Whitham-Broer-Kaup 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.

Fractional view analysis of coupled Whitham- Broer-Kaup and Jaulent-Miodek equations

In this study, I systematically investigate the fractional Whitham-Broer-Kaup (WBK) system and the fractional Coupled Jaulent-Miodek (CJM) equation under the Caputo fractional calculus. The nonlinear fractional differential equation systems are investigated via the Aboodh transform iteration method and the Aboodh residual power series method, thus offering a thorough analytical investigation. The dynamics of the fractional WBK system are accomplished using the Aboodh transform iteration method, and the Aboodh residual power series method is employed to study the behavior of the fractional CJM equation. Using established solutions, we completely analyze their dynamics employing both symbolic computations and numerical simulations. Consequently, novel solutions are identified, and the behavior of these fractional systems in the Caputo operator sense are clarified. The results obtained show good agreement of convergence of the numerical and analytical solutions, highlighting the effectiveness of the schemes adopted for unraveling the complex dynamics of fractional nonlinear systems.

Modification of the Differential Transform Method With Search Direction Along the Spatial Axis

This paper presents a modification of the Differential Transform Method (ModDTM) formulated so that the search direction of the solution is along a spatial axis (x). It is shown that the method is applicable to a wide spectrum of (1 + 1) partial differential equations, integro partial differential equations, and integral equations. The solutions obtained are in the form of a Taylor series, the coefficients of which are determined by recursively operating a differential transform of the equation. These solutions are either exact and in closed form, or are good approximations. To illustrate the application of the method, as well as to show its versatility, examples are chosen from all the categories mentioned earlier. These include equations such as the nonlinear Fisher, combined KdV‐mKdV, Hunter–Saxton (H‐S), Fornberg–Whitham (FW), coupled systems such as the Whitham–Broer–Kaup (WBK) and the sine‐Gordon (s‐G) equations, integral equations, Volterra partial integro‐differential equations (PIDE), and linear and nonlinear versions of the (complex) Schröedinger equation. The Tables and 3D‐plots show the good rate of convergence of the obtained solutions with the exact ones. It is thus found that the new scheme is successful in producing satisfactory results as any other method in which the conventional approach of searching for solutions along the temporal axis is followed. Further, the procedures involved are simple, and hence may be operated with great ease.

Numerical solution of the Whitham-Broer-Kaup shallow water equation by quartic B-spline collocation method

This paper introduces a novel quartic B-spline collocation method to address the coupled Whitham–Broer–Kaup (WBK) problem. The WBK problem is a topic of interest in the study of nonlinear wave phenomena and has applications in various fields, including fluid dynamics, plasma physics, and nonlinear optics. The method combines spatial quartic B-spline scheme discretization, and Crank–Nicolson temporal discretization. It is unconditionally stable as proven by the Von-Neumann technique. Numerical examples demonstrate the method’s superior accuracy compared to existing solutions. Error analysis employs and norms, while the method exhibits high computational efficiency. The nonlinearity is managed through Rubin-Graves linearization. Comparisons with prior approaches highlight its efficiency, stability, adaptability to complex problems. The quartic B-spline method is well-suited for simulating fluid flow phenomena in shallow water scenarios, offering high accuracy and low computational cost.

Analytical behavior of soliton solutions to the couple type fractional-order nonlinear evolution equations utilizing a novel technique

Shallow water waves are one of the most prominent and widely used waves in coastal areas. The anomalous bi-directional transmission of long waves over shallow water is described by the nonlinear fractional partial differential equations (NLFPDEs), namely space-time-fractional coupled Whitham-Broer-Kaup (WBK) and coupled approximate long water (ALW) wave equations. To achieve new analytical solutions, the enhanced extended tanh-function technique for the mentioned fractional equations in the sense of conformable derivative is used. Some well-known wave shapes of solitons, namely, kink, bell-shape, and other types, through the suggested technique are achieved, and the 3D, contour, list point plots, and vector plots of the solutions are sketched to further illustrate the wave profile. The results achieved in this study have been validated using the computational software Maple and found to be accurate. It is observed that the method is realistic, and dependable to investigate more generalized analytical wave solutions.

