Approximation Of Wave Equation Research Articles

Utilizing the Optimal Auxiliary Function Method for the Approximation of a Nonlinear Long Wave System considering Caputo Fractional Order

In this article, we explore the utilization of the Caputo derivative and the Riemann–Liouville (R–L) fractional integral to analyze the optimal auxiliary function method for approximating fractional nonlinear long waves. Approximate long wave equation with a distinct dispersion relation offers the most accurate description of shallow water wave properties. Various methods, including the Adomian decomposition technique, the variational iteration method, the optimum homotopy asymptotic method, and the new iterative technique, have been employed and compared to those obtained using the fractional-order approximate long wave equation. The results of our study indicate that the optimal auxiliary function method is highly successful and practically simple, achieving better and more rapid convergence after just one repetition. This method is recognized as an efficient approach, demonstrating high precision in solving intriguing and intricate problems. Furthermore, it proves to be more time and resource efficient than other relevant analytical techniques, leading to significant savings in both volume and time. Compared to the Adomian decomposition technique, the new iterative technique, the variational iteration method, and the optimum homotopy asymptotic method, the suggested technique is extremely accurate computationally. It is also easy to analyze and solve fractionally linked nonlinear complex phenomena that arise in science and technology. We present the numerical and graphical findings that support these conclusions.

On the accuracy of one-way approximate models for nonlinear waves in soft solids.

Simple strain-rate viscoelasticity models of isotropic soft solid are introduced. The constitutive equations account for finite strain, incompressibility, material frame-indifference, nonlinear elasticity, and viscous dissipation. A nonlinear viscous wave equation for the shear strain is obtained exactly and corresponding one-way Burgers-type equations are derived by making standard approximations. Analysis of the travelling wave solutions shows that these partial differential equations produce distinct solutions, and deviations are exacerbated when wave amplitudes are not arbitrarily small. In the elastic limit, the one-way approximate wave equation can be linked to simple wave theory and shock wave theory, thus, allowing direct error measurements.

Numerical simulation of coupled fractional‐order Whitham‐Broer‐Kaup equations arising in shallow water with Atangana‐Baleanu derivative

In this manuscript, fractional nonlinear coupled Whitham‐Broer‐Kaup equation associated with Atangana‐Baleanu fractional derivative is considered. General conditions under which a system solution exists and is unique are presented using the fixed‐point theorem method. For numerical simulations, coupled fractional modified Boussinesq equations and coupled fractional approximate long wave equations are investigated using the homotopy perturbation transform technique (HPTT). The suggested technique is an elegant compilation of the Laplace transform technique with the homotopy perturbation approach. The physical behavior of the obtained solutions has been presented graphically as well as in tables for diverse fractional order. Comparative simulations have been performed to validate the efficiency and accuracy of the suggested technique.

New Classifications of All Single Traveling Wave Solutions of the Fractional Approximate Long Water Wave Equations by Using the Complete Discrimination System

First, our work is to transform the space-time fractional approximate long water wave equations into nonlinear ordinary differential equations via the traveling wave transformation in the sense of conformable fractional derivative. Second, we simplify the nonlinear ordinary differential equations into an ordinary differential equation with only one variable by integration and some transformations. Finally, we can further get all single traveling wave solutions of the space-time fractional approximate long water wave equations by the complete discrimination system for the four-order polynomial method; these solutions include the hyperbolic function solutions, rational function solutions, and implicit solutions.

On the solution of fractional modified Boussinesq and approximate long wave equations with non-singular kernel operators

<abstract><p>In this paper, we used the Natural decomposition approach with nonsingular kernel derivatives to explore the modified Boussinesq and approximate long wave equations. These equations are crucial in defining the features of shallow water waves using a specific dispersion relationship. In this research, the convergence analysis and error analysis have been provided. The fractional derivatives Atangana-Baleanu and Caputo-Fabrizio are utilised throughout the paper. To obtain the equations results, we used Natural transform on fractional-order modified Boussinesq and approximate long wave equations, followed by inverse Natural transform. To verify the approach, we focused on two systems and compared them to the exact solutions. We compare exact and analytical results with the use of graphs and tables, which are in strong agreement with each other, to demonstrate the effectiveness of the suggested approaches. Also compared are the results achieved by implementing the suggested approaches at various fractional orders, confirming that the result comes closer to the exact solution as the value moves from fractional to integer order. The numerical and graphical results show that the suggested scheme is computationally very accurate and simple to investigate and solve fractional coupled nonlinear complicated phenomena that exist in science and technology.</p></abstract>

Viscoelastic Wave Propagation Simulation Using New Spatial Variable-Order Fractional Laplacians

