Fractional Boussinesq Equation Research Articles

Solving Time-Space Fractional Boussinesq Equation Using Homotopy Perturbation Method

This paper aims to implement the homotopy perturbation technique to solve the time-space fractional Boussinesq equation, a significant model in the analysis of nonlinear wave propagation. Through the application of the homotopy perturbation technique, we derive analytical expressions for the solutions of the time-space fractional Boussinesq equation and validate these solutions through comparisons with numerical methods. Obtained results demonstrate the efficiency and accuracy of the homotopy perturbation method in solving the time-space fractional Boussinesq equation.

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Bright-dark solitary wave solutions to the nonlinear fractional Boussinesq and breaking soliton equations

Convergence rates and central limit theorem for 3-D stochastic fractional Boussinesq equations with transport noise

This work investigates the 3-D stochastic modified fractional Boussinesq equations driven by transport noise on the torus. The existence of weak solutions is initially established through Galerkin approximation and compactness techniques. Subsequently, we demonstrate that the weak solutions of stochastic modified fractional Boussinesq equations converge to the unique solution of a deterministic Boussinesq model with fractional dissipation, subject to an appropriate noise scaling, along with providing the quantitative estimates with explicit convergent rates. Finally, the central limit theorem with an explicit convergence rate is established based on the aforementioned scaling limit. It is noteworthy that the scaling limit result of stochastic modified fractional Boussinesq equations to deterministic Boussinesq models with additional fractional dissipation implies that the transport noise regularizes the modified fractional Boussinesq equations, thus leading to “approximate weak uniqueness”.

Well-posedness and limit behavior of stochastic fractional Boussinesq equation driven by nonlinear noise

This paper primarily explores the well-posedness and limit behavior of the stochastic fractional Boussinesq equation driven by periodic force and nonlinear noise. Firstly, we establish the existence and uniqueness of the strong solution in probabilistic sense. Subsequently, the existence of weak mean random attractor and evolution systems of measures will be proved. Furthermore, we investigate the asymptotic behavior of the invariant measure as the noise intensity approaches zero. To this end, we utilize the Wentzell-Freidlin large deviation principle by demonstrating the compactness of level sets and the weak convergence between stochastic and controlled equations. Finally, we establish the ergodicity and exponential mixing by employing the Harris-like theorem under periodic force.

SOLITON SOLUTIONS FOR THE TWO-DIMENSIONAL LOCAL FRACTIONAL BOUSSINESQ EQUATION

In this work we study the two-dimensional local fractional Boussinesq equation. Based on the basic definitions and properties of the local fractional derivatives and bilinear form, we studied the soliton solutions of non-differentiable type with the generalized functions defined on Cantor sets by using bilinear method. Meanwhile, we discuss the result when fractal dimension is 1, and compare it with the result when fractal dimension is [Formula: see text].

Innovative approaches of a time-fractional system of Boussinesq equations within a Mohand transform

<p>This paper investigated the application of analytical methods, specifically the Mohand transform iterative method (MTIM) and the Mohand residual power series method (MRPSM), to solve the fractional Boussinesq equation. Utilizing the Caputo operator to manage fractional derivatives, these semi-analytical approaches provide accurate solutions to complex fractional differential equations. Through convergence analysis and error estimation, the study validated the efficacy of these methods by comparing numerical solutions to known exact solutions. Graphical and tabular representations illustrated the accuracy of the proposed methods, highlighting their performance for varying fractional orders. The findings demonstrated that both MTIM and MRPSM offer reliable, efficient solutions, making them valuable tools for addressing fractional differential systems in fields such as applied mathematics, engineering, and physics.</p>

Generalized exponential rational function method to the fractional shallow water wave phenomena

In this study we introduce the fractional shallow water wave phenomena, namely, the new extended (2 + 1)-dimensional fractional Boussinesq equation and in the following we apply the generalized exponential rational function for extracting the soliton solutions to the Ex-FBe (extended (2 + 1)-dimensional fractional Boussinesq equation) by using Maple tools for calculating and Mathematica for graphical behaviors. Many types of exact soliton solutions for the fswwp equation have been derived by the erf method. Physical interpretations of some of the obtained solutions are also included. It can be easily concluded that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

Analytical solutions and mathematical simulation of traveling wave solutions to fractional order nonlinear equations

We investigate the tanh–coth method’s usefulness in understanding space–time fractional nonlinear evolution problems. To use this method and then examine their plots, we take into consideration three significant equations: the space–time fractional Burgers equation, the space–time fractional regularized long wave equation, and the space–time fractional Boussinesq equation. The modified Riemann–Liouville derivatives are used to define the fractional derivative. The approach provides a linearization- or small perturbation-free analytical solution in the form of a convergent series with easily calculable components. For the aforementioned nonlinear fractional equations, we have discovered their precise solutions. We utilize a generalized fractional complex transform to translate these to ordinary differential equations which subsequently resulted into number of exact solutions. The method provides us abundant analytical traveling wave solutions containing fewer number parameters with the aid of Mathematica and MATLAB.

