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Keywords = time-fractional KdV–Burgers equation

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13 pages, 871 KiB  
Article
High-Accuracy Solutions to the Time-Fractional KdV–Burgers Equation Using Rational Non-Polynomial Splines
by Miguel Vivas-Cortez, Majeed A. Yousif, Bewar A. Mahmood, Pshtiwan Othman Mohammed, Nejmeddine Chorfi and Alina Alb Lupas
Symmetry 2025, 17(1), 16; https://doi.org/10.3390/sym17010016 - 25 Dec 2024
Cited by 10 | Viewed by 869
Abstract
A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. [...] Read more.
A non-polynomial spline is a technique that utilizes information from symmetric functions to solve mathematical or physical models numerically. This paper introduces a novel non-polynomial spline construct incorporating a rational function term to develop an efficient numerical scheme for solving time-fractional differential equations. The proposed method is specifically applied to the time-fractional KdV–Burgers (TFKdV) equation. and time-fractional differential equations are crucial in physics as they provide a more accurate description of various complex processes, such as anomalous diffusion and wave propagation, by capturing memory effects and non-local interactions. Using Taylor expansion and truncation error analysis, the convergence order of the numerical scheme is derived. Stability is analyzed through the Fourier stability criterion, confirming its conditional stability. The accuracy and efficiency of the rational non-polynomial spline (RNPS) method are validated by comparing numerical results from a test example with analytical and previous solutions, using norm errors. Results are presented in 2D and 3D graphical formats, accompanied by tables highlighting performance metrics. Furthermore, the influences of time and the fractional derivative are examined through graphical analysis. Overall, the RNPS method has demonstrated to be a reliable and effective approach for solving time-fractional differential equations. Full article
(This article belongs to the Special Issue Symmetry in Mathematical Theory and Simulation Methods for Backward Problems)
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18 pages, 2638 KiB  
Article
Radical Petrov–Galerkin Approach for the Time-Fractional KdV–Burgers’ Equation
by Youssri Hassan Youssri and Ahmed Gamal Atta
Math. Comput. Appl. 2024, 29(6), 107; https://doi.org/10.3390/mca29060107 - 21 Nov 2024
Cited by 13 | Viewed by 1295
Abstract
This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers’ equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of [...] Read more.
This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers’ equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of solving the differential problem together with the conditions, we solve a system of algebraic equations. The method can handle complex boundary conditions. Some numerical experiments are exhibited to demonstrate that this approach is highly efficient and produces results that are better than some existing numerical methods in the literature. This technique offers more advanced solutions for time-fractional problems in various fields. Full article
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