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This work aims to develop a Legendre spectral operational matrix method for solving two-dimensional integro-differential equations.

May 21, 2026MathematicsOpen Access

A Legendre Spectral Operational Matrix Method with Convergence Analysis for Two-Dimensional Integro-Differential Equations

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Randomized trial reveals a new method for solving two-dimensional integro-differential equations, indicating improved accuracy.

This work aims to develop a Legendre spectral operational matrix method for solving two-dimensional integro-differential equations.
Utilizes tensor-product Legendre polynomial expansions to approximate solutions.
Constructs exact operational matrices for differentiation and integration in a finite-dimensional system.
Establishes convergence estimates in L2 and L∞ norms for a specialized linear problem.
Demonstrates rapid error decay as polynomial degree increases with numerical errors nearing machine precision.
Confirms accuracy and reliability of the Legendre spectral operational matrix framework.
Showcases performance through benchmark examples of linear and nonlinear integro-differential equations.

Ishtiaq Ali (2026) studied this question.

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