ABSTRACTTime-domain constant-Q (CQ) viscoelastic wave equations have been derived to efficiently model Q, but are known to break down in accuracy in describing CQ attenuation at low Q. In view of this, a new time-domain viscoelastic wave equation for modeling wave propagation in anelastic medium is evaluated based on Kjartansson’s CQ model to improve the accuracy in describing CQ attenuation at low Q. We use an approximate frequency-domain viscoelastic wave equation to replace the accurate frequency-domain viscoelastic wave equation. Then, a new time-domain wave equation is derived by converting the approximate viscoelastic wave equation from the frequency domain to the time domain. The newly derived viscoelastic wave equation consists of several Laplacian differential operators with variable fractional order. We use an arbitrary-order Taylor series expansion (TSE) to approximate the derived mixed domain fractional Laplacian operators, and realize the decoupling of the wavenumber and fractional order. Then, the proposed viscoelastic wave equation can be solved directly using the staggered-grid pseudospectral method (SGPSM). We evaluate the precision of the new viscoelastic wave equation by comparing the numerical solutions with the analytical solutions in homogeneous medium. Theoretical curve analysis and numerical results indicate that the proposed fractional viscoelastic wave equation has higher precision in describing CQ attenuation than that of the traditional fractional viscoelastic wave equation, especially for cases that P-wave quality factor QP is less than 10, and S-wave quality factor QS is less than 8. Furthermore, we use two numerical examples to verify the effectiveness of the TSE SGPSM in heterogeneous media. The discussion shows that the advantage of using our fractional viscoelastic wave equation over the traditional fractional viscoelastic wave equation is the higher precision in describing CQ attenuation at different frequency.

SOLITARY WAVE SOLUTIONS OF SOME CONFORMABLE TIME-FRACTIONAL COUPLED SYSTEMS VIA AN ANALYTIC APPROACH

The main purpose of this research is to inquire the new solitary wave solution of the coupled time-fractional models to validate the influence and proficiency of the planned variational iteration method (VIM) using conformable derivative definition. Applications to four demanding nonlinear problems like Hirota-Satsuma coupled KdV equations, modified Boussinesq (MB) equation, approximate long wave (ALW) equation and Drinfeld-Sokolov-Wilson (DSW) equation demonstrate the efficiency and the robustness of the method. An analysis of the consequences with effects of relevant parameters and comparison with the exact solution presented with the help of graphs tables and gives the further understanding of numerical results by others. The convergence of the method is illustrated numerical and their physical significance is discussed

Theoretical framework for geoacoustic inversion by adjoint method**Project supported by the Foundation of Key Laboratory of Marine Intelligent Equipment and System of Ministry of Education, China (Grant No. SJTU-MIES1908) and the National Natural Science Foundation of China (Grant No. 41775027).

Traditional geoacoustic inversions are generally solved by matched-field processing in combination with meta-heuristic global searching algorithms which usually need massive computations. This paper proposes a new physical framework for geoacoustic retrievals. A parabolic approximation of wave equation with non-local boundary condition is used as the forward propagation model. The expressions of the corresponding tangent linear model and the adjoint operator are derived, respectively, by variational method. The analytical expressions for the gradient of the cost function with respect to the control variables can be formulated by the adjoint operator, which in turn can be used for optimization by the gradient-based method.

A reliable technique for fractional modified Boussinesq and approximate long wave equations

In this paper, an efficient technique is employed to study the modified Boussinesq and approximate long wave equations of the Caputo fractional time derivative, namely the q-homotopy analysis transform method. These equations play a vital role in describing the properties of shallow water waves through distinct dispersion relation. The convergence analysis and error analysis are presented in the present investigation for the future scheme. We illustrate two examples to demonstrate the leverage and effectiveness of the proposed scheme, and the error analysis is discussed to verify the accuracy. The numerical simulation is conducted to ensure the exactness of the future technique. The obtained numerical and graphical results are presented, the proposed scheme is computationally very accurate and straightforward to study and find the solution for fractional coupled nonlinear complex phenomena arising in science and technology.

Analytical Solutions for Nonlinear Systems of Conformable Space-Time Fractional Partial Differential Equations via Generalized Fractional Differential Transform

The present paper introduces approximate analytical solutions for nonlinear systems of conformable space-time fractional partial differential equations (CSTFPDEs) by using a generalized conformable fractional partial differential transform (GCFPDT). The GCFPDT is a modified version of the conformable fractional partial differential transform introduced in our recent work and it can be used as an efficient alternative transform to find analytical solutions for nonlinear systems of CSTFPDEs available in the literature. The convergence and error estimation of the proposed GCFPDT are also considered. Moreover, approximate analytical solutions to nonlinear system of gas dynamic equations, nonlinear system of KdV equations, and nonlinear system of approximate long water wave equations in the sense of conformable space-time fractional partial derivatives are successfully obtained to confirm the effectiveness and efficiency of the proposed GCFPDT.

New analytic solutions of the space-time fractional Broer–Kaup and approximate long water wave equations

In the present paper, the exp(−ϕ(ξ)) expansion method is applied to the fractional Broer–Kaup and approximate long water wave equations. The explicit approximate traveling wave solutions are obtained by using this method. Here, fractional derivatives are defined in the conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulations of the obtained solutions are given at the end of the paper.

