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Open AccessArticle

Dynamical Analysis and Analytical Solutions of the Fractional Benjamin–Bona–Mahony–Burger Equation

by Beenish, Mohammed M. Babatin and Mohamed A. Abdelkawy

Symmetry 2026, 18(4), 634; https://doi.org/10.3390/sym18040634 - 9 Apr 2026

Cited by 1 | Viewed by 632

In this paper, we study the dynamical analysis and solutions of the fractional Benjamin–Bona–Mahony–Burger equation. We demonstrate various derived solutions using different definitions of fractional derivatives, namely the

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-derivative, conformable derivative, and M-truncated derivative, to examine their kinetic characteristics. Firstly, we find [...] Read more.

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-derivative, conformable derivative, and M-truncated derivative, to examine their kinetic characteristics. Firstly, we find the solution of the fractional Benjamin–Bona–Mahony–Burger equation using two different approaches. We then discuss the effects of the fractional derivative on the solutions using 3D graphical discussion. Finally, we discuss the dynamical analysis using sensitivity and chaos analysis. We also discuss the chaos analysis using permutation entropy, 2D and 3D phase portrait, fractal dimension, time analysis, return map, Lyapunov exponent, and multistability through Poincare map and basins of attraction. To explore a diverse range of phenomena across the fields of physical science and engineering, this study highlights the computational strength and flexibility of the proposed method. Full article

(This article belongs to the Special Issue Symmetries and Fractional Differential Equations: Theory and Applications)

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20 pages, 1619 KB

Open AccessArticle

Hybrid Shifted Gegenbauer Integral–Pseudospectral Method for Solving Time-Fractional Benjamin–Bona–Mahony–Burgers Equation

by Kareem T. Elgindy

Mathematics 2025, 13(16), 2678; https://doi.org/10.3390/math13162678 - 20 Aug 2025

Viewed by 1053

Abstract

This paper introduces a novel hybrid shifted Gegenbauer integral–pseudospectral (HSG-IPS) method to solve the time-fractional Benjamin–Bona–Mahony–Burgers (FBBMB) equation with high accuracy. The approach transforms the equation into a form with only a first-order derivative, which is approximated using a stable shifted Gegenbauer differentiation matrix (SGDM), while other terms are computed with precise quadrature rules. By integrating advanced techniques such as the shifted Gegenbauer pseudospectral method (SGPS), fractional derivative and integral approximations, and barycentric integration matrices, the HSG-IPS method achieves spectral accuracy. Numerical results show it reduces average absolute errors (AAEs) by up to 99.99% compared to methods like Crank–Nicolson linearized difference scheme (CNLDS) and finite integration method using Chebyshev polynomial (FIM-CBS), with computational times as low as 0.04–0.05 s. The method’s stability is improved by avoiding ill-conditioned high-order derivative approximations, and its efficiency is boosted by precomputed matrices and Kronecker product structures. Robust across various fractional orders, the HSG-IPS method offers a powerful tool for modeling wave propagation and nonlinear phenomena in fractional calculus applications. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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22 pages, 6282 KB

Open AccessArticle

Quadrature Solution for Fractional Benjamin–Bona–Mahony–Burger Equations

by Waleed Mohammed Abdelfattah, Ola Ragb, Mokhtar Mohamed, Mohamed Salah and Abdelfattah Mustafa

Fractal Fract. 2024, 8(12), 685; https://doi.org/10.3390/fractalfract8120685 - 22 Nov 2024

Cited by 1 | Viewed by 1385

Abstract

In this work, we present various novelty methods by employing the fractional differential quadrature technique to solve the time and space fractional nonlinear Benjamin–Bona–Mahony equation and the Benjamin–Bona–Mahony–Burger equation. The novelty of these methods is based on the generalized Caputo sense, classical differential quadrature method, and discrete singular convolution methods based on two different kernels. Also, the solution strategy is to apply perturbation analysis or an iterative method to reduce the problem to a series of linear initial boundary value problems. Consequently, we apply these suggested techniques to reduce the nonlinear fractional PDEs into ordinary differential equations. Hence, to validate the suggested techniques, a solution to this problem was obtained by designing a MATLAB code for each method. Also, we compare this solution with the exact ones. Furthermore, more figures and tables have been investigated to illustrate the high accuracy and rapid convergence of these novel techniques. From the obtained solutions, it was found that the suggested techniques are easily applicable and effective, which can help in the study of the other higher-D nonlinear fractional PDEs emerging in mathematical physics. Full article

