Second-kind Chebyshev spectral Petrov-Galerkin scheme for solving the time-fractional Euler-Bernoulli beam equation

Document Type : Research Article

Authors

Youssri Hassan Youssri ¹

Maha Muftah Muttardi ²

¹ Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt.

² Mathematics and Actuarial Science Department, School of Science and Engineering, The American University in Cairo, New Cairo 11835, Egypt.

10.22067/ijnao.2026.97767.1839

Abstract

This work introduces a novel Petrov-Galerkin spectral technique designed to investigate vibrational behavior in fourth-order time-fractional Euler--Bernoulli beams with pinned-pinned boundary conditions. A tailored basis comprising shifted second-kind Chebyshev polynomials is developed, which automatically fulfills homogeneous boundary constraints, thus lowering computational demands. By selecting test functions from the same Chebyshev family, the Petrov-Galerkin framework converts the fractional partial differential equation into a structured algebraic system that can be efficiently resolved through Gaussian elimination. The proposed scheme delivers exponential convergence for sufficiently smooth solutions, exhibits low numerical dispersion, and remains straightforward to implement even for high-order problems. Rigorous error estimates in both $L_\infty$ and $L_2$ norms ensure theoretical reliability. Computational tests, including a case lacking a closed-form exact solution, validate the precision and computational efficiency of the approach.

Keywords

Time-fractional fourth-order partial differential equations

Chebyshev polynomials

Petrov-Galerkin method

Error analysis

Subjects

• Numerical Analysis

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