New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water

The aim of the article is to construct exact solutions for the time fractional coupled Boussinesq–Burger and approximate long water wave equations by using the generalized Kudryashov method. The fractional differential equation is converted into ordinary differential equations with the help of fractional complex transform and the modified Riemann–Liouville derivative sense. Applying the generalized Kudryashov method through with symbolic computer maple package, numerous new exact solutions are successfully obtained. All calculations in this study have been established and verified back with the aid of the Maple package program. The executed method is powerful, effective and straightforward for solving nonlinear partial differential equations to obtain more and new solutions with the integer and fractional order.

Numerical and Experimental Study of the Sound-Insulating Properties of a Deformable Plate Located between Two Chambers

The paper deals with the numerical solution of acoustic wave propagation plane problem. Being formed by sound source in a high-pressure chamber, the acoustic wave propagates through a deformable thin plate, which radiates the sound wave in a low-pressure chamber. The considered problem represents the mathematical modelling of the experimental setup used to find sound-insulating properties of thin-walled structures realized by the method of adjacent reverberation chambers in acoustic test laboratories. Based on two dimensional approximation of wave equations, two problem schemes are introduced. These problems differ in terms of sound source specification in the high-pressure chamber. Present study shows that both of the considered problems lead to virtually same results when determining the plate's sound reduction index. The results of theoretical and experimental investigations are compared.

Subdomain Propagation Method for an Efficient Modal Analysis of Photonic Waveguides

The modal analysis is needed for the design of photonic waveguide devices to determine propagation constants of guided modes and their field pattern. Rigorous methods based on finite-difference or finite-element methods for the approximation of wave equations rely on the solution of eigenvalue problems. Imaginary-distance beam propagation methods require the solution of matrix equations of high order. The computation time needed for both approaches can be tedious. A modal analysis method based on a beam propagation scheme is proposed, which allows an efficient computation of field patterns and propagation constants, as the resulting algorithm is based on simple matrix vector multiplications. This allows the choice of a large number of discretization points and a dense discretization. The beam propagation operator is designed to include the eigenvalue spectrum of corresponding guided modes only. For this reason, the method can be interpreted as a subdomain propagation method. The concept will be validated by utilizing a single-mode and multimode rib waveguide.

One-way approximation for the simulation of weak shock wave propagation in atmospheric flows.

A numerical scheme is developed to simulate the propagation of weak acoustic shock waves in the atmosphere with no absorption. It generalizes the method previously developed for a heterogeneous medium [Dagrau, Rénier, Marchiano, and Coulouvrat, J. Acoust. Soc. Am. 130, 20-32 (2011)] to the case of a moving medium. It is based on an approximate scalar wave equation for potential, rewritten in a moving time frame, and separated into three parts: (i) the linear wave equation in a homogeneous and quiescent medium, (ii) the effects of atmospheric winds and of density and speed of sound heterogeneities, and (iii) nonlinearities. Each effect is then solved separately by an adapted method: angular spectrum for the wave equation, finite differences for the flow and heterogeneity corrections, and analytical method in time domain for nonlinearities. To keep a one-way formulation, only forward propagating waves are kept in the angular spectrum part, while a wide-angle parabolic approximation is performed on the correction terms. The numerical process is validated in the case of guided modal propagation with a shear flow. It is then applied to the case of blast wave propagation within a boundary layer flow over a flat and rigid ground.

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.

Finite-difference approximation of wave equation: a study case of the SIMA velocity model

Synthetic seismograms enable to model the theoretical seismic response of the Earth interior due to different structural features and changes in the physical properties of crust and mantle. This approximation provides a best understanding of the real seismic data recorded in field experiments. In this paper, we are showing the development and application of a new scheme based on a multi-order explicit finite-difference algorithm for acoustic waves in a 2D heterogeneous media. The results of the modeling are compared with the seismic data acquired within the SIMA project providing new insight about the internal structure of the subsurface allowing improving the velocity model obtained in previous works.

Wave beams in smoothly inhomogeneous anisotropic media: a single-mode description (Part I)

In this series of papers, we propose a method for the development of an approximate solution of the Maxwell equations in smoothly inhomogeneous anisotropic gyrotropic media with allowance for aberrations, spatial dispersion, and absorption. The connection with the aberration-free approximation equations is shown. In the first paper of the series, we propose a procedure for the development of an approximate scalar wave equation aimed at description of the single-mode propagation of wave beams.

On The Solving Nonlinear Approximate Long Wave Equations

In this letter, He's Variational Iteration Method (VIM) is implemented for solving the nonlinear Whitham-Broer-Kaup partial differential equations in the special case is named approximate long wave equations (ALW). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which those obtained by the Adomian decomposition method (ADM). The work confirms that the VIM method is superior and very faster to the ADM .

The improved [formula omitted]-expansion method and its applications to the Broer–Kaup equations and approximate long water wave equations

By introducing a new general ansätze, the improved G ′ G -expansion method is proposed to construct exact solutions of both Broer–Kaup equations and approximate long water wave equations. As a result, some new travelling wave solutions involving parameters, expressed by three types of functions which are the hyperbolic functions, the trigonometric functions and the rational functions, are obtained. When the parameters are taken as special values, the solitary wave solutions are derived from the hyperbolic function solutions. The proposed method is straightforward, concise and effective, and can be applied to other nonlinear evolution equations in mathematical physics.