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Keywords = barycentric Lagrange polynomial

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33 pages, 1375 KB  
Article
The Numerical Approximation of Caputo Fractional Derivatives of Higher Orders Using a Shifted Gegenbauer Pseudospectral Method: A Case Study of Two-Point Boundary Value Problems of the Bagley–Torvik Type
by Kareem T. Elgindy
Mathematics 2025, 13(11), 1793; https://doi.org/10.3390/math13111793 - 27 May 2025
Cited by 1 | Viewed by 879
Abstract
This paper introduces a novel Shifted Gegenbauer Pseudospectral (SGPS) method for approximating Caputo fractional derivatives (FDs) of an arbitrary positive order. The method employs a strategic variable transformation to express the Caputo FD as a scaled integral of the mth-derivative of the [...] Read more.
This paper introduces a novel Shifted Gegenbauer Pseudospectral (SGPS) method for approximating Caputo fractional derivatives (FDs) of an arbitrary positive order. The method employs a strategic variable transformation to express the Caputo FD as a scaled integral of the mth-derivative of the Lagrange interpolating polynomial, thereby mitigating singularities and improving numerical stability. Key innovations include the use of shifted Gegenbauer (SG) polynomials to link mth-derivatives with lower-degree polynomials for precise integration via SG quadratures. The developed fractional SG integration matrix (FSGIM) enables efficient, pre-computable Caputo FD computations through matrix–vector multiplications. Unlike Chebyshev or wavelet-based approaches, the SGPS method offers tunable clustering and employs SG quadratures in barycentric forms for optimal accuracy. It also demonstrates exponential convergence, achieving superior accuracy in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley–Torvik type. The method unifies interpolation and integration within a single SG polynomial framework and is extensible to multidimensional fractional problems. Full article
(This article belongs to the Special Issue Advances in Computational Mathematics and Applied Mathematics)
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19 pages, 372 KB  
Article
Numerical Solutions of Volterra Integral Equations of Third Kind and Its Convergence Analysis
by Imtiyaz Ahmad Bhat and Lakshmi Narayan Mishra
Symmetry 2022, 14(12), 2600; https://doi.org/10.3390/sym14122600 - 8 Dec 2022
Cited by 19 | Viewed by 3806
Abstract
The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in [...] Read more.
The current work suggests a method for the numerical solution of the third type of Volterra integral equations (VIEs), based on Lagrange polynomial, modified Lagrange polynomial, and barycentric Lagrange polynomial approximations. To do this, the interpolation of the unknown function is considered in terms of the above polynomials with unknown coefficients. By substituting this approximation into the considered equation, a system of linear algebraic equations is obtained. Then, we demonstrate the method’s convergence and error estimations. The proposed approaches retain the possible singularity of the solution. To the best of the authors’ knowledge, the singularity case has not been addressed by researchers yet. To illustrate the applicability, effectiveness, and correctness of new methods for the proposed integral equation, examples with both types of kernels, symmetric as well as non-symmetric, are provided at the end. Full article
(This article belongs to the Section Mathematics)
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