High-Precision Matrix Computational Technique for Fractional Differential Equations using Generalized Symmetric Polynomials

Yasser A. Amer; Kamal R. Raslan; Mohamed A. El Salam; Aziza A. Mahdi

English
العربيّة

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High-Precision Matrix Computational Technique for Fractional Differential Equations using Generalized Symmetric Polynomials

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Article info

In Press

Received 2025-07-12

Received in revised form

Accepted 2025-10-20

Availabe online 2025-11-27

Pages None - None

Keywords

Chebyshev polynomials
Symmetric orthogonal polynomials
Spectral Tau method
Monic and shifted monic forms

Abstract

In this work, a generalized main class of symmetric orthogonal polynomials is proposed and utilized within a spectral frame-work for solving fractional differential equations (FDEs) using tau method. Emphasis is placed on the monic symmetric and shifted monic symmetric representations of several kinds Chebyshev olynomials. These polynomials are employed to develop general unified operational differentiation matrices, which convert the original differential equations into equivalent systems of algebraic equations. The symmetry and orthogonality properties inherent to these polynomial families enhance both the precision and stability of the numerical schemes. A comparative study is conducted to assess the performance of each type in terms of accuracy, convergence behavior, and computational efficiency. The results confirm that the proposed method provides a reliable and adaptable framework for effectively solving a wide range of linear and nonlinear differential equations within the context of numerical analysis and applied mathematics.

High-Precision Matrix Computational Technique for Fractional Differential Equations using Generalized Symmetric Polynomials

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معلومات المقال

قيد النشر

تاريخ التسليم 2025-07-12

تاريخ تسليم النسخة المعدلة

تاريخ القبول 2025-10-20

على شبكة الإنترنت 2025-11-27

الصفحات None - None

الكلمات الإفتتاحية

Chebyshev polynomials
Symmetric orthogonal polynomials
Spectral Tau method
Monic and shifted monic forms

الملخص

In this work, a generalized main class of symmetric orthogonal polynomials is proposed and utilized within a spectral frame-work for solving fractional differential equations (FDEs) using tau method. Emphasis is placed on the monic symmetric and shifted monic symmetric representations of several kinds Chebyshev olynomials. These polynomials are employed to develop general unified operational differentiation matrices, which convert the original differential equations into equivalent systems of algebraic equations. The symmetry and orthogonality properties inherent to these polynomial families enhance both the precision and stability of the numerical schemes. A comparative study is conducted to assess the performance of each type in terms of accuracy, convergence behavior, and computational efficiency. The results confirm that the proposed method provides a reliable and adaptable framework for effectively solving a wide range of linear and nonlinear differential equations within the context of numerical analysis and applied mathematics.