In this paper, a three-dimensional Legendre polynomial (3D-LPs) is used for solving the nonlinear three-dimensional Volterra integral equations (VIEs). Converting the main problem to a nonlinear algebraic system using 3D-LPs, which can be generalized to equations in higher dimensions, then the nonlinear system will be solved. Some results concerning the error analysis have been achieved. Several examples are included to demonstrate the validity and applicability of the method. Moreover, we prove a theorem and a corollary about a sufficient condition for the minimum of mean square error under the Legendre coefficients and the uniqueness of the solution of the nonlinear VIEs. In addition, illustrative examples are included to demonstrate the validity and applicability of the presented method.
[1] N. H. Ali, S. A. Mohammed, and J. Manafian, Study on the simplified MCH equation and the combined KdVmKdV equations with solitary wave solutions, Partial Diff. Eq. Appl. Math., 9 (2024), 100599.
[2] M. U. Alibubin, J. Sulaiman, F. A. Muhiddin, A. Sunarto, and G. B. Ekal, A Caputo-based nonlocal arithmeticmean discretization for solving nonlinear time-fractional diffusion equation using half-sweep KSOR, Edelweiss Appl. Sci. Tech., 9(1) (2025), 896-911.
[3] M. Bakhshi, M. Asghari Larimi, and M. Asghari Larimi, Three-dimensional differential transform method for solving nonlinear three-dimensional volterra integral equations, J. Math. Comput. Sci., 4(2) (2012), 246-256.
[4] J. Biazar and M. Eslami, Modified HPM for solving systems of Volterra integral equations of the second kind, J. King Saud Univ., 23 (2011), 35-39.
[5] F. Bouchaala, M. Y. Ali, and A. Farid, Estimation of compressional seismic wave attenuation of carbonate rocks in Abu Dhabi, United Arab Emirates. Comptes Rendus. Geosci., 346(7-8) (2014), 169-178.
[6] F. Bouchaala, M. Y. Ali, and J. Matsushima, Attenuation modes from vertical seismic profiling and sonic waveform in a carbonate reservoir, Abu Dhabi, United Arab Emirates, Geophys. Prospecting, 64 (2016), 1030-1047.
[7] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equation by collocation methods, SIAM J. Numer. Anal., 27 (1990), 978-1000.
[8] C. Canuto, A. Quarteroni, M. Y. Hussaini, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer Series in Computational Physics, 1988.
[9] C. Cattani and A. Kudreyko, Harmonic wavelet method towards solution of the Fredholm type integral equations of the second kind, Appl. Math. Comput., 12(166) (2010), 4164-4171.
[10] R. Y. Chang and M. L. Wang, Shifted Legendre direct method for variational problems, J. Optim. Theory. Appl., 39 (1983), 299-307.
[11] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Num. Meth. Partial Diff. Eq J., 26 (2010), 448-479.
[12] M. Dehghan, J. Manafian, and A. Saadatmandi, The solution of the linear fractional partial differential equations using the homotopy analysis method, Z. Naturforsch, 65(a) (2010), 935-949.
[13] H. Du and M. Cui, A method of solving nonlinear mixed Volterra-Fredholm integral equation, Appl. Math. Sci., 1 (2007), 2505-2516.
[14] N. Ebrahimi and J. Rashidinia, Collocation method for linear and nonlinear Fredholm and Volterra integral equations, Appl. Math. Comput., 270 (2015), 156-164.
[15] A.M.A. El-Sayed, H.H.G. Hashem, and E.A.A. Ziada, Picard and Adomian decomposition methods for a quadratic integral equation of fractional order, Comput. Appl. Math., 33(1) (2014), 95-109.
[16] F. Erdogan, A second order numerical method for singularly perturbed Volterra integro-differential equations with delay, Int. J. Math. Comput. Eng., 2(1) (2024), 85-96.
