Application of Combine Natural Transform and Adomian Decomposition Method in Volterra Integro-Differential Equations

In this paper we introduced the combine Natural transform and Adomian decomposition method to solve the nonlinear voltrra integro-differential equations of first kind and second kind.To illustrate the method some examples are solved by using the above said method

Laplace Adomian Decomposition Method to study Chemical ion transport through soil

The paper deals with a theoretical study of chemical ion transport in soil under a uniform external force in the transverse direction, where the soil is taken as porous medium. The problem is formulated in terms of boundary value problem that consists of a set of partial differential equations, which is subsequently converted to a system of ordinary differential equations by applying similarity transformation along with boundary layer approximation. The equations hence obtained are solved by utilizing Laplace Adomian Decomposition Method (LADM). The merit of this method lies in the fact that much of simplifying assumptions need not be made to solve the non-linear problem. The decomposition parameter is used only for grouping the terms, therefore, the nonlinearities is handled easily in the operator equation and accurate approximate solution are obtained for the said physical problem. The computational outcomes are introduced graphically. By utilizing parametric variety, it has been demonstrated that the intensity of the external pressure extensively influences the flow behavior

Analytical solutions of some special nonlinear partial differential equations using Elzaki-Adomian decomposition method

We apply the Elzaki-Adomian Decomposition Method (EADM) in this study to solve nonlinear Benjamin-Bona-Mahony (BBM) and Fisher's partial differential equations (PDE). This method, being an integral transform, is a hybrid of two well-known and efficient methods: the Elzaki transform and the Adomian decomposition method. The method is demonstrated by solving two special cases of the BBM Equation and one special case of Fisher's partial differential equation. Because of its high convergence rate in approximating exact solutions, this approach is very dependable. The method can also produce numerical solutions without the usage of restrictive assumptions or the discretization typical of numerical methods; making it free of round-off errors. The Elzaki-Adomian Decomposition method employs a straightforward computation that leads to effectiveness. The efficiency of EADM is demonstrated in the significant reduction of number of numerical computations. The effectiveness and efficiency of EADM account for its broad application, particularly for higher order PDEs

Laplace Decomposition Method for the System of Linear and Non-Linear Ordinary Differential Equations

In this paper we use Modified form of Adomian’s Decomposition Method Laplace, which is a mixture of Laplace transforms and Adomian’s Decomposition Method called the Laplace Decomposition Method (LDM) to solve the system of ordinary differential equation of the first order and an ordinary differential equation of any order by converting it into a system of differential equation of order one. Some examples are presented to show the ability of the method for linear and non-linear systems of differential equations also present the comparison of their solution with the exact solution through graphically. Keywords: Laplace Transformation, Adomian’s Decomposition Method (ADM), System of differential equation, linear differential equation and non-linear ordinary differential equation

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems

SOLUSI SEMI ANALITIK PERSAMAAN BURGERS MENGGUNAKAN METODE DEKOMPOSISI ADOMIAN LAPLACE

Persamaan Burgers adalah persamaan diferensial parsial yang penting pada mekanika fluida. Karena mempunyai bentuk nonlinear dalam persamaannya, solusi eksak dari persamaan tersebut sulit untuk dicari sehingga terus dikembangkan beragam metode untuk mencari solusi hampiran yang dapat mendekati solusi eksaknya. Dalam penelitian ini, metode dekomposisi Adomian Laplace digunakan untuk mencari solusi hampiran dari persamaan Burgers. Metode tersebut adalah metode semi analitik dalam penyelesaian persamaan diferensial nonlinear. Solusi hampiran diperoleh dalam bentuk deret tak hingga berdasarkan kondisi awal yang diberikan dengan menguraikan bagian nonlinear dalam persamaan menggunakan polinomial Adomian dan dikombinasikan dengan penggunaan transformasi Laplace. Metode tersebut telah digunakan dalam penyelesaian persamaan Burgers. Solusi hampiran yang diperoleh disimulasi menggunakan perangkat lunak Maple dan dibandingkan dengan solusi eksak serta penelitian sebelumnya yang menggunakan metode dekomposisi Adomian tanpa transformasi Laplace. Diperoleh hasil bahwa solusi hampiran yang diperoleh dari metode ini dapat mendekati solusi eksak dengan jumlah galat mutlak dan relatif yang lebih kecil dibandingkan solusi hampiran dengan metode dekomposisi Adomian, sehingga dapat disimpulkan metode dekomposisi Adomian Laplace lebih akurat dibandingkan dengan metode dekomposisi Adomian dalam menghampiri solusi eksak dari persamaan Burgers. Burgers equation is a partial differential equation which has important rule in fluid mechanics. Because it has nonlinear term inside the equation, the exact solution is complicated to find, therefore many methods have been developed to find the approximate solution that can estimate the exact solution. In this research, the Laplace Adomian decomposition method is applied to calculate the approximate solution of Burgers equation. The method is a semi analytical method to resolve nonlinear differential equation. The approximate solution is acquired in the form of infinite series according to the initial condition, by decompose the nonlinear term in the equation with Adomian polynomial and combined with the use of the Laplace transform. The proposed method has been used to resolve the Burgers equation. The approximate solution is simulated using Maple software and compared with the exact solution also with the previous research which used the Adomian decomposition method without the Laplace transform. The approximate solution which obtained by this method can estimate the exact solution with the sum of absolute and relative error less than the approximate solution obtained by the Adomian decomposition method, therefore the Laplace Adomian decomposition method is more accurate than the Adomian decomposition method in order to estimate the exact solution of the Burgers equation