A spectral operational matrix method using shifted fractional Legendre functions for nonlinear fractional Volterra integro-differential equations
DOI: https://doi.org/10.3846/mma.2026.24683Abstract
This paper proposes an operational matrix method based on shifted fractional Legendre functions for solving nonlinear fractional Volterra integro-differential equations. The existence and uniqueness of the solution are established via the contraction mapping theorem. The method begins by constructing shifted Legendre functions of fractional order, followed by deriving a fractional integration operational matrix via the Laplace transform. These tools are then employed to convert the original nonlinear equation into an equivalent algebraic system, thereby enhancing computational efficiency. This formulation effectively addresses equations with weakly singular kernels and solutions exhibiting endpoint singularities or low regularity. The nonlinear term is handled directly through fixed-point iteration without requiring linearization or decomposition. An error analysis is conducted, and an upper bound in the \( L^2_w \)-norm is derived. The results show that the error tends to zero as the number of basis functions increases, under appropriate smoothness conditions. Compared with traditional integer-order basis functions, fractional functions better accommodate the endpoint singularities and low-regularity features of the solution. The superior adaptability yields improved accuracy and more efficient convergence.
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fractional calculus, operational matrix method, fractional Legendre functions, Laplace transform applications, error analysisHow to Cite
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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Copyright (c) 2026 The Author(s). Published by Vilnius Gediminas Technical University.
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