Search Results (13)

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

In this study, we investigate the position and momentum Shannon entropy, denoted as $S_x$ and $S_p$, respectively, in the context of the fractional Schrödinger equation (FSE) for a hyperbolic double well potential (HDWP). We explore various values of the fractional derivative represented by k in our analysis. Our findings reveal intriguing behavior concerning the localization properties of the position entropy density, ${\rho }_s\left(x\right)$, and the momentum entropy density, ${\rho }_s\left(p\right)$, for low-lying states. Specifically, as the fractional derivative k decreases, ${\rho }_s\left(x\right)$ becomes more localized, whereas ${\rho }_s\left(p\right)$ becomes more delocalized. Moreover, we observe that as the derivative k decreases, the position entropy $S_x$ decreases, while the momentum entropy $S_p$ increases. In particular, the sum of these entropies consistently increases with decreasing fractional derivative k. It is noteworthy that, despite the increase in position Shannon entropy $S_x$ and the decrease in momentum Shannon entropy $S_p$ with an increase in the depth u of the HDWP, the Beckner–Bialynicki-Birula–Mycielski (BBM) inequality relation remains satisfied. Furthermore, we examine the Fisher entropy and its dependence on the depth u of the HDWP and the fractional derivative k. Our results indicate that the Fisher entropy increases as the depth u of the HDWP is increased and the fractional derivative k is decreased. 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

In the current study, we investigate the stochastic Benjamin–Bona–Mahony equation with beta derivative (SBBME-BD). The considered stochastic term is the multiplicative noise in the Itô sense. By combining the $\mathcal{F}$-expansion approach with two separate equations, such as the Riccati and elliptic equations, new hyperbolic, trigonometric, rational, and Jacobi elliptic solutions for SBBME-BD can be generated. The solutions to the Benjamin–Bona–Mahony equation are useful in understanding various scientific phenomena, including Rossby waves in spinning fluids and drift waves in plasma. Our results are presented using MATLAB, with numerous 3D and 2D figures illustrating the impacts of white noise and the beta derivative on the obtained solutions of SBBME-BD. Full article

The accurate assessment of forest biomass is vital to climate change mitigation. Based on forest survey data, stand biomass models can effectively assess forest biomass carbon at large scales. However, traditional stand biomass models have ignored the potential effects of the climate on stand biomass estimation. There is still a lack of research on whether or not and in what ways the effects of the climate reduce uncertainty in biomass estimation and carbon accounting. Therefore, two types of stand biomass models, including basic stand biomass models (BBMs) and climate-sensitive stand biomass models (CBMs), were developed and tested using 311 plantation plots of Korean pine (Pinus koraiensis Siebold & Zucc.), Korean larch (Larix olgensisi A. Henry), and Mongolian pine (Pinus sylvestris var. mongolica Litv.) in Northeast China. The two types of models were developed by applying simultaneous equations based on nonlinear, seemingly unrelated, regression (NSUR) to ensure additivity of the stand total and components biomass (root, stem, branch, and needle). The results of fitting and leave-one-out cross-validation (LOOCV) indicated that the CBMs performed better than the corresponding BBMs. The RMSEs of the stand total biomass decreased by 3.5% to 10.6% for the three conifer species. The influence of temperature-related climate variables on the biomass of stand components was greater than that of precipitation-related climate variables. The sensitivity of the three conifer species to climate variables was ranked as Korean pine > Mongolian pine > Korean larch. This study emphasizes the importance of combining climate variables in stand biomass models to reduce the uncertainty and climate effects in forest biomass estimation, which will play a role in carbon accounting for forest ecosystems. Full article

In this work we have studied the Shannon information entropy for two hyperbolic single-well potentials in the fractional Schrödinger equation (the fractional derivative number $(0<n\le 2)$ by calculating position and momentum entropy. We find that the wave function will move towards the origin as the fractional derivative number n decreases and the position entropy density becomes more severely localized in more fractional system, i.e., for smaller values of n, but the momentum probability density becomes more delocalized. And then we study the Beckner Bialynicki-Birula–Mycieslki (BBM) inequality and notice that the Shannon entropies still satisfy this inequality for different depth u even though this inequality decreases (or increases) gradually as the depth u of the hyperbolic potential $U_1$ (or $U_2$) increases. Finally, we also carry out the Fisher entropy and observe that the Fisher entropy increases as the depth u of the potential wells increases, while the fractional derivative number n decreases. Full article

In this work, we study the bifurcation and the numerical analysis of the nonlinear Benjamin-Bona-Mahony-KdV equation. According to the bifurcation theory of a dynamic system, the various kinds of traveling wave profiles are obtained including the behavior of solitary and periodic waves. Additionally, a two-level linear implicit finite difference algorithm is implemented for investigating the Benjamin-Bona-Mahony-KdV model. The application of a priori estimation for the approximate solution also provides the convergence and stability analysis. It was demonstrated that the current approach is singularly solvable and that both time and space convergence are of second-order precision. To confirm the computational effectiveness, two numerical simulations are prepared. The findings show that the current technique performs admirably in terms of delivering second-order accuracy in both time and space with the maximum norm while outperforming prior schemes. Full article

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters. Full article

We consider the Benjamin–Bona–Mahony (BBM) equation of the form $u_t+u_x+u{u}_x-u_{xxt}=0,(x,t)\in \mathcal{M}\times \mathbb{R}$ where $\mathcal{M}=\mathbb{T}$ or $\mathbb{R}.$ We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in $H^s\left(\mathbb{T}\right)$ established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in $H^s\left(\mathbb{R}\right).$ Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces $M_s^{2,1}\left(\mathbb{R}\right)$ for $s\ge 0$. Full article

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical). Full article

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its $S-$integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability. Full article

We study the realized power variations for the fourth order linearized Kuramoto–Sivashinsky (LKS) SPDEs and their gradient, driven by the space–time white noise in one-to-three dimensional spaces, in time, have infinite quadratic variation and dimension-dependent Gaussian asymptotic distributions. This class was introduced-with Brownian-time-type kernel formulations by Allouba in a series of articles starting in 2006. He proved the existence, uniqueness, and sharp spatio-temporal Hölder regularity for the above class of equations in $d=1,2,3$. We use the relationship between LKS-SPDEs and the Houdré–Villaa bifractional Brownian motion (BBM), yielding temporal central limit theorems for LKS-SPDEs and their gradient. We use the underlying explicit kernels and spectral/harmonic analysis to prove our results. On one hand, this work builds on the recent works on the delicate analysis of variations of general Gaussian processes and stochastic heat equation driven by the space–time white noise. On the other hand, it builds on and complements Allouba’s earlier works on the LKS-SPDEs and their gradient. Full article

In this paper, we investigate the Wick-type stochastic (3+1)-dimensional modified Benjamin–Bona–Mahony (BBM) equations. We present a generalised version of the modified tanh–coth method. Using the generalised, modified tanh–coth method, white noise theory, and Hermite transform, we produce a new set of exact travelling wave solutions for the (3+1)-dimensional modified BBM equations. This set includes solutions of exponential, hyperbolic, and trigonometric types. With the help of inverse Hermite transform, we obtained stochastic travelling wave solutions for the Wick-type stochastic (3+1)-dimensional modified BBM equations. Eventually, by application example, we show how the stochastic solutions can be given as white noise functional solutions. Full article