Laplace Decomposition Method Research Articles

A Note on Double Conformable Laplace Transform Method and Singular One Dimensional Conformable Pseudohyperbolic Equations

The purpose of this article is to obtain the exact and approximate numerical solutions of linear and nonlinear singular conformable pseudohyperbolic equations and conformable coupled pseudohyperbolic equations through the conformable double Laplace decomposition method. Further, the numerical examples were provided in order to demonstrate the efficiency, high accuracy, and the simplicity of present method.

On Some Initial and Initial Boundary Value Problems for Linear and Nonlinear Boussinesq Models

The main concern of this paper is to apply the modified double Laplace decomposition method to a singular generalized modified linear Boussinesq equation and to a singular nonlinear Boussinesq equation. An a priori estimate for the solution is also derived. Some examples are given to validate and illustrate the method.

An approximate solution of a singular fractional telegraph equation by double Laplace decomposition method

An approximate solution of a singular fractional telegraph equation by double Laplace decomposition method

Numerical simulation of the fractional Lienard’s equation

PurposeThe purpose of this paper is to apply an efficient semi-analytical method for the approximate solution of Lienard’s equation of fractional order.Design/methodology/approachA Laplace decomposition method (LDM) is implemented for the nonlinear fractional Lienard’s equation that is complemented with initial conditions. The nonlinear term is decomposed and then a recursive algorithm is constructed for the determination of the proposed infinite series solution.FindingsA number of examples are tested to explicate the efficiency of the proposed technique. The results confirm that this approach is convergent and highly accurate by using only few iterations of the proposed scheme.Originality/valueThe approach is original and is of value because it is the first time that this approach is used successfully to tackle fractional differential equations, which are of great interest for authors in the recent years.

Comparison of Numerical Methods for SWW Equations

Abstract: In this paper we consider three methods of approximation for the nonlinear water wave equation. In particular we are interested of KdV equation as a stationary water wave. The first is the method of approximation with a polynomial, the second method is the finite–volume method and the third method is Laplace decomposition method (LDM). A comparison between the methods is mentioned in this article.
 We treat the considered methods comparing the obtained solutions with the exact ones. We give in particular the numerical results compared with the analytical results. We show that the used methods are effective and convenient for solving the water wave equations. We can propose and sure that the method of approximation with a polynomial gives accurate results.

New approach to solving the atmospheric pollutant dispersion equation using fractional derivatives

In atmospheric environments, traditional differential equations do not adequately describe the problem of turbulent diffusion because the usual derivatives are not well defined in the non-differentiable behaviour introduced by turbulence, where the fractional calculation has become a very useful tool for studying anomalous dispersion and other transportation processes. Considering a new direction, this paper presents an analytical series solution of a three-dimensional advection–diffusion equation of fractional order, in the Caputo sense, applied to the dispersion of atmospheric pollutants. The solution is obtained by applying the generalised integral transform technique (GITT), solving the transformed problem by the Laplace decomposition method (LDM), and considering the lateral and vertical turbulent diffusion dependence on the longitudinal distance from the source, as well as a fractional parameter. The fractional solution is more general than the traditional solution in the sense that consideration of the integer order of the fractional parameter yields the traditional solution. The solution considers the memory effect in eddy diffusivity and in the fractional derivative, and it is simple, easy to implement, and converges rapidly. Numerical simulations were conducted to compare the performance of the proposed fractional solution to the traditional solution using an experimental dataset and other models, which also made it possible to find a better parametrisation for use in Gaussian models. The best results are obtained with the fractional order of the derivative.

Solving Helmholtz Equation with Local Fractional Derivative Operators

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

Numerical solution of nonlinear fractional Zakharov–Kuznetsov equation arising in ion-acoustic waves

The main purpose of this work is to suggest an efficient hybrid computational technique, namely the q-homotopy analysis transform method (q-HATM) to find the solution of the nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. The uniqueness and convergence analysis of the nonlinear time-FZK equation is presented. The Laplace decomposition method (LDM) is also employed to get the approximate solution of the nonlinear FZK equation. We implemented these techniques on two numerical examples, plotted the solution and compared the absolute error with the variational iteration technique and homotopy perturbation transform technique to show the efficiency of these techniques.

Solution of singular one-dimensional Boussinesq equation by using double conformable Laplace decomposition method

In this study the method which was obtained from a combination of the conformable fractional double Laplace transform method and the Adomian decomposition method has been successfully applied to solve linear and nonlinear singular conformable fractional Boussinesq equations. Two examples are given to illustrate our method. Furthermore, the results show that the proposed method is effective and is easy to use for certain problems in physics and engineering.

Approximate Solutions of the Damped Wave Equation and Dissipative Wave Equation in Fractal Strings

In this paper, we apply the local fractional Laplace variational iteration method (LFLVIM) and the local fractional Laplace decomposition method (LFLDM) to obtain approximate solutions for solving the damped wave equation and dissipative wave equation within local fractional derivative operators (LFDOs). The efficiency of the considered methods are illustrated by some examples. The results obtained by LFLVIM and LFLDM are compared with the results obtained by LFVIM. The results reveal that the suggested algorithms are very effective and simple, and can be applied for linear and nonlinear problems in sciences and engineering.

Aboodh Homotopy Perturbation Method of solving Burgers Equation.

