Time Fractional Boussinesq Equation Research Articles

A Note on the Application of the Double Sumudu–Generalized Laplace Decomposition Method and 1+1- and 2+1-Dimensional Time-Fractional Boussinesq Equations

The current paper concentrates on discovering the exact solutions of the singular time-fractional Boussinesq equation and coupled time-fractional Boussinesq equation by presenting a new technique known as the double Sumudu–generalized Laplace and Adomian decomposition method. Here, two main theorems are addressed that are very useful in this work. Moreover, the mentioned method is effective in solving several problems. Some examples are presented to check the precision and symmetry of the technique. The outcomes show that the proposed technique is precise and gives better solutions as compared to existing methods in the literature.

Numerical approximations of groundwater flow problem using fractional variational iteration method with fractional derivative of singular and nonsingular kernels

This study aims to present the numerical approximations of groundwater flow problem, namely the time-fractional Boussinesq equation using the fractional variational iteration method. Fractional derivatives in time variable are taken in Caputo and Caputo–Fabrizio sense. This work compares the solutions of the time-fractional Boussinesq equation for the Caputo and Caputo–Fabrizio fractional operators. Graphical representation of the water table head’s behavior for different time values is shown.

Time-fractional Boussinesq equation for heavy-nucleus-acoustic wave excitations in highly dense relativistically degenerate quantum plasmas

The dynamics of nonlinear propagation of the heavy-nucleus-acoustic waves (HNAW) is theoretically investigated in a plasma comprising non-degenerate inertial heavy ions, degenerate non-relativistic non-inertial light ions, and degenerate electrons-positrons with relativistic velocity. At first, the classical Boussinesq equation is obtained, then using Agrawal’s method, it is transformed into the time-fractional Boussinesq equation to observe the effects of time-fractional parameter on the propagation dynamics of HNAW, where the notion of the Riesz fractional derivative refers to the time-fractional term. The variational iteration method is employed to construct the heavy-nucleus-acoustic solitary wave solution which later turns into an oscillatory wave pulse. It comes out that the heavy-nucleus acoustic waves’ amplitude is greatly altered by the time-fractional parameter. We have briefly explored the impact of plasma parameters on the creation and propagation of heavy-nucleus acoustic waves. The significance of our findings is emphasized in relation to the highly dense relativistic astrophysical plasmas usually observed in dense stars.

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.

Fractional Analysis of Nonlinear Boussinesq Equation under Atangana–Baleanu–Caputo Operator

This article proposed two novel techniques for solving the fractional-order Boussinesq equation. Several new approximate analytical solutions of the second- and fourth-order time-fractional Boussinesq equation are derived using the Laplace transform and the Atangana–Baleanu fractional derivative operator. We give some graphical and tabular representations of the exact and proposed method results, which strongly agree with each other, to demonstrate the trustworthiness of the suggested methods. In addition, the solutions we obtain by applying the proposed approaches at different fractional orders are compared, confirming that as the value trends from the fractional order to the integer order, the result gets closer to the exact solution. The current technique is interesting, and the basic methodology suggests that it might be used to solve various fractional-order nonlinear partial differential equations.

The modified simplest equation procedure for conformable time-fractional Boussinesq equations

In this paper, we used the traveling wave transformation to find out the exact solutions of the conformable time fractional differential equations with the help of the Modified simplest equation procedure. We applied the given procedure to two different equations which are conformable time-fractional Boussinesq and coupled Boussinesq equations. First, we reduced the given equations to ordinary differential equations using wave transformations. Then, we applied the procedure to the given equations. We obtained hyperbolic, trigonometric and rational solutions. We gave the 3D plots, 2D plots and contour plots for some obtained solutions. According to our results, we can say the given method is effective, powerful and gives simple, reliable solutions compared to the other methods. All obtained solutions were checked with the help of Mathematica.

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.

Constructing analytical estimates of the fuzzy fractional-order Boussinesq model and their application in oceanography

The main idea of this article is the investigation of atmospheric internal waves, often known as gravity waves. This arises within the ocean rather than at the interface. A shallow fluid assumption is illustrated by a series of nonlinear partial differential equations in the framework. Because the waves are scattered over a wide geographical region, this system can precisely replicate atmospheric internal waves. In this research, the numerical solutions to the fuzzy fourth-order time-fractional Boussinesq equation (BSe) are determined for the case of the aquifer propagation of long waves having small amplitude on the surface of water from a channel. The novel scheme, namely the generalized integral transform (proposed by H. Jafari [35]) coupled with the Adomian decomposition method (GIADM), is used to extract the fuzzy fractional BSe in R,Rn and (2nth)-order including gH-differentiability. To have a clear understanding of the physical phenomena of the projected solutions, several algebraic aspects of the generalized integral transform in the fuzzy Caputo and Atangana-Baleanu fractional derivative operators are discussed. The confrontation between the findings by Caputo and ABC fractional derivatives under generalized Hukuhara differentiability are presented with appropriate values for the fractional order and uncertainty parameters ℘∈[0,1] were depicted in diagrams. According to proposed findings, hydraulic engineers, being analysts in drainage or in water management, might access adequate storage volume quantity with an uncertainty level.

