional differential equations, IAENG Int. J. Appl. Math., 51(1) (2021), 1-10.
  12. R. Almeida, A Gronwall inequality for a general Caputo fractional operator, https://doi.org/10.48550/arXiv.1705.10079, 2017.
  13. U. Arshad, M. Sultana, A.H. Ali, O. Bazighifan, A.A. Al-moneef and K. Nonlaopon, Numerical Solutions of Fractional-Order Electrical RLC Circuit Equations via Three Numerical Techniques, Mathematics, 10(17) (2022), 3071.
  14. I. M. Batiha, N. Alamarat, S. Alshorm, O. Y. Ababneh and S. Moman, Semi-analytical solution to a coupled linear incommensurate system of fractional differential equations, Nonlinear Funct. Anal. Appl, 28(2) (2023), 449-471. https://doi.org/10.1109/ICIT58056.2023.10225807
  15. D.V. Bayram and A. Dascpmoglu, A method for fractional Volterra integro-differential equations by Laguerre polynomials, Adv. Differ. Equ., 2018(1) (2018), 466.
  16. C. da Vanterler, J. Sousa and E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. https://doi.org/10.1016/j.cnsns.2018.01.005
  17. P. Das, S. Rana and H. Ramos, Homotopy perturbation method for solving Caputo-type fractional order Volterra-Fredholm integro-differential equations, Comput. Math. Meth., 1(5) (2019), e1047.
  18. P. Das, S. Rana and H. Ramos, A perturbation-based approach for solving fractional-order Volterra-Fredholm integro differential equations and its convergence analysis, Int. J. Comput. Math., (2019), 1-21.
  19. J.S. Duan, R. Rach, D. Baleanu and A.M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Frac. Calc., 3(2) (2012), 73-99.
  20. K.M. Furati, M.D. Kassim and N.E. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64 (2012), 1616-1626. https://doi.org/10.1016/j.camwa.2012.01.009
  21. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  22. R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys., 284 (2002), 399-408. https://doi.org/10.1016/S0301-0104(02)00670-5
  23. H.N.A. Ismail, I.K. Youssef and T.M. Rageh, Modification on Adomian decomposition method for solving fractional Riccati differential equation, Int. Adv. Research J. Sci. Eng. Tec., 4(12) (2017), 1-11.
  24. M.B. Jeelani, A.S. Alnahdi, M.A. Almalahi, M.S. Abdo, H.A. Wahash and N.H. Alharthi, Qualitative Analyses of Fractional Integrodifferential Equations with a Variable Order under the Mittag-Leffler Power Law, J. Funct. Spaces, 2022.
  25. A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, 204, Elsevier Science, Amsterdam, 2006.
  26. N. Limpanukorn, P. Sa Ngiamsunthorn, D. Songsanga and A. Suechoei, On the stability of differential systems involving ψ-Hilfer fractional derivative, Nonlinear Funct. Anal. Appl, 27(3) (2022), 513-532.
  27. R. Mittal and R. Nigam, Solution of fractional integro-differential equations by Adomian decomposition method, Int. J. Appl. Math. Mech., 4(2) (2008), 87-94.
  28. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego 1999.
  29. R. Rach, On the Adomian decomposition method and comparisons with Picard's method, J. Math. Anal. Appl., 128(2) (1987), 480-483. https://doi.org/10.1016/0022-247X(87)90199-5
  30. S. Redhwan and S.L. Shaikh, Implicit fractional differential equation with nonlocal integral-multipoint boundary conditions in the frame of Hilfer fractional derivative, J. Math. Anal. Mode., 2(1) (2021), 62-71. https://doi.org/10.48185/jmam.v2i1.176
  31. S.S. Redhwan, S.L. Shaikh, M.S. Abdo, W. Shatanawi, K. Abodayeh, M.A. Almalahi and T. Aljaaidi, Investigating a generalized Hilfer-type fractional differential equation with two-point and integral boundary conditions, AIMS Mathematics, 7(2) (2022), 1856-1872. https://doi.org/10.3934/math.2022107
  32. S.S. Redhwan, A.M. Suad, S. Shaikh and A. Mohammed, A coupled non-separated system of Hadamard-type fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 6(1) (2021), 33-44. https://doi.org/10.31197/atnaa.925365
  33. S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Switzerland, 1993.
  34. D.R. Smart, Fixed Point Theorems, Cambridge Univ. Press 66, 1980.
  35. J.V.D.C. Sousa and E.C. de Oliveira, A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator, (2017), https://doi.org/10.48550/arXiv.1709.03634.
  36. J. Vanterler da C. Sousa and E. Capelas de Oliveira, On the ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72-91. https://doi.org/10.1016/j.cnsns.2018.01.005
  37. A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput., 102(1) (1999), 77-86. https://doi.org/10.1016/S0096-3003(98)10024-3
  38. E.A.A. Ziada, Solution of coupled system of Cauchy problem of nonlocal differential equations, Electronic J. Math. Anal. Appl., 8(2) (2020), 220-230. https://doi.org/10.21608/ejmaa.2020.312851