Whitham Broer Kaup Equations Research Articles

Computational Study of Coupled Whitham Broer Kaup equations via Interpolation technique

The main purpose of this study is to investigate the numerical solution of coupled Whitham Broer Kaup (WBK) equation via Quintic B-Spline interpolation technique. This problem is of significant interest in the study of nonlinear wave phenomena because of its applications in various fields, such as fluid dynamics, plasma physics, and nonlinear optics and climate modeling. For the temporal derivative, the forward difference technique and a quadrature rule are utilized to deal the integer and fractional models, respectively, while spatial operators and the solutions are then obtained using Quintic B-spline. Furthermore, the non-linear terms are linearized using Quasi-linearization technique. Absolute error, L2 and L∞ error norms are computed to check the accuracy of the proposed method. The computed results are represented graphically and compared with the exact solution. It is found that our method is efficient due to less computational cost and proffer better accuracy. Stability of the proposed method is discussed using Von-Neumann stability which identifies that the scheme is conditionally stable.

Similarity wave solutions of Whitham–Broer–Kaup equations in the oceanic shallow water

Similarity wave solutions of Whitham–Broer–Kaup equations in the oceanic shallow water

Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain

Fourier spectral method for the fractional-in-space coupled Whitham–Broer–Kaup equations on unbounded domain

Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas.

Similarity Reductions Related to Nonlocal Symmetries for Whitham–Broer–Kaup Equations with Variable Coefficients

Similarity Reductions Related to Nonlocal Symmetries for Whitham–Broer–Kaup Equations with Variable Coefficients

Numerical Solutions of Time-Fractional Whitham–Broer–Kaup Equations via Sumudu Decomposition Method

In this paper, the coupled system of Whitham–Broer–Kaup equations of the Caputo fractional derivative (CFD) is studied using the Sumudu decomposition method (SDM). Using different dispersion relations, these equations are needed to describe the properties of waves in shallow water. The current investigation for the future scheme includes convergence and error analysis. We use two examples to demonstrate the leverage and effectiveness of the proposed scheme, and the error analysis is discussed to ensure its accuracy. The numerical simulation is carried out to ensure the accuracy of the future technique. The obtained numerical and graphical results are presented, and the proposed scheme is computationally very accurate and simple to study and solve fractionally coupled nonlinear complex phenomena encountered in science and technology.

Two Novel Methods for Fractional Nonlinear Whitham–Broer–Kaup Equations Arising in Shallow Water

Two Novel Methods for Fractional Nonlinear Whitham–Broer–Kaup Equations Arising in Shallow Water

Abundant variant wave patterns by coupled Boussinesq–Whitham–Broer–Kaup equations

This paper focus on searching for abundant wave patterns to time-fractional coupled Boussinesq–Whitham–Broer–Kaup equations, which plays a crucial role in obtaining the novel insights for propagational dynamics of shallow water. Through the innovative application of complete discrimination system for polynomial method (CDSPM) and conformable fractional transformation, the whole types of solutions for the fractional equations presented in the existing literatures are obtained. In particular, we obtain the solitary wave type solutions and Jacobian elliptic functions type solutions, and the later are initially founded. The existence and applicability of the obtained solutions are proved by concrete examples. Finally, the nature of wave propagation and the fractional characteristics are intuitively revealed by graphical simulations.

Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations

Painlevé integrability and superposition wave solutions of Whitham–Broer–Kaup equations

Fractional-order partial differential equations describing propagation of shallow water waves depending on power and Mittag-Leffler memory

<abstract><p>In this research, the $ \bar{\mathbf{q}} $-homotopy analysis transform method ($ \bar{\mathbf{q}} $-HATM) is employed to identify fractional-order Whitham–Broer–Kaup equation (WBKE) solutions. The WBKE is extensively employed to examine tsunami waves. With the aid of Caputo and Atangana-Baleanu fractional derivative operators, to obtain the analytical findings of WBKE, the predicted algorithm employs a combination of $ \bar{\mathbf{q}} $-HAM and the Aboodh transform. The fractional operators are applied in this work to show how important they are in generalizing the frameworks connected with kernels of singularity and non-singularity. To demonstrate the applicability of the suggested methodology, various relevant problems are solved. Graphical and tabular results are used to display and assess the findings of the suggested approach. In addition, the findings of our recommended approach were analyzed in relation to existing methods. The projected approach has fewer processing requirements and a better accuracy rate. Ultimately, the obtained results reveal that the improved strategy is both trustworthy and meticulous when it comes to assessing the influence of nonlinear systems of both integer and fractional order.</p></abstract>

Multiple-soliton and periodic solutions to space–time fractional Whitham–Broer–Kaup equations

In this paper, the space–time fractional Whitham–Broer–Kaup equations are investigated. By means of new fractional scaling transformations, the fractional nonlinear system of different time and space orders is transformed to the integer one. The multiple solitary solutions and periodic solutions are obtained, respectively. All those solutions are given exactly by introducing new scaling transformations, which makes our study unique and different from most existing studies. It is expected that exact solutions for nonlinear wave system of fractional order can be handled in the similar way.

