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.
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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