[17] Y. Gu, S. Malmir, J. Manafian, O. A. Ilhan, A. A. Alizadeh, and A. J. Othman, Variety interaction between k-lump and k-kink solutions for the (3+1)-D Burger system by bilinear analysis, Results Phys., 43 (2022), 106032.
[18] C. Hwang and M. Y. Chen, A direct approach using the shifted Legendre series expansion for near optimum control of linear time-varying systems with multiple state and control delays, Int. J. Control, 43 (1986), 1673-1692.
[19] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley and Sons, 1989.
[20] M. Lakestani and J. Manafian, Analytical treatment of nonlinear conformable timefractional Boussinesq equations by three integration methods, Opt. Quant. Elec., 50(4) (2018), 1-31.
[21] M. Lakestani and J. Manafian, Analytical treatments of the space-time fractional coupled nonlinear Schr¨odinger equations, Opt. Quant. Elec., 50(396) (2018), 1-33.
[22] M. Lakestani, J. Manafian, A. R. Najafizadeh, and M. Partohaghighi, Some new soliton solutions for the nonlinear the fifth-order integrable equations, Comput. Meth. Diff. Equ., 10(2) (2022), 445-460.
[23] M. Lakestani, B. N. Saray, and M. Dehghan, Numerical solution for the weakly singular Fredholm integrodifferential equations using Legendre multi-wavelets, J. Comput. Appl. Math., 235 (2011), 3291-3303.
[24] U. Lepik and E. Tamme, Solution of nonlinear Fredholm integral equations via the Haar wavelet method, Proc. Estonian Acad. Sci. Phys. Math., 56 (2007), 11-27.
[25] P. Linz, Analytical and numerical methods for Volterra equations, SIAM Studiesin Applied Mathematics, 1985.
[26] Y. Liu, Application of Legendre Polynomials in Solving Volterra Integral Equations of the Second Kind, Appl. Math., 3(5) (2013), 157-159.
[27] K. Maleknejad and M. T. Kajani, Solving second kind integral equations by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput., 145 (2003), 623-629.
[28] K. Maleknejad and S. Sohrabi, Legendre polynomial solution of nonlinear volterra-fredholm integarl equations, IUST Int. J. Eng. Sci., 19 (2008), 49-52.
[29] J. Manafian, Solving the integro-differential equations using the modified Laplace Adomian decomposition method, J. Math. Ext., 6 (2012), 1-15.
[30] J. Manafian and P. Bolghar, Numerical solutions of nonlinear three-dimensional Volterra integral-differential equations with 3D-block-pulse functions, Math. Meth. Appl. Sci. J., 41(12) (2018), 4867-4876.
[31] J. Manafian, L. A. Dawood, and M. Lakestani, New solutions to a generalized fifth-order KdV like equation with prime number p = 3 via a generalized bilinear differential operator, Partial Diff. Equ. Appl. Math., 9 (2024), 100600.
[32] J. Manafian and M. Lakestani, N-lump and interaction solutions of localized waves to the (2+ 1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation, J. Geom. Phys., 150 (2020), 103598.
[33] J. Manafian and M. Lakestani, Application of tan(ϕ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity, Optik, 127(4) (2016), 2040-2054.
[34] J. Manafian, and M. Lakestani, Optical soliton solutions for the Gerdjikov-Ivanov model via tan(ϕ/2)-expansion method, Optik, 127(20) (2016), 9603-9620.
[35] J. Manafian and M. Lakestani, Abundant soliton solutions for the Kundu-Eckhaus equation via tan(ϕ(ξ))expansion method, Optik, 127(14) (2016), 5543-5551.
[36] J. Manafian and M. Lakestani, A new analytical approach to solve some the fractional-order partial differential equations, Indian J. Phys., 91 (2017), 243-258. 168-175.
[37] B. N. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Appl. Math. Comput., 190(2)(2007), 1707-1716.
[38] S. Mashayekhi, M. Razzaghi, and O. Tripak, Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials, Sci. World J., 2014 (2014), 1-8.