Modelling and Analysis of Complex Viscous Fluid in Thin Elastic Tubes

Cardiovascular disease is a major threat to human health. The study on the pathogenesis and prevention of cardiovascular disease has received special attention. In this paper, we have contributed to the derivation of a mathematical model for the nonlinear waves in an artery. From the Navier–Stokes equations and continuity equation, the vorticity equation satisfied by the blood flow is established. And based on the multiscale analysis and perturbation method, a new model of the Boussinesq equation with viscous term is derived to describe the propagation of a viscous fluid through a thin tube. In order to be more consistent with the flow of the fluid, the time-fractional Boussinesq equation with viscous term is deduced by employing the semi-inverse method and the fractional variational principle. Moreover, the approximate analytical solution of the fractional equation is obtained, and the effect of viscosity on the amplitude and width of the wave is studied. Finally, the effects of the fractional order parameters and vessel radius on blood flow volume are discussed and analyzed.

A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel method

In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, “good” Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.

Homotopy Analysis Method for the Time-Fractional Boussinesq Equation

In this paper, the exact and approximate analytical solutions to the time-fractional Boussinesq equation are constructed using the homotopy analysis method. Several examples about the fourth-order and sixth-order time-fractional Boussinesq equations show the flexibility and efficiency of the method. Furthermore, by choosing an appropriate value for the auxiliary parameter $h$, we can obtain the $N$-term approximate solution with improved accuracy.

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.

Space time fractional Boussinesq equation with singular and non singular kernels

Space time fractional Boussinesq equation with singular and non singular kernels

Analytic Solution of Time Fractional Boussinesq Equation for Groundwater Flow in Unconfined Aquifer

An approximate analytical solution to the nonlinear time fractional Boussinesq equation is presented here. Derivative with respect to time variable is replaced with Caputo fractional derivative. The Natural transform method and Adomian decomposition method are employed to obtain the solution. Some test problems are solved to show the accuracy of the proposed method. Behavior of water table head is depicted graphically for various time values.

Spectral method for solving the time fractional Boussinesq equation

In this paper, Fourier spectral approximation for the time fractional Boussinesq equation with periodic boundary condition is considered. The space is discretized by the Fourier spectral method and the Crank–Nicolson scheme is used to discretize the Caputo time fractional derivative. Stability and convergence analysis of the numerical method are proven. Some numerical examples are included to testify the effectiveness of our given method. Based on the presented numerical results, the Fourier spectral method is shown to be effective for solving the time fractional Boussinesq equation.

Time-fractional generalized Boussinesq equation for Rossby solitary waves with dissipation effect in stratified fluid and conservation laws as well as exact solutions

Construct fractional order model to describe Rossby solitary waves can provide more pronounced effects and deeper insight for comprehending generalization and evolution of Rossby solitary waves in stratified fluid. In the paper, from the quasi-geostrophic vorticity equation with dissipation effect and complete Coriolis force, based on the multi-scale analysis and perturbation method, a classical generalized Boussinesq equation is derived to describe the Rossby solitary waves in stratified fluid. Further, by employing the reduction perturbation method, the semi-inverse method, the Agrawal method, we derive the Euler–lagrangian equation of classical generalized Boussinesq equation and obtain the time-fractional generalized Boussinesq equation. Without dissipation effect, by using Lie group analysis method, the conservation laws of time-fractional Boussinesq equation are given. Finally, with the help of the improved (G′/G) expansion method, the exact solutions of the above equation are generated. Meanwhile, in order to consider the dissipation effect, we have to derive the approximate solutions by adopting the New Iterative Method. We remark that the fractional order model can open up a new window for better understanding the waves in fluid.

A new modification in the exponential rational function method for nonlinear fractional differential equations

We have modified the traditional exponential rational function method (ERFM) and have used it to find the exact solutions of two different fractional partial differential equations, one is the time fractional Boussinesq equation and the other is the (2+1)-dimensional time fractional Zoomeron equation. In both the cases it is observed that the modified scheme provides more types of solutions than the traditional one. Moreover, a comparison of the recent solutions is made with some already existing solutions. We can confidently conclude that the modified scheme works better and provides more types of solutions with almost similar computational cost. Our generalized solutions include periodic, soliton-like, singular soliton and kink solutions. A graphical simulation of all types of solutions is provided and the correctness of the solution is verified by direct substitution. The extended version of the solutions is expected to provide more flexibility to scientists working in the relevant field to test their simulation data.

Analytical treatment of nonlinear conformable time-fractional Boussinesq equations by three integration methods

Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the $$\exp (-\Omega (\eta ))$$ -expansion method and the $$\tan (\phi (\eta )/2)$$ -expansion method are used in examining the analytical solution of the nonlinear fractional equations. The proposed methods are based on the integration method and a wave transformation. The fractional derivative in the sense of conformable time-fractional derivative is defined. Fractional complex transform is implemented to change fractional differential equations into ordinary differential equations in this paper. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of nonlinear conformable time-fractional Boussinesq equations.

Symmetry properties, similarity reduction and exact solutions of fractional Boussinesq equation

In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.