Usage of numerical methods to solve nonlinear mixed Volterra-Fredholm integral equations and their system

Yıl 2021, Cilt: 4 Sayı: 4, 244 - 262, 31.12.2021

Fawziah Al-saar Kirtiwant Ghadle

https://doi.org/10.53006/rna.988774

Cited By: 1

Öz

In this paper, we apply the homotopy perturbation method (HPM), modified homotopy perturbation method
(MHPM), variational iteration method (VIM), Adomian decomposition method (ADM), and modified Adomian decomposition method (MADM) to solve nonlinear mixed Volterra-Fredholm integral equations and its
system. We investigate the approximate solution of this equation and its system via proposed methods. The
validity and efficiency of these methods are demonstrated through various numerical examples that illustrate
the efficiency, accuracy, and simplicity of the proposed methods. Moreover, the convergence and uniqueness
of the solution of the suggested methods are confirmed and compared with the exact solutions.

Anahtar Kelimeler

Nonlinear mixed Volterra-Fredholm integral equations, ADM, MADM, VIM, HPM, MHPM.

Kaynakça

• [1] M.S. Abdo, and S.K. Panchal, Some new uniqueness results of solutions to nonlinear fractional integro-differential equations. APAM, 2018, 16, 345-352.
• [2] M.S. Abdo, A.M. Saeed, H.A. Wahash and S.K. Panchal, On nonlocal problems for fractional integro-differential equation in Banach space, Eur. J. Sci. Res. 2019, 151, 320-334.
• [3] Q.M. Al-Mdallal, Monotone iterative sequences for nonlinear integro-differential equations of second order, Nonlinear Anal. Real World Appl. 2011, 12, 3665-3673.
• [4] F.M. Al-Saar, K.P. Ghadle, and P.A. Pathade, The approximate solutions of Fredholm integral equations by Adomian decomposition method and its modi?cation, Int. J. Math. Appl. 2018, 6, 327-336.
• [5] F.M. Al-Saar and K.P. Ghadle, An approximate solution for solving the system of Fredholm integral equations of the second kind, Bull. Pure Appl. Sci. Math. 2019, 1, 208-215.
• [6] M. Asif, I. Khan, N. Haider, Q. Al-Mdallal, Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations, Alex. Eng. J. 2020, 59, 5099-5109.
• [7] E. Babolian and J. Biazar, Solution of a system of linear Volterra equations by Adomian decomposition method, Far East J. Math. Sci. 2002, 2, 935-945.
• [8] S.S. Behzadi, The use of iterative methods to solve two-dimensional nonlinear Volterra-Fredholm integro-differential equa- tions, Commun. Math. Anal. 2012, 2012, 1-20.
• [9] J. Biazar, B. Ghanbari, M. Porshokouhi and M. Porshokouhi, He's homotopy perturbation method: a strongly promising method for solving non-linear systems of the mixed Volterra-Fredholm integral equations, Comput. Math. with Appl. 2011, 61, 1016-1023.
• [10] C. Dong, Z. Chen and W. Jiang, A modified homotopy perturbation method for solving the nonlinear mixed Volterra- Fredholm integral equation, J. Comput. Appl. Math. 2013, 239, 359-366.
• [11] A.A. Hamoud and K.P. Ghadle, Modified Adomian decomposition method for solving fuzzy Volterra-Fredholm integral equations, J. Indian Math. Soc. 2018, 85, 53-69.
• [12] A.A. Hamoud and K.P. Ghadle, Recent advances on reliable methods for solving Volterra-Fredholm integral and integro- di?erential equations, Asian J. Math. Comput. Res. (2018), 24, 128-157.
• [13] A.A. Hamoud and K.P. Ghadle, The combined modified Laplace with Adomian decomposition method for solving the nonlinear Volterra-Fredholm integro-differential equations, J. Korean Soc. Ind. Appl. Math. (2017),21, 17-28.
• [14] A.A. Hamoud and K.P. Ghadle, The approximate solutions of fractional Volterra-Fredholm integro-differential equations by using analytical techniques, Probl. Anal. Issues Anal. (2018),25, 41-58.
• [15] J.H. He, Homotopy perturbation technique, Comput. Methods Appl. Mech. Engrg. (1999),178, 257-262.
• [16] F.A. Hendi, and M.M. Al-Qarni, Numerical treatment of nonlinear Volterra-Fredholm integral equation with a generalized singular kernel, Am. J. Comput. Math. (2016), 6, 245-250.
• [17] F.A. Hendia and M.M. Al-Qarnib, Numerical solution of nonlinear mixed integral equations with singular Volterra kernel, Int. J. Adv. Appl. Math. and Mech. 2016, 3, 41-48.
• [18] I. Khan, M. Asif, Q. Al-Mdallal, F. Jarad, On a new method for finding numerical solutions to integro-differential equations based on Legendre multi-wavelets collocation, Alex. Eng. J. 2021, 1-13.
• [19] K. Maleknejad, K. Nouri and L. Torkzadeh, Comparison projection method with Adomian's decomposition method for solving system of integral equations, Bull. Malays. Math. Sci. Soc. (2011),34, 379-388.
• [20] F. Mirzaee and A.A. Hoseini, Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of block- pulse functions and Taylor series, Alex. Eng. J. 2013, 52, 551-555.
• [21] M.H. Saleh, D.S. Mohamed, and R.A. Taher, Variational Iteration Method for Solving Two Dimensional Volterra-Fredholm Nonlinear Integral Equations, Int. J. Comput. Appl. 2016, 152, 29-33.
• [22] N.A. Sulaiman, Some numerical methods to solve a system of Fredholm integral equations of the 2nd kind with symmetric kernel, Kirkuk Univ. J. Sci. Stud. (2009), 4, 108-116.
• [23] R.K. Pandey, O.P. Singh and V.K. Singh, Efifcient algorithms to solve singular integral equations of Abel type, Comput. Math. with Appl. (2009), 57, 664-676.
• [24] M. Rabbani and R. Jamali, Solving nonlinear system of mixed Volterra-Fredholm integral equations by using variational iteration method, J. Math. Comput. Sci. 2012, 5, 280-287.
• [25] H.A. Wahash, M.S. Abdo and S.K. Panchal, An existence result for fractional integro-differential equations on Banach space, J. Math. Ext. 2019, 13, 19-33.
• [26] H.A. Wahash, M.S. Abdo, A.M. Saeed and S.K. Panchal, Singular fractional differential equations with ψ -Caputo operator and modi?ed Picard's iterative method, Appl. Math. E-Notes, 2020, 20, 215-229.
• [27] A.M. Wazwaz, A reliable treatment for mixed Volterra-Fredholm integral equations, Appl. Math. and Comput. 2002, 127, 405-414.
• [28] S.A. Yousefi, A. Lotfi and M. Dehghan, He's variational iteration method for solving nonlinear mixed Volterra-Fredholm integral equations, Comput. Math. with Appl. 2009, 58, 2172-2176.