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.

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.

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 multiple soliton similariton‐pair solutions, conservation laws via multiplier and stability analysis for the Whitham–Broer–Kaup equations in weakly dispersive media

In this article, the generalized unified method (GUM) is used for finding multiwave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equation with variable coefficients. Which describes the propagation of of shallow water waves. Here, we study the effects of the indirect nonlinear interaction of one‐, two‐ and three‐solitonic similaritons on the behavior of propagation of waves, in quasi‐periodic distributed system. This study can unable us to control the dynamics of type soliton (soliton, anti‐soliton) similaritons waves in dispersive waveguides. To give more physical insight to the obtained solutions, they are shown graphically. Their different structures are depicted by taking appropriate arbitrary functions. Further, with the suitable parameters, the indirect nonlinear interaction between two and three‐soliton waves are shown weal, in the sense that their amplitude does not blow up. Moreover, because of the importance of conservation laws Cls and stability analysis SA in the investigation of integrability, internal properties, existence, and uniqueness of a differential equation, we compute the Cls via multiplier technique and stability analysis via the concept of linear stability analysis for the WBK equations using the constant coefficients.

Symmetry Reduction Related by Nonlocal Symmetry and Explicit Solutions for the Whitham-Broer-Kaup System

The nonlocal symmetry for the Whitham-Broer-Kaup (WBK) equation is obtained by the Painleve analysis. With the introduction of two auxiliary dependent variables, the nonlocal symmetry is localized to the Lie point symmetry. The reciprocal transformation is established by the localization nonlocal symmetry procedure and can be used to construct a link between the WBK equation and the usual Burgers equation. By using the similarity reductions related with the nonlocal symmetry of the Burgers equation, we derive the symmetry invariant solutions for the WBK equation. Furthermore, the power-series solutions are computed by using the power-series theory.

Some exact solutions of the nonlinear space–time fractional differential equations

ABSTRACTIn this article, we construct exact soliton solutions of the nonlinear space–time fractional Klein–Gordon equation, Hirota–Satsuma coupled KdV (HSC KdV) system, coupled Whitham–Broer–Kaup (WBK) system, and KdV–Zakharov–Kuznetsov (KdV-ZK) equation using exponential rational function method. Our solutions include soliton-like, singular soliton, and kink solutions. Two-dimensional and three-dimensional graphical representation of all solutions is also presented to understand the accuracy of the technique. Moreover, a comparison of the recent solutions is made with some already existing solutions. It has been shown that the suggested method will be helpful in finding the solutions of different kind of nonlinear fractional differential equations emerging in mathematical physics. Throughout the paper, all the calculations are made with the aid of the Maple software.

New solitary solutions of the Gardner equation and Whitham–Broer–Kaup equations by the modified simplest equation method

In this paper, the modified simplest equation method (SEM) is employed to handle the Gardner equation and Whitham-Broer-Kaup (WBK) equations, respectively. Five new solitary solutions are accordingly obtained. The paper proceeds step-by-step with increasing detail about derivation process, first illustrating the algorithm of the modified SEM, which the simplest equation and its exact solution are not pre-defined. Then, new solutions are used to illustrate the connection between the Gardner equation and WBK equations. In addition, new solutions are reconstructed as solutions for the VBEs, ALWEs, the dispersive long wave equations, and the modified KdV equation.

A novel method for finding new multi-soliton wave solutions of the completely integrable equations

This paper presents a novel simplest equation method (SEM) to construct multi-soliton wave solutions. The paper proceeds step-by-step with increasing detail about derivation process, first illustrating the algorithm of the SEM and then exploiting an even deeper connection between Burgers’ equation and multi-soliton solutions and the simplest equation. Comparing with the Hirota’s method, our method provides the possibility for constructing multi-soliton solutions of the completely integrable equations with variable coefficients concisely. Applying such a method to the Gardner equation and the Whitham-Broer-Kaup (WBK) shallow water model, new general forms of multi-soliton solutions are obtained.