In this paper, we present a reliable combination of Aboodh Transform and Homotopy perturbation method to determine the exact solution of one dimensional Burgers equation which is a nonlinear partial differential equation. Some cases of one dimensional nonlinear partial differential equations are considered to illustrate the capability and reliability of Aboodh Homotopy perturbation method. We have compared the analytical solution obtained with the available Laplace decomposition method. The solution which is found to be exactly the same. The result revealed that the combination of the Aboodh Transform and homotopy perturbation method is quite capable, practically well appropriate for use in such problems. The method is tested on three examples and results show that the new method is more effective and convenient to use and high accuracy of it is evident.

On Conformable Double Laplace Transform and One Dimensional Fractional Coupled Burgers’ Equation

In the present work we introduced a new method and name it the conformable double Laplace decomposition method to solve one dimensional regular and singular conformable functional Burger’s equation. We studied the existence condition for the conformable double Laplace transform. In order to obtain the exact solution for nonlinear fractional problems, then we modified the double Laplace transform and combined it with the Adomian decomposition method. Later, we applied the new method to solve regular and singular conformable fractional coupled Burgers’ equations. Further, in order to illustrate the effectiveness of present method, we provide some examples.

Laplace Decomposition Method for the System of Non Linear PDEs

The Laplace Decomposition Method [1] [2] [3] is applied to a system of nonlinear partial differential equations to demonstrate potential applicability to such systems.

Nonlinear Partial Differential Equations, Laplace Decomposition Method, Differential Transform Method, Runge-Kutta Method of 6th Order

Nonlinear Partial Differential Equations, Laplace Decomposition Method, Differential Transform Method, Runge-Kutta Method of 6th Order

Computational methods based laplace decomposition for solving nonlinear system of fractional order differential equations

In this paper, we considered nonlinear systems of fractional order differential equations. They have been solved by a computational methods which are so-called Laplace Adomian decomposition method (LADM) and modified Laplace decomposition method (MLDM). The fractional derivatives are described in the Caputo sense. The (LADM) and the (MLDM) are a combination of the Laplace transform and the Adomian decomposition method and iterative method respectively. These techniques were applied for some illustrative examples in order to solve nonlinear systems of fractional order differential equations. From the results of the illustrative examples we conclude that these methods are computationally efficient.

Mathematical Model Using Fractional Derivatives Applied to the Dispersion of Pollutants in the Planetary Boundary Layer

We present an analytical solution of the advection–diffusion equation of integer and fractional order applied to the dispersion of pollutants in the planetary boundary layer. The solution is obtained using the Laplace decomposition method, and the perturbation is obtained by homotopy, considering the Caputo derivative in the fractional case. To obtain the solution, two types of eddy diffusivities are used: in the integer-order equation, the eddy diffusivity is dependent on the longitudinal distance from the source (K $$ \propto $$ x and K $$ \propto $$ x2); in the fractional-order equation, the eddy diffusivity is constant. To validate the model, the results are compared with experimental data from the literature (Copenhagen and Prairie Grass). In the Copenhagen experiment, which was conducted under moderately unstable conditions, the best results are obtained under the influence of the memory effect with the eddy diffusivity dependent upon the source distance as K $$ \propto $$ x (with constant eddy diffusivity in the equation with a fractional derivative). However, in the strongly convective case of the Prairie Grass experiment, the best results are obtained only when the eddy diffusivity depends on the source distance as K $$ \propto $$ x2.

Solving Nonlinear Equations Using the Laplace Decomposition Method: An Application to Projectile Motion with Resistive Term

Objectives: An approximate solution of the problem of two-dimensional projectile motion with linear and quadratic air resistance obtained using the Laplace decomposition method LDM. Methods/Statistical Analysis: In this study, we examined the Laplace Decomposition Method for obtaining a solution of the trajectory problem of a body in linear and quadratic resistant medium. Findings: The solutions are obtained by using developed technique that combine Laplace Transform and the Adomian Decomposition Method ADM. The all obtained solutions are plotted to compare the effect of the nonlinear term on the basic parameters like range and maximum high and other parameters. Application/Improvements: The Laplace Decomposition Method improves the results to achieve more precision results compared to the previous related works. Keywords: Application, Laplace Decomposition Method, Nonlinear Equations, Projectile Motion, Resistive Term

A Model Using Fractional Derivatives with Vertical Eddy Diffusivity Depending on the Source Distance Applied to the Dispersion of Atmospheric Pollutants

This work presents an analytical solution of the two-dimensional advection–diffusion equation of fractional order, in the sense of Caputo and applied it to the dispersion of atmospheric pollutants. The solution is obtained using Laplace decomposition and homotopy perturbation methods, and it considers the vertical eddy diffusivity dependency on the longitudinal distance of the source with fractional exponents of the same order of the fractional derivative (K $$\propto$$ xα). For validation of the model, simulations were compared with data from Copenhagen experiments considering moderately unstable conditions. The best results were obtained with α = 0.98, considering wind measured at 10 m, and α = 0.94 with wind measured at a height of 115 m.

Existence result and approximate solutions for quadratic integro-differential equations of fractional order

This research outlines a reliable strategy for finding a solutions of nonlinear quadratic integro-differential equation of fractional order (FQIDEs). Local and global existence theorems of solutions of the FQIDEs have been obtained by using Schauder’s and Tychonoff fixed point theorems. The fractional derivative is described in the Caputo sense. The Laplace decomposition method (LDM) and modified Adomian decomposition method (MADM) are described to be fast and accurate. Illustrative examples are included to demonstrate the efficiency and reliability of presented techniques.

A note on double Laplace decomposition method for solving singular one dimensional pseudo thermo-elasticity coupled system

A note on double Laplace decomposition method for solving singular one dimensional pseudo thermo-elasticity coupled system