Solitary waves of the fractal Whitham–Broer–Kaup equation in shallow water

Solitary waves of the fractal Whitham–Broer–Kaup equation in shallow water

Analytical new soliton wave solutions of the nonlinear conformable time-fractional coupled Whitham–Broer–Kaup equations

The study of nonlinear physical and abstract systems is greatly important in order to determine the behavior of the solutions for Fractional Partial Differential Equations (FPDEs). In this paper, we study the analytical wave solutions of the time-fractional coupled Whitham–Broer–Kaup (WBK) equations under the meaning of conformal fractional derivative. These solutions are derived using the modified extended tanh-function method. Accordingly, different new forms of the solutions are obtained. In order to understand its behavior under varying parameters, we give the visual representations of all the solutions. Finally, the graphs are discussed and a conclusion is given.

Numerical Investigation of the Time-Fractional Whitham–Broer–Kaup Equation Involving without Singular Kernel Operators

This paper aims to implement an analytical method, known as the Laplace homotopy perturbation transform technique, for the result of fractional-order Whitham–Broer–Kaup equations. The technique is a mixture of the Laplace transformation and homotopy perturbation technique. Fractional derivatives with Mittag-Leffler and exponential laws in sense of Caputo are considered. Moreover, this paper aims to show the Whitham–Broer–Kaup equations with both derivatives to see their difference in a real-world problem. The efficiency of both operators is confirmed by the outcome of the actual results of the Whitham–Broer–Kaup equations. Some problems have been presented to compare the solutions achieved with both fractional-order derivatives.

VARIATIONAL PRINCIPLES FOR FRACTAL WHITHAM–BROER–KAUP EQUATIONS IN SHALLOW WATER

In this paper, we mainly consider the Whitham–Broer–Kaup equations with unsmooth boundaries, and a new fractal derivative is adopted to construct the fractal model. The variational principles of the fractal Whitham–Broer–Kaup equations are successfully constructed by fractal Semi-inverse method, which is helpful to study the symmetry, discover the conserved quantity, and has a wide application prospect in numerical simulation. The results are discussed by the two-scale transform method. The approximate analytical solutions of the fractal Whitham–Broer–Kaup equations are obtained according to the variational iteration method and two-scale transform method.

On the approximate solution of fractional-order Whitham–Broer–Kaup equations

In this paper, the Homotopy perturbation Laplace method is implemented to investigate the solution of fractional-order Whitham–Broer–Kaup equations. The derivative of fractional-order is described in Caputo’s sense. To show the reliability of the suggested method, the solution of certain illustrative examples are presented. The results of the suggested method are shown and explained with the help of its graphical representation. The solutions of fractional-order problems as well as integer-order problems are determined by using the present technique. It has been observed that the obtained solutions are in significant agreement with the actual solutions to the targeted problems. Computationally, it has been analyzed that the solutions at different fractional-orders have a higher rate of convergence to the solution at integer-order of the derivative. Due to the analytical analysis of the problems, this study can further modify the solution of other fractional-order problems.

The analytical solution of fractional‐order Whitham–Broer–Kaup equations by an Elzaki decomposition method

AbstractIn this article, the Elzaki decomposition method is used to evaluate the solution of fractional‐order Whitham–Broer–Kaup equations. With the help of Elzaki transform coupled with Adomian decomposition method, an iterative procedure is established to investigate approximate solution to the suggested coupled scheme of nonlinear partial fractional differential equations. The solution of some illustrative examples is presented to show the validity of the proposed technique. The results of the proposed technique are shown and analyzed with the help of figures and tables. Also, we compared the results of our proposed technique with the homotopy perturbation method. The present method has less computational costs and a higher rate of convergence. The suggested algorithm is therefore very helpful to solve other fractional‐order systems of partial differential equations.

Application of New Iterative Method to Time Fractional Whitham–Broer–Kaup Equations

This article presents the Fractional Laplace transform with the help of new iterative method for finding the approximate solutions of coupled system of fractional order partial differential equations. The time fractional Whitham-Broer-Kaup system is taken as test example The fractional derivatives are described in the Caputo sense. Numerical results obtained by the proposed method are compared with that of Adomian Decomposition Method (ADM), Variational Iteration Method (VIM) and Optimal Homotopy Asymptotic Method (OHAM).Numerical results show that the proposed method is reliable and efficient for solution of fractional order coupled system partial differential equations. The accuracy of the method increases by taking higher order approximations.

Exact solutions to the space-time fraction Whitham–Broer–Kaup equation

The objective work of this paper is to transform the nonlinear space-time fraction Whitham–Broer–Kaup equation into ordinary differential equation by using the conformal fractional derivative, and find the exact solutions through the complete polynomial discriminant system. At the same time, we build the appropriate solution for the identified parameters to show the existence of the solution. In addition, we provide the 3D and 2D graphics to show that the solutions are real and effective.

Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches

The fractional traveling wave solution of important Whitham–Broer–Kaup equations was investigated by using the q-homotopy analysis transform method and natural decomposition method. The Caputo definition of fractional derivatives is used to describe the fractional operator. The obtained results, using the suggested methods are compared with each other as well as with the exact results of the problems. The comparison shows the best agreement of solutions with each other and with the exact solution as well. Moreover, the proposed methods are found to be accurate, effective, and straightforward while dealing with the fractional-order system of partial differential equations and therefore can be generalized to other fractional order complex problems from engineering and science.