Numerical Analysis of the Time-Fractional Boussinesq Equation in Gradient Unconfined Aquifers with the Mittag-Leffler Derivative

In this study, two numerical methods—the variational iteration transform method (VITM) and the Adomian decomposition (ADM) method—were used to solve the second- and fourth-order fractional Boussinesq equations. Both methods are helpful in approximating non-linear problems effectively, easily, and accurately. The fractional Atangana–Baleanu operator and ZZ transform were utilized to derive solutions for the equation. Two examples are discussed to validate the methods and solutions. The results demonstrate that both the VITM and ADM methods are effective in obtaining accurate and reliable solutions for the time-fractional Boussinesq equation.

Group analysis, invariance results, exact solutions and conservation laws of the perturbed fractional Boussinesq equation

The scope of this work is based on symmetry analysis of perturbed fractional Boussinesq equation. For beginning the group formalism, invariance properties and conservation laws of the nonlinear perturbed fractional Boussinesq equation have been explored generally. This method was first described by Lukashchuk [Commun. Nonlinear Sci. Numer. Simul. 68 (2019) 147–159]. The key subject is that when the order of fractional derivative in a fractional differential equation (FDE) is nearly integers, we can approximate it to a perturbed integer-order differential equation with a small perturbation parameter. For obtaining the results, perturbed and unperturbed symmetries are computed. Then, the methodology of reduction is applied for finding some new solutions by the symmetry operators of the equation. These solutions are obtained by the similarity transformations of the symmetries. Another exact solution will be found with constructing one-dimensional optimal system of the symmetries. Finally, the meaning of nonlinear self-adjointness concept is attended in order to find conservation laws with informal Lagrangians.

Further results about the exact solutions of conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation

Seeking for the exact solutions of fractional nonlinear partial differential equations (FNPDE) has penetrated into almost every discipline of the natural, engineering, mathematics, and social sciences. In this article, with the aid of Maple software, using the (G′G2)-expansion method and the unified F-expansion method, we successfully investigate the exact solutions about conformable space–time fractional Boussinesq equation (FBE) and breaking soliton (Calogero) equation. Consequently, three types traveling wave solutions are found such as Jacobian double periodic elliptic, simply periodic and the rational function solutions. The (G′G2)-expansion and unified F-expansion methods are relatively complex, but we can find more solutions that have not been seen before. We give the some specific examples solutions such as double-periodic Jacobian elliptic functions which can be molted into single-periodic functions. Also we point the bridge linking among the Weierstrass elliptic, hyperbolic, trigonometric functions and Jacobian elliptic functions. The connection and distinction between these three formal solutions are also indicated in our paper.

On exact solutions of some important nonlinear conformable time-fractional differential equations

The nonlinear fractional Boussinesq equations are known as the fractional differential equation class that has an important place in mathematical physics. In this study, a method called $$\Big (\frac{G'}{G^2}\Big )$$ -extension method which works well and reveals exact solutions is used to examine nonlinear Boussinesq equations with conformable time-fractional derivative. This method is a very useful approach and extremely utility compared to other analytical methods. With the proposed method, there are three unique types of solutions such as hyperbolic, trigonometric and rational solutions. This approach can similarly be applied to other nonlinear fractional models.

Stability of hydrostatic equilibrium to the 2D fractional Boussinesq equations

<p style='text-indent:20px;'>Stability problem on perturbations near the hydrostatic balance is one of the important issues for Boussinesq equations. This paper focuses on the asymptotic stability and large-time behavior problem of perturbations of the 2D fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity. Since the linear portion of the Boussinesq equations plays a crucial role in the stability properties, we firstly study the linearized fractional Boussinesq equations with only fractional velocity dissipation or fractional thermal diffusivity and complete the following work: 1) assessing the stability and obtaining the precise large-time asymptotic behavior for solutions to the linearized system satisfied the perturbation; 2) understanding the spectral property of the linearization; 3) showing the <inline-formula><tex-math id="M1">\begin{document}$ H^2 $\end{document}</tex-math></inline-formula>-stability for the linearized system, and prove that the <inline-formula><tex-math id="M2">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M3">\begin{document}$ \nabla{u} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \Delta{u} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M5">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \Delta\theta $\end{document}</tex-math></inline-formula>), the <inline-formula><tex-math id="M7">\begin{document}$ L^\varrho $\end{document}</tex-math></inline-formula>-norm <inline-formula><tex-math id="M8">\begin{document}$ (2<\varrho<\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M9">\begin{document}$ u $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ \nabla{u} $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M11">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula>) are all approaching to zero as <inline-formula><tex-math id="M13">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M14">\begin{document}$ \alpha = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ \eta = 0 $\end{document}</tex-math></inline-formula> (or <inline-formula><tex-math id="M16">\begin{document}$ \nu = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \beta = 1 $\end{document}</tex-math></inline-formula>). Secondly, we obtain the <inline-formula><tex-math id="M18">\begin{document}$ H^1 $\end{document}</tex-math></inline-formula>-stability for the full nonlinear system and prove the <inline-formula><tex-math id="M19">\begin{document}$ L^\varrho $\end{document}</tex-math></inline-formula>-norm <inline-formula><tex-math id="M20">\begin{document}$ (2<\varrho<\infty) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M21">\begin{document}$ \theta $\end{document}</tex-math></inline-formula> and the <inline-formula><tex-math id="M22">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of <inline-formula><tex-math id="M23">\begin{document}$ \nabla\theta $\end{document}</tex-math></inline-formula> approaching to zero as <inline-formula><tex-math id="M24">\begin{document}$ t\rightarrow\infty $\end{document}</tex-math></inline-formula>.</p>

