Explicit collocation algorithm for the nonlinear fractional Duffing equation via third-kind Chebyshev polynomials

Document Type : Research Article

Authors

• Y.H. Youssri ¹
• A.G. Atta ²
• M.O. Moustafa ³
• Z.Y. Abu Waar ⁴

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

² Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo 11341, Egypt.

³ Department of Physics, School of Science and Engineering, The American University in Cairo (AUC), New Cairo 11835, Egypt.

⁴ Department of Physics, College of Science, The University of Jordan, Amman, 11942, Jordan.

10.22067/ijnao.2025.90483.1543

Abstract

Herein, we propose an accurate algorithm to approximate the solution of the nonlinear fractional-order Duffing equation (NFDE). The algorithm is based on using shifted Chebyshev polynomials of the third-kind as basis functions and the spectral collocation method as a solver. We study the error analysis of the method in-depth, and we exhibit some numerical test problems to check the applicability of the method. Also, we compare it with other existing techniques to show the superiority of our proposed numerical scheme. Our results show that the method employed provides a useful tool to simulate the solution of the NFDE. The main advantages of the proposed method are that it does not require a huge number of retained modes, simply a few terms, and does not exhaust the machine used to render the codes.

Keywords

• Chebyshev polynomials
• Spectral methods
• Nonlinear fractional-order Duffing equation
• Convergence analysis

Main Subjects

• Numerical methods for differential equations and its applications

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