(This article belongs to the Section Numerical and Computational Methods)

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15 pages, 3655 KB

Open AccessArticle

Dynamical Behavior of the Fractional BBMB Equation on Unbounded Domain

by Wei Zhang, Haijing Wang, Haolu Zhang, Zhiyuan Li and Xiaoyu Li

Fractal Fract. 2024, 8(7), 383; https://doi.org/10.3390/fractalfract8070383 - 28 Jun 2024

Cited by 2 | Viewed by 1680

Abstract

The fractional-order Benjamin-Bona-Mahony-Burgers (BBMB) equation is a generalization of the classical BBMB equation. It’s dynamic behaviors is much more complex than that of the corresponding integer-order BBMB equation. The main purpose of this paper is to explore the dynamic behaviors of the fractional-order BBMB equations by using the Fourier spectral method. Firstly, the numerical solution is compared with the exact solution. It is proved that the proposed method is effective and high precision for solving the spatial fractional order BBMB equation. Then, some dynamical behaviors of fractional order BBMB equations are obtained by using the present method, and some novel fractal waves of the the fractional-order BBMB equation on unbounded domain are shown. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

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18 pages, 7937 KB

Open AccessArticle

A New Hybrid Optimal Auxiliary Function Method for Approximate Solutions of Non-Linear Fractional Partial Differential Equations

by Rashid Ashraf, Rashid Nawaz, Osama Alabdali, Nicholas Fewster-Young, Ali Hasan Ali, Firas Ghanim and Alina Alb Lupaş

Fractal Fract. 2023, 7(9), 673; https://doi.org/10.3390/fractalfract7090673 - 7 Sep 2023

Cited by 22 | Viewed by 3364

Abstract

This study uses the optimal auxiliary function method to approximate solutions for fractional-order non-linear partial differential equations, utilizing Riemann–Liouville’s fractional integral and the Caputo derivative. This approach eliminates the need for assumptions about parameter magnitudes, offering a significant advantage. We validate our approach using the time-fractional Cahn–Hilliard, fractional Burgers–Poisson, and Benjamin–Bona–Mahony–Burger equations. Comparative testing shows that our method outperforms new iterative, homotopy perturbation, homotopy analysis, and residual power series methods. These examples highlight our method’s effectiveness in obtaining precise solutions for non-linear fractional differential equations, showcasing its superiority in accuracy and consistency. We underscore its potential for revealing elusive exact solutions by demonstrating success across various examples. Our methodology advances fractional differential equation research and equips practitioners with a tool for solving non-linear equations. A key feature is its ability to avoid parameter assumptions, enhancing its applicability to a broader range of problems and expanding the scope of problems addressable using fractional calculus techniques. Full article

(This article belongs to the Section General Mathematics, Analysis)

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23 pages, 17240 KB

Open AccessArticle

The Investigation of Dynamical Behavior of Benjamin–Bona–Mahony–Burger Equation with Different Differential Operators Using Two Analytical Approaches

by Xiaoming Wang, Rimsha Ansar, Muhammad Abbas, Farah Aini Abdullah and Khadijah M. Abualnaja

Axioms 2023, 12(6), 599; https://doi.org/10.3390/axioms12060599 - 16 Jun 2023

Cited by 8 | Viewed by 2007

Abstract

The dynamic behavior variation of the Benjamin–Bona–Mahony–Burger (BBM-Burger) equation has been investigated in this paper. The modified auxiliary equation method (MAEM) and Ricatti–Bernoulli (RB) sub-ODE method, two of the most reliable and useful analytical approaches, are used to construct soliton solutions for the [...] Read more.

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-derivative, conformable derivative (CD), and M-truncated derivatives (M-TD) to understand their dynamic behavior. The hyperbolic and trigonometric functions are used to derive the analytical solutions for the given model. As a consequence, dark, bell-shaped, anti-bell, M-shaped, W-shaped, kink soliton, and solitary wave soliton solutions are obtained. We observe the fractional parameter impact of the derivatives on physical phenomena. The BBM-Burger equation is functional in describing the propagation of long unidirectional waves in many nonlinear diffusive systems. The 2D and 3D graphs have been presented to confirm the behavior of analytical wave solutions. Full article

(This article belongs to the Special Issue Special Topics in Differential Equations with Applications)

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