The Numerical Approximation of Caputo Fractional Derivative of Higher Orders Using A Shifted Gegenbauer Pseudospectral Method: Two-Point Boundary Value Problems of the Bagley Torvik Type Case Study

TL;DR

This work develops a shifted Gegenbauer pseudospectral framework (SGPS) to numerically approximate Caputo fractional derivatives of order $\alpha>0$, transforming $^{c}D_x^{\alpha}f$ into a scaled integral of the $m$th derivative with $m=\lceil\alpha\rceil$ to mitigate near-zero singularities. It introduces the fractional SG integration matrix (FSGIM) derived from Gegenbauer polynomials and SG quadratures, enabling efficient, precomputable matrix–vector evaluations at arbitrary points and providing rigorous error analysis with exponential convergence for smooth functions. The method is demonstrated on Caputo TPBVPs of Bagley–Torvik type, achieving near-machine-precision accuracy with modest discretizations and fast runtimes, while offering practical parameter guidelines and stability considerations. The SGPS framework outperforms many existing approaches and is flexible enough to extend to multidimensional fractional problems, offering a robust, scalable tool for high-precision fractional differential equation solving.

Abstract

This work presents a new framework for approximating Caputo fractional derivatives (FDs) of any positive order using a shifted Gegenbauer pseudospectral (SGPS) method. By transforming the Caputo FD into a scaled integral of the $m$th-derivative of the Lagrange interpolating polynomial (with $m$ being the ceiling of the fractional order $α$), we mitigate the singularity near zero, improving stability and accuracy. The method links $m$th-derivatives of shifted Gegenbauer (SG) polynomials with SG polynomials of lower degrees, allowing for precise integration using SG quadratures. We employ orthogonal collocation and SG quadratures in barycentric form to obtain an accurate and efficient approach for solving fractional differential equations. We provide error analysis showing that the SGPS method is convergent in a semi-analytic framework and conditionally convergent with exponential rate for smooth functions in finite-precision arithmetic. This exponential convergence improves accuracy compared to wavelet-based, operational matrix, and finite difference methods. The SGPS method is flexible, with adjustable SG parameters for optimal performance. A key contribution is the fractional SG integration matrix (FSGIM), which enables efficient computation of Caputo FDs via matrix-vector multiplications and accelerates the SGPS method through pre-computation and storage. The method remains within double-precision limits, making it computationally efficient. It handles any positive fractional order $α$ and outperforms existing schemes in solving Caputo fractional two-point boundary value problems (TPBVPs) of the Bagley-Torvik type.

Figures (6)

• Figure 1: Logarithmic absolute errors of the Caputo FD approximations of $f_1$ at $t = 0.5$, for $N = 2:10, \alpha = 1.5$, using Gegenbauer interpolants of degrees $n = N+1:N+4$ and parameter values $\lambda = -0.1, 0, 0.5, 1, 2$, together with a $(15,0.5)$-SGPS quadrature. Each curve within a plot corresponds to a different power $N$ of the function $t^N$, ranging from $2$ to $10$.
• Figure 2: Logarithmic absolute errors of the Caputo FD approximations of $f_2$ at $t = 0.5$, for $\beta = 0.1, \alpha = 1.5$, using Gegenbauer interpolants of degrees $n = 3:7$ and parameter values $\lambda = -0.1, 0, 0.5, 1, 2$, together with a $(15,0.5)$-SGPS quadrature.
• Figure 3: Log-lin plots of ${\hat{\vartheta} _{m,\lambda }}$ for $\lambda = -0.1, 0, 0.5, 1, 2$, and $m = 2:10$.
• Figure 4: The exact solution of Example 1 and its approximation on $\mathbf{\Omega}_1$ (upper) and the absolute errors at the collocation points (lower). The approximate solution was obtained using the SGPS method with parameters $n = n_q = 4$ and $\lambda = \lambda_q = 1.1$.
• Figure 5: The exact solution of Example 2 and its approximation on $\mathbf{\Omega}_1$ (upper) and the absolute errors at the collocation points (lower). The approximate solution was obtained using the SGPS method with parameters $n = n_q = \lambda = \lambda_q = 2$.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof
• Theorem 4.3
• proof
• Theorem 4.4
• proof