Numerical Solution of Nonlinear Whitham-Broer-Kaup Shallow Water Model Using Finite Difference Methods

In this paper, we presented finite difference methods for solving nonlinear Whitham-Broer-Kaup (WBK) shallow water model numerically. We first subdivided the domain of the model by a net with a finite number of mesh points, and the derivative at each point replaced by explicit, Crank-Nicolson, and exponential finite difference approximations. The result is the system of algebraic equations which when solved, provide an approximation to the solutions of WBK model at the selected grid points. Also, a comparison has been made between the approximate solutions obtained by the proposed methods and the exact solutions. Numerical results represented in tables and figures with the help of MATLAB R2015a.

Nonlocal Symmetry, CTE Solvability and Interaction Solutions of Whitham–Broer–Kaup Equations**Supported by the Key Foundation of Anhui Education Bureau under Grant No. KJ2013A028, the 211 Project of Anhhui University under Grant No. J18520104, Scientific Training Project for University Students, National Natural Science Foundation of China under Grant Nos. 11471015, 11571016, and Natural Science Foundation of Anhui Province under Grant No. 1408085MA02

Whitham–Broer–Kaup (WBK) equations in the shallow water small-amplitude regime is hereby under investigation. Nonlocal symmetry and Bäcklund transformation are presented via the truncated Painlevé expansion. This residual symmetry is localised to Lie point symmetry by the properly enlarged system. The finite symmetry transformation of the prolonged system is computed. Based on the CTE method, WBK equations are linearized and new analytic interaction solutions between solitary waves and cnoidal waves are given with the aid of solutions for the linear equation.

New Double Wronskian Solutions of the Whitham-Broer-Kaup System: Asymptotic Analysis and Resonant Soliton Interactions

In this paper, by the Darboux transformation together with the Wronskian technique, we construct new double Wronskian solutions for the Whitham-Broer-Kaup (WBK) system. Some new determinant identities are developed in the verification of the solutions. Based on analyzing the asymptotic behavior of new double Wronskian functions as t → ±∞, we make a complete characterization of asymptotic solitons for the non-singular, non-trivial and irreducible soliton solutions. It turns out that the solutions are the linear superposition of two fully-resonant multi-soliton configurations, in each of which the amplitudes, velocities and numbers of asymptotic solitons are in general not equal as t → ±∞. To illustrate, we present the figures for several examples of soliton interactions occurring in the WBK system.

Residual symmetries and interaction solutions for the Whitham–Broer–Kaup equation

The nonlocal residual symmetry for the Whitham–Broer–Kaup (WBK) equation is derived by the truncated Painleve analysis. The nonlocal residual symmetry is localized to the Lie point symmetry by introducing the auxiliary dependent variables. By using Lie’s first theorem, we obtain the finite transformation for the localized residual symmetry. Based on the consistent tanh expansion method (CTE), some exact interaction solutions among different nonlinear excitations are explicitly presented. Some special interaction solutions are investigated in both analytical and graphical ways.

Conservation Laws and Hamiltonian Symmetries of Whitham-Broer-Kaup Equations

In the present paper, conservation laws of the tri-Hamiltonain system of equations Whitham-Broer-Kaup (WBK) are investigated by applying the first homotopy formula. Hamiltonian symmetries of the system are constructed by using the corresponding Hamiltonian operators and the conserved densities.

A New Fractional Projective Riccati Equation Method for Solving Fractional Partial Differential Equations

In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space-time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained.

Solution of coupled Whitham–Broer–Kaup equations using optimal homotopy asymptotic method

In this work, an approximate method called Optimal Homotopy Asymptotic Method (OHAM) is proposed and implemented for the numerical solution of Whitham–Broer–Kaup (WBK) equations with blow-up and periodic solutions. Results obtained through OHAM are compared with the exact as well as with the results available in the literature. It was revealed that only second-order OHAM solutions are sufficient to achieve the desired accuracy in comparison to the higher order solutions of the methods we have made comparison with.