scholarly journals Solutions of Volterra integral and integro-differential equations using modified Laplace Adomian decomposition method

2019 ◽  
Vol 15 (1) ◽  
pp. 5-18
Author(s):  
D. Rani ◽  
V. Mishra
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Convergence Analysis ◽  
Decomposition Method ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Modified Technique ◽  
Adomian Decomposition

Abstract In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.

scholarly journals Application of Sumudu Decomposition Method to Solve Nonlinear System of Partial Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Hassan Eltayeb ◽  
Adem Kılıçman
Keyword(s):  
Partial Differential Equations ◽  
Nonlinear System ◽  
Differential Equations ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Adomian Polynomials ◽  
Adomian Decomposition ◽  
Sumudu Transform

We develop a method to obtain approximate solutions of nonlinear system of partial differential equations with the help of Sumudu decomposition method (SDM). The technique is based on the application of Sumudu transform to nonlinear coupled partial differential equations. The nonlinear term can easily be handled with the help of Adomian polynomials. We illustrate this technique with the help of three examples, and results of the present technique have close agreement with approximate solutions obtained with the help of Adomian decomposition method (ADM).

scholarly journals Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

2021 ◽  
pp. 3679-3689
Author(s):  
Eman Mohmmed Namah
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Decomposition Method ◽  
Integral Transform ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Fractional Delay ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

     The idea of the paper is to consolidate Mahgoub transform and variational iteration method (MTVIM) to solve fractional delay differential equations (FDDEs). The fractional derivative was in Caputo sense. The convergences of approximate solutions to exact solution were quick. The MTVIM is characterized by ease of application in various problems and is capable of simplifying the size of computational operations.  Several non-linear (FDDEs) were analytically solved as illustrative examples and the results were compared numerically. The results for accentuating the efficiency, performance, and activity of suggested method were shown by comparisons with Adomian Decomposition Method (ADM), Laplace Adomian Decomposition Method (LADM), Modified Adomian Decomposition Method (MADM) and Homotopy Analysis Method (HAM).

scholarly journals Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Abdelhalim Ebaid
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Applied Mathematics ◽  
Mathematical Formulation ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Stationary Solutions ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
The Mathematical Model

The formation of liver zones is modeled by a system of two integropartial differential equations. In this research, we introduce the mathematical formulation of these integro-partial differential equations obtained by Bass et al. in 1987. For better understanding of this mathematical formulation, we present a medical introduction for the liver in order to make the formulation as clear as possible. In applied mathematics, the Adomian decomposition method is an effective procedure to obtain analytic and approximate solutions for different types of operator equations. This Adomian decomposition method is used in this work to solve the proposed model analytically. The stationary solutions (as time tends to infinity) are also obtained through it, which are in full agreement with those obtained by Bass et al. in 1987.

scholarly journals Solving Ordinary Differential Equation of Higher Order by Adomian Decomposition Method

2020 ◽  
pp. 44-53
Author(s):  
Sumayah Ghaleb Othman ◽  
Yahya Qaid Hasan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Higher Order ◽  
Initial Value ◽  
Adomian Decomposition ◽  
Different Types

Aims/ Objectives: In this article, we use Adomian Decomposition method (ADM) for solving initial value problems in the higher order ordinary differential equations. Many researchers have used the ADM in order to find convergent as well as exact solutions of different types of equations. Therefore, the ADM is considered as an effective and successful method for solving differential equations. In this paper, we presented some suggested amendments to the ADM by using a new differential operator in order to find solutions for higher order types of equations. We demonstrated the effectiveness of this method through many examples and we find out that we get an approximate solutions using the proposed amendments. We can conclude that the suggested modification of ADM is afftective and produces reliable results.

scholarly journals Adomian Decomposition Method with Modified Bernstein Polynomials for Solving Ordinary and Partial Differential Equations

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
Ahmed Farooq Qasim ◽  
Ekhlass S. AL-Rawi
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Bernstein Polynomials ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Adomian Decomposition ◽  
Productive System ◽  
Linear And Nonlinear

In this paper, we used Bernstein polynomials to modify the Adomian decomposition method which can be used to solve linear and nonlinear equations. This scheme is tested for four examples from ordinary and partial differential equations; furthermore, the obtained results demonstrate reliability and activity of the proposed technique. This strategy gives a precise and productive system in comparison with other traditional techniques and the arrangements methodology is extremely straightforward and few emphasis prompts high exact solution. The numerical outcomes showed that the acquired estimated solutions were in appropriate concurrence with the correct solution.

scholarly journals Adomian Decomposition Method with Orthogonal Polynomials: Laguerre Polynomials and the Second Kind of Chebyshev Polynomials

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1796
Author(s):  
Yingying Xie ◽  
Lingfei Li ◽  
Mancang Wang
Keyword(s):  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Chebyshev Polynomials ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Laguerre Polynomials ◽  
Adomian Decomposition Method ◽  
Numerical Results ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

In this paper, a new efficient and practical modification of the Adomian decomposition method is proposed with Laguerre polynomials and the second kind of Chebyshev polynomials which has not been introduced in other articles to the best of our knowledge. This approach can be utilized to approximately solve linear and nonlinear differential equations. The proposed formulations are examined by a representative example and the numerical results confirm their efficiency and accuracy.

scholarly journals Approximate Solutions for Systems of Volterra Integro-differential Equations Using Laplace –Adomian Method

2020 ◽  
pp. 2655-2662
Author(s):  
Firas S. Ahmed
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Iterative Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Approximate Solutions ◽  
Adomian Decomposition ◽  
Adomian Method ◽  
The Given

Some modified techniques are used in this article in order to have approximate solutions for systems of Volterra integro-differential equations. The suggested techniques are the so called Laplace-Adomian decomposition method and Laplace iterative method. The proposed methods are robust and accurate as can be seen from the given illustrative examples and from the comparison that are made with the exact solution.

Numerical Solution of Nonlinear Volterra Integro-Differential Equations of Fractional Order by Using Adomian Decomposition Method

2014 ◽  
Vol 635-637 ◽  
pp. 1582-1585
Author(s):  
Li Feng Wang ◽  
Yun Peng Ma ◽  
Yong Qiang Yang
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Convergence Analysis ◽  
Fractional Order ◽  
Decomposition Method ◽  
Series Solution ◽  
Adomian Decomposition Method ◽  
Computational Method ◽  
Numerical Example ◽  
Adomian Decomposition

In this work we present a computational method for for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on Adomian Decom-position Method. Convergence analysis is dependable enough to estimate the maximum absolute truncated error of the Adomian series solution. Numerical example is included to demonstrate the validity and applicability of the method.

scholarly journals Approximate Solutions of Fractional Riccati Equations Using the Adomian Decomposition Method

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Fei Wu ◽  
Lan-Lan Huang
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Riccati Equations ◽  
Crucial Step ◽  
Adomian Decomposition

The fractional derivative equation has extensively appeared in various applied nonlinear problems and methods for finding the model become a popular topic. Very recently, a novel way was proposed by Duan (2010) to calculate the Adomian series which is a crucial step of the Adomian decomposition method. In this paper, it was used to solve some fractional nonlinear differential equations.

Sign in / Sign up

Export Citation Format

Share Document