Classification of All Single Traveling Wave Solutions of Fractional Coupled Boussinesq Equations via the Complete Discrimination System Method

In this paper, the complete discrimination system method is used to construct the exact traveling wave solutions for fractional coupled Boussinesq equations in the sense of conformable fractional derivatives. As a result, we get the exact traveling wave solutions of fractional coupled Boussinesq equations, which include rational function solutions, Jacobian elliptic function solutions, implicit solutions, hyperbolic function solutions, and trigonometric function solutions. Finally, the obtained solution is compared with the existing literature.

Asymptotic properties of the 2D stochastic fractional Boussinesq equations driven by degenerate noise

The current paper is devoted to stochastic fractional Boussinesq equations (FBEs) on the Torus T2=[0,2π]×[0,2π] with degenerate random forcing. We first show the existence and pathwise uniqueness of strong solutions by virtue of the Yamada-Watanabe Theorem, and then we investigate the asymptotic properties of invariant measures in two aspects: the first is, focusing on the stochastic systems, to prove the existence of an invariant measure in the case of multiplicative noise, and then prove that the invariant measure can be unique and exponentially mixing if we are restricted to degenerate additive noise. The second is, from the perspective of stochastic stability, to prove that the zero-noise limits of stationary measures for the perturbed stochastic systems are invariant for the deterministic system and their supports are concentrated on the Birkhoff center.

New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with <i>0</i> < α ≤ <i>1</i> or <i>1</i> < α ≤ <i>2</i>. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

Coupled Fractional Traveling Wave Solutions of the Extended Boussinesq–Whitham–Broer–Kaup-Type Equations with Variable Coefficients and Fractional Order

In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

Symmetry classification and exact solutions of (3 + 1)-dimensional fractional nonlinear incompressible non-hydrostatic coupled Boussinesq equations

The symmetry classifications of two fractional higher dimensional nonlinear systems, namely, (3 + 1)-dimensional incompressible non-hydrostatic Boussinesq equations and (3 + 1)-dimensional Boussinesq equations with viscosity, are discussed. Both the fractional Boussinesq equations are considered to have derivatives with respect to all variables of fractional type, and some exact solutions are reported along with graphical illustrations.

Analytical solutions of space fractional Boussinesq equation to simulate water table profiles between two parallel drainpipes under different initial conditions

This work presents the first attempt to derive analytical solutions of a space fractional Boussinesq equation (SFBE) for parabola type 1, parabola type 2, and elliptical initial water table profiles, denoted as SFBE-P1, SFBE-P2, and SFBE-E, respectively. Laboratory and field data published in literature were used to evaluate the performances of the fractional models in homogeneous and heterogeneous soils. Besides, the performances of the proposed fractional models were compared with that of a fractional analytical solution developed in the literature for a flat initial water table condition (SFBE-F). According to the results of the sensitivity analysis, heterogeneity degree of soil (α) had a more effect on outputs of the fractional models. Also, the results indicated that the α parameter in the SFBE-P1, SFBE-E, and SFBE-F was capable of describing well the heterogeneity degree of soil. The α value was estimated almost 2 in relatively homogenous soil, while its values were respectively appraised almost 1.3 and 1.1 in soils with average and relative heterogeneity degrees. The fractional models reduced to their counterpart classical ones (BE-P1, BE-P2, BE-E, and BE-F) in the relatively homogenous soil. The measurement and prediction results demonstrated the best and similar performance for the SFBE-E and SFBE-F models, while the SFBE-P2 models was the weakest. This behavior was also observed by using the classical models. Compared to the BE-E, the SFBE-E provided better prediction results in the heterogeneous soils. Overall, the SFBE-E and SFBE-F can be applied as practical models to simulate water table profile between two parallel drainpipes for both homogeneous and heterogeneous soils.

Space time fractional Boussinesq equation with singular and non singular kernels

Recently, many authors have found analytical and numerical solutions of fractional Boussinesq equation by applying various fractional operators with singular kernels. Motivated by recently introduced fractional operators with non-singular kernels, in this paper a comparison of the solution of linearised fractional Boussinesq equation has been made for the fractional operators Caputo (with singular kernel) and Caputo-Fabrizio (with non-singular kernel). Linearised Boussinesq equation is derived by assuming that the average thickness of saturated layer of an aquifer is constant.