Search Results (30)

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

In this work we propose a new second-order implicit–explicit difference scheme for a pseudoparabolic equation with a nonlinear flux term. The proposed method is evaluated in comparison with two known numerical approaches. The significance of this study stems from its relevance to physical and mechanical models, including the Benjamin–Bona–Mahony (–Burgers) equations, which arise as specific instances of the considered equation. A stability analysis of the constructed scheme is conducted. Furthermore, the method is generalized to address a two-dimensional case. Several numerical experiments are carried out to assess the accuracy and efficiency of the proposed scheme. Full article

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma. Full article

In this paper, we consider a modified Benjamin–Bona–Mahony (BBM) equation, which, for example, arises in shallow-water models. We discuss the well-posedness of the Dirichlet initial-boundary-value problem for the BBM equation. Our focus is on identifying a time-dependent source based on integral observation. First, we reformulate this inverse problem as an equivalent direct (forward) problem for a nonlinear loaded pseudoparabolic equation. Next, we develop and implement two efficient numerical methods for solving the resulting loaded equation problem. Finally, we analyze and discuss computational test examples. Full article

This paper consists of an exploration of the wave structures of the Benjamin–Ono equation along with a $\beta$-time fractional derivative. The model concerned is utilized to demonstrate internal waves of deep-stratified fluids. Bright, rational, periodic, and many more kinds of solutions for waves are achieved by utilizing the extended sinh-Gordon equation expansion (EShGEE) technique and the improved $G^{\prime }/G$-expansion scheme. An influence of fractional-order derivatives was also explored which gives the non-existing results. The Mathematica tool is utilized to gain and verify the results. The results are represented by 3-D, 2-D, and contour graphs. A stability analysis is utilized to confirm that results are precise as well as exact. Modulation instability (MI) is also performed for the steady-state solutions to the concerned model. Full article

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

In this work, we study the large-x asymptotic of classical solutions to the non-linear Benjamin equation modeling propagation of small amplitude internal waves in a two fluid system. In our analysis, we extend known $H^s$-well-posedness results to the case of the variable-weight Sobolev spaces. The spaces provide a direct control over the asymptotics of classical solutions and their weak derivatives, and permit us to compute the bulk large-x asymptotic of classical solutions explicitly in terms of input data. The asymptotic formula provides a precise description of the qualitative behaviour of classical solutions in weighted spaces and yields a number of weighted persistence and continuation results automatically. Full article

The Wazwaz–Benjamin–Bona–Mahony (WBBM) equation is a well-known regularized long-wave model that examines the propagation kinematics of water waves. The current work employs an effective approach, called the Riccati Modified Extended Simple Equation Method (RMESEM), to effectively and precisely derive the propagating soliton solutions to the (3+1)-dimensional WBBM equation. By using this upgraded approach, we are able to find a greater diversity of families of propagating soliton solutions for the WBBM model in the form of exponential, rational, hyperbolic, periodic, and rational hyperbolic functions. To further graphically represent the propagating behavior of acquired solitons, we additionally provide 3D, 2D, and contour graphics which clearly demonstrate the presence of kink solitons, including solitary kink, anti-kink, twinning kink, bright kink, bifurcated kink, lump-like kink, and other multiple kinks in the realm of WBBM. Furthermore, by producing new and precise propagating soliton solutions, our RMESEM demonstrates its significance in revealing important details about the model behavior and provides indications regarding possible applications in the field of water waves. Full article

This paper investigates the explicit, accurate soliton and dynamic strategies in the resolution of the Wazwaz–Benjamin–Bona–Mahony (WBBM) equations. By exploiting the ensuing wave events, these equations find applications in fluid dynamics, ocean engineering, water wave mechanics, and scientific inquiry. The two main goals of the study are as follows: Firstly, using the dynamic perspective, examine the chaos, bifurcation, Lyapunov spectrum, Poincaré section, return map, power spectrum, sensitivity, fractal dimension, and other properties of the governing equation. Secondly, we use a generalized rational exponential function (GREF) technique to provide a large number of analytical solutions to nonlinear partial differential equations (NLPDEs) that have periodic, trigonometric, and hyperbolic properties. We examining the wave phenomena using 2D and 3D diagrams along with a projection of contour plots. Through the use of the computational program Mathematica, the research confirms the computed solutions to the WBBM equations. Full article

In this article, we analyzed the time fractional higher-dimensional nonlinear modified model of wave propagation, namely the (3 + 1)-dimensional Benjamin–Bona–Mahony-type equation. The fractional sense was defined by the classical Riemann–Liouville fractional derivative. We derived firstly the existence of symmetry of the time fractional higher-dimensional equation. Next, we constructed the one-dimensional optimal system to the time fractional higher-dimensional nonlinear modified model of wave propagation. Subsequently, it was reduced into the lower-dimensional fractional differential equation. Meanwhile, on the basis of the reduced equation, we obtained its similarity solution. Through a series of analyses of the time fractional high-dimensional model and the results of the above obtained, we can gain a further understanding of its essence. Full article

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

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 proposed model. We demonstrate some of the extracted solutions using definitions of the $\beta$-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

A new local fractional modified Benjamin–Bona–Mahony equation is proposed within the local fractional derivative in this study for the first time. By defining some elementary functions via the Mittag–Leffler function (MLF) on the Cantor sets (CSs), a set of nonlinear local fractional ordinary differential equations (NLFODEs) is constructed. Then, a fast algorithm namely Yang’s special function method is employed to find the non-differentiable (ND) exact solutions. By this method, we can extract abundant exact solutions in just one step. Finally, the obtained solutions on the CS are outlined in the form of the 3-D plot. The whole calculation process clearly shows that Yang’s special function method is simple and effective, and can be applied to investigate the exact ND solutions of the other local fractional PDEs. Full article

It has long been known that plane wave solutions of the cubic nonlinear Schrödinger Equation (NLS) are linearly unstable. This fact is widely known as modulation instability (MI), and sometimes referred to as Benjamin–Feir instability in the context of water waves. In 1978, I.E. Alber introduced a methodology to perform an analogous linear stability analysis around a sea state with a known power spectrum, instead of around a plane wave. This analysis applies to second moments, and yields a stability criterion for power spectra. Asymptotically, it predicts that sufficiently narrow and high-intensity spectra are unstable, while sufficiently broad and low-intensity spectra are stable, which is consistent with empirical observations. The bifurcation between unstable and stable behaviour has no counterpart in the classical MI (where all plane waves are unstable), and we call it Landau–Alber bifurcation because the stable regime has been shown to be a case of Landau damping. In this paper, we work with the realistic power spectra of ocean waves, and for the first time, we produce clear, direct evidence for an abrupt bifurcation as the spectrum becomes narrow/intense enough. A fundamental ingredient of this work was to look directly at the nonlinear evolution of small, localised inhomogeneities, and whether these can grow dramatically. Indeed, one of the issues affecting previous investigations of this bifurcation seem to have been that they mostly looked for the indirect evidence of instability, such as an increase in overall extreme events. It is also found that a sufficiently large computational domain is crucial for the bifurcation to manifest. Full article

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied. Full article