Main Article Content
In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model.
Mathematical Subject Classification (2010): 45P05, 45G10, 47H99
Received 19/12/2019, Accepted 27/4/2020, Published Online First 11/1/2021
This work is licensed under a Creative Commons Attribution 4.0 International License.
Maleknejad K, Saeedipoor E. Convergence analysis of hybrid functions method for two-dimensional nonlinear Volterra–Fredholm integral equations. J Comput Appl Math. 2019 Oct 28:112533.
Sen M, Saha D, Agarwal RP. A Darbo fixed point theory approach towards the existence of a functional integral equation in a Banach algebra. Appl Math Comput . 2019 Oct 1; 358:111-8.
Saffarzadeh M, Loghmani GB, Heydari M. An iterative technique for the numerical solution of nonlinear stochastic Itô–Volterra integral equations. J Comput Appl Math. 2018 May 1;333:74-86.
Mirzaee F, Samadyar N. Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra–Fredholm integral equations. Trans. A. Razmadze Math. Inst.. 2018 Dec 1;172(3):631-41.
Kazemi M, Ezzati R. Existence of solution for some nonlinear two-dimensional Volterra integral equations via measures of noncompactness. Appl Math Comput. 2016 Feb 15; 275:165-71.
Małolepszy T. A new upper estimation for the blow-up time of solutions of a Volterra integral equation and its application to the modeling of the formation of shear bands. J Math Anal Appl. 2017 Apr 15;448(2):786-96.
Almasieh H, Meleh JN. Numerical solution of a class of mixed two-dimensional nonlinear Volterra–Fredholm integral equations using multiquadric radial basis functions. J Comput Appl Math. 2014 Apr 1; 260:173-9.
Maleknejad K, Khodabin M, Rostami M. A numerical method for solving m-dimensional stochastic Itô–Volterra integral equations by stochastic operational matrix. Comput Math Appl. 2012 Jan 1; 63(1):133-43.
Khodabin M, Maleknejad K, Rostami M, Nouri M. Numerical approach for solving stochastic Volterra–Fredholm integral equations by stochastic operational matrix. Comput Math Appl. 2012 Sep 1; 64(6):1903-13.
Maleknejad K, JafariBehbahani Z. Applications of two-dimensional triangular functions for solving nonlinear class of mixed Volterra–Fredholm integral equations. Math Comput Model. 2012 Mar 1; 55(5-6):1833-44.
Ioannou Y, Fyrillas MM, Doumanidis C. Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer. Eng Anal Bound Elem. 2012 Aug 1; 36(8):1278-83.
Feng WZ, Li HY, Gao LF, Qian W, Yang K. Hypersingular flux interface integral equation for multi-medium heat transfer analysis. Int J Heat Mass Tran. 2019 Aug 1; 138: 852-65.
Wei T, Xu M. An integral equation approach to the unsteady convection–diffusion equations. Appl Math Comput. 2016 Feb 1; 274: 55-64.
Hong YB, Wu CY. Integral equation solutions using radial basis functions for radiative heat transfer in higher-dimensional refractive media. Int J Heat Mass Tran. 2018 Mar 1; 118: 1180-9.
Maleknejad K, Shahabi M. Application of hybrid functions operational matrices in the numerical solution of two-dimensional nonlinear integral equations. Appl Numer Math. 2019 Feb 1; 136: 46-65.
March NH, Nagy Á. Formally exact integral equation theory of the exchange-only potential in density functional theory: Refined closure approximation. Phys Lett A. 2006 Jan 2; 348(3-6):374-8.
Mirzaee F, Hadadiyan E, Bimesl S. Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via three-dimensional block-pulse functions. Appl Math Comput. 2014 Jun 15; 237: 168-75.
Mirzaee F, Hadadiyan E. Applying the modified block-pulse functions to solve the three-dimensional Volterra–Fredholm integral equations. Appl Math Comput. 2015 Aug 15; 265: 759-67.
Mirzaee F, Hadadiyan E. Three-dimensional triangular functions and their applications for solving nonlinear mixed Volterra–Fredholm integral equations. ALEX ENG J. 2016 Sep 1; 55(3): 2943-52.
Maleknejad K, Rashidinia J, Eftekhari T. Numerical solution of three-dimensional Volterra–Fredholm integral equations of the first and second kinds based on Bernstein’s approximation. Appl Math Comput. 2018 Dec 15; 339: 272-85.
Hameed HH, Eshkuvatov ZK, Muminov Z, Kilicman A. Solving system of nonlinear integral equations by Newton-Kantorovich method. InAIP Conference Proceedings 2014 Jul 10 (Vol. 1605, No. 1, pp. 518-523). AIP.
Eshkuvatov ZK, Hameed HH, Taib BM, Long NN. General 2× 2 system of nonlinear integral equations and its approximate solution. J Comput Appl Math. 2019 Dec 1; 361: 528-46.
Eshkuvatov ZK, Hameed HH, Long NN. One dimensional nonlinear integral operator with Newton–Kantorovich method. J King Saud Univ.y-Sci.. 2016 Apr 1; 28(2):172-7.
Hameed HH, Eshkuvatov ZK, Long NN. an approximate solution of two dimensional nonlinear volterra integral equation using newton-kantorovich method. Malaysian J. Sci. 2016 Jun 30; 35(1):37-43.
Hameed HH, Eshkuvatov ZK, Long NM. On the Solution of Multi-Dimensional Nonlinear Integral Equation with Modified Newton Method. J Comput Theor Nanos. 2017 Nov 1; 14(11): 5298-303.
Ezquerro JA, Hernández-Verón MA. The majorant principle applied to Hammerstein integral equations. Appl Math Lett. 2018 Jan 1; 75:50-8.
Amer SM, Dardery S. The method of Kantorovich majorants to nonlinear singular integral equation with shift. Appl Math Comput. 2009 Dec 15; 215(8): 2799-805.
Argyros IK, Hilout S. Improved local convergence of Newton’s method under weak majorant condition. J Comput Appl Math. 2012 Jan 1; 236(7):1892-902.
Atkinson Ke. the numerical solution of integral equations of the second kind. Cambridge University, Cambridge. 1997 552p.
Zeidler E. Applied functional analysis: main principles and their applications. Springer Science & Business Media; 2012 Dec 6. 420 p.
Husam Hameed H, Eshkuvatov ZK, Ahmedov A, Nik Long NM. On Newton-Kantorovich method for solving the nonlinear operator equation. Abstr Appl Anal 2015 (Vol. 2015). Hindawi.
Leonid VK, Gleb PA. Functional Analysis. Pergamon Press Ltd. & "Nauka" publishers; 1982. 596 p.
Mirzaee F, Hadadiyan E, Bimesl S. Numerical solution for three-dimensional nonlinear mixed Volterra–Fredholm integral equations via three-dimensional block-pulse functions. Appl Math Comput. 2014 Jun 15; 237:168-75.
Ziqan A, Armiti S, Suwan I. Solving three-dimensional Volterra integral equation by the reduced differential transform method. International Journal of Applied Mathematics Research. 2016 Apr 1; 5(2):103.