[39] F. Mirzaee and E. Hadadiyan, A computational method for nonlinear mixed Volterra-Fredholm integral equations, Caspian J. Math. Sci., 2 (2013), 113-123.
[40] F. Mirzaee, E. Hadadiyan, and S. Bimesl, Numerical solution for three-dimensional nonlinear mixed Volterra-Fredholm integral equations via three-dimensional block-pulse functions, Appl. Math. Comput., 237 (2014),
[41] D. S. Mohamed and R. A. Taher, Comparison of Chebyshev and Legendre polynomials methods for solving two-dimensional Volterra-Fredholm integral equations, J. Egyptian Math. Soc., 25 (2017), 302-307.
[42] S. R. Moosavi, N. Taghizadeh, and J. Manafian, Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method, Comput. Meth. Diff. Equ., 8(3) 2020, 537-552.
[43] S. Nemati, P. M. Lima, and Y. Ordokhani, Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials, J. Comput. Appl. Math., 242 (2013), 53-69.
[44] S. Nemati and Y. Ordokhani, Numerical solution of two-dimensional nonlinear Volterra integral equations by the Legendre polynomials, J. Sci. Tarbiat Moallem University, 11 (2012), 1-16.
[45] A. Rivaz, S. Jahanara, and F. Yousefi, Two-dimensional Chebyshev Polynomials for Solving Two-dimensional Integro-Differential Equations, C¸ankaya University J.Sci. Eng., 12 (2015), 001-011.
[46] P. k. Sahu and S. S. Ray, Numerical solutions for the system of Fredholm integral equations of second kind by a new approach in volving semi orthogonal B-spline wavelet collocation method, Appl. Math. Comput., 234 (2014), 368-379.
[47] B. N. Saray, M. Lakestani, and M. Razzaghi, Sparse Representation of System of Fredholm Integro-Differential Equations by using Alpert Multi-wavelets, Comput. Math. Math. Phys., 55(9) (2015), 1468-1483.
[48] I. Singh and S. Kumar, Haar wavelet method for some nonlinear Volterra integral equations of the first kind, J. Comput. Appl. Math., 292 (2016), 541-552.
[49] A. Tari, On the Existence, Uniqueness and Solution of the Nonlinear Volterra Partial Integro-Differential Equations, Int. J. Nonlinear Sci., 16 (2013), 152-163.
[50] C. Yang, Chebyshev polynomial solution of nonlinear integral equations, J. Franklin Inst., 349 (2012), 947-956.
[51] M. Zarebnia and J. Rashidinia, Convergence of the Sinc method applied to Volterra integral equations, Appl. Appl. Math., 5(1) (2010), 198-216.
[52] M. Zhang, X. Xie, J. Manafian, O. A. Ilhan, and G. Singh, Characteristics of the new multiple rogue wave solutions to the fractional generalized CBS-BK equation, J. Adv. Res., 38 (2022), 131-142.
Manafian, J. and Bolgar, P. (2026). Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials. Computational Methods for Differential Equations, 14(1), 361-373. doi: 10.22034/cmde.2025.65630.3028
MLA
Manafian, J. , and Bolgar, P. . "Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials", Computational Methods for Differential Equations, 14, 1, 2026, 361-373. doi: 10.22034/cmde.2025.65630.3028
HARVARD
Manafian, J., Bolgar, P. (2026). 'Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials', Computational Methods for Differential Equations, 14(1), pp. 361-373. doi: 10.22034/cmde.2025.65630.3028
CHICAGO
J. Manafian and P. Bolgar, "Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials," Computational Methods for Differential Equations, 14 1 (2026): 361-373, doi: 10.22034/cmde.2025.65630.3028
VANCOUVER
Manafian, J., Bolgar, P. Numerical idea to solve three-dimensional nonlinear Volterra integral equations with 3D-Legendre polynomials. Computational Methods for Differential Equations, 2026; 14(1): 361-373. doi: 10.22034/cmde.2025.65630.3028
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