Modified Adomian Decomposition Method Research Articles

Free vibration analysis of non-uniform Bernoulli beam by using Laplace Adomian decomposition method

This paper discusses an investigation into the influence of the cone ratio and axial force on the vibration problem in a non-uniform cantilever Euler–Bernoulli beam. In the analysis, the governing equation for the non-uniform cantilever Euler–Bernoulli beam is solved using the Laplace Adomian decomposition method (LADM). The LADM is used to convert the governing equation into a characteristic equation of a non-uniform Euler–Bernoulli beam, and a simple calculation is performed to obtain the natural frequencies and corresponding modals. The obtained numerical results are verified using a comparison with the results reported in previous studies. The present study speeds up the convergent rate and the accuracy of calculation by comparing the results using the modified Adomian decomposition method (MADM) and differential transformation method (DTM). The main power and advantage of the LADM are providing an analytical approximation to a nonlinear differential equation without linearization, perturbation, approximation, and discretization, all of which lead to huge numerical computation. The numerical methods demonstrated that the natural frequency increases with increasing the rotating spring modulus and moving spring modulus, and the moving spring modulus has a greater influence on the natural frequency. The effects of the cone ratio and axial force are presented for non-uniform Euler–Bernoulli beams. The numerical results show that the LADM is a suitable technique for analyzing the behavioral characteristics of beams.

Efficient decomposition method for integro-differential equations

Different methods have been used in the solution of integro-differential equations. Many of these methods such as Standard Adomian Decomposition Method (SADM) take several iterations which might be difficult to solve and also consume time before getting an approximation. This present study developed a new Modified Adomian Decomposition Method (MADM) for Integro-Differential Equations. The modification was carried out by decomposing the source term function into series. The newly modified Adomian decomposition method (MADM) accelerates the convergence of the solution (MADM) faster the Standard Adomian Decomposition Method (SADM). This study recommends the use of MADM for solving Integro-Differential Equations.

Inelastic materials and mathematical variables for obstacle bridge problem evaluation

Nonlinear issues are tough to tackle, but this novel method simplifies the process. For this, many sorts of challenges were studied, including 4th Order Homogeneous Impediment, Homogeneous 2nd Order Continuous-Support Cable-Stayed Bridge and 8th Order Homogenous Multi-Support Cable-Stayed bridge. A simple supported continuous beam with four supports and varying loading conditions is used for this purpose. To improve the accurateness and conjunction of the problems stated above, a novel technique called the Modified Adomian Decomposition Method was applied to multi-support cable-stayed and continuous bridge problems. With the help of novel approaches such as MADM, ongoing support bridge problems can be quickly resolved. The Modified Adomian Decomposition Method results congregate very hastily and are closer to the exact solution. The paper employs the novel technique method to various initial value problems such as emotional problems. This method has been adapted in multiple fields of Engineering. Limited research has been done in the dynamics field using this method. So, an attempt has been made to examine the validity of such a method. The technique gave exact solutions for most of the problems.

Usage of numerical methods to solve nonlinear mixed Volterra-Fredholm integral equations and their system

In this paper, we apply the homotopy perturbation method (HPM), modified homotopy perturbation method(MHPM), variational iteration method (VIM), Adomian decomposition method (ADM), and modified Adomian decomposition method (MADM) to solve nonlinear mixed Volterra-Fredholm integral equations and itssystem. We investigate the approximate solution of this equation and its system via proposed methods. Thevalidity and efficiency of these methods are demonstrated through various numerical examples that illustratethe efficiency, accuracy, and simplicity of the proposed methods. Moreover, the convergence and uniquenessof the solution of the suggested methods are confirmed and compared with the exact solutions.

Variational Approximate Solutions of Fractional Delay Differential Equations with Integral Transform

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).

On a method for solving nonlinear integro differential equation of order

This work is concerned with the study of a general class of nonlinear integro-differential equations of order n. Using a suitable transformation, we derive an equivalent nonlinear Fredholm-Volterra integral equation (NF-VIE) to this class of equations. The existence of continuous solutions for the NF-VIE is investigated subject to the verification of some sufficient conditions. We apply the modified Adomian's decomposition method (MADM) and homotopy analysis method (HAM) to solve this NF-VIE. The convergence and error estimate of the approximate solution are also studied. The numerical results in this article show that the HAM technique may give an approximate solution with high accuracy and convergence rate faster than the one obtained using the MADM technique provided the convergence control parameter \(\hbar\) is properly chosen when applying the HAM.

Numerical solution of Drinfeld–Sokolov–Wilso system by using modified Adomian decomposition method

<p class="Char">In this paper, the modified Adomian decomposition method (MADM) is usedto solve different types of differential equations, one of the numerical analysis methods for solving non linear partial differential equations (Drinfeld–Sokolov–Wilson system) and short (DSWS) that occur in shallow water flows. A Genetic Algorithm was used to find the optimal value for the parameter (a). We numerically solved the system (DSWS) and compared the result to the exact solution. When the value of it is low and close to zero, the MADM provides an excellent approximation to the exact solution. As well as the lower value of leads to the numerical algorithm of (MADM) approaching the real solution. Finally, found the optimal value when a=-10 by using the Genetic Algorithm (G-MADM). All the computations were carried out with the aid of Maple 18 and Matlab to find the parameter value (a) by using the genetic algorithm as well as to figures drawing. The errors in this paper resulted from cut errors and mean square errors.</p>

Numerical Solution of Integral Equation by using New Modified Adomian Decomposition Method and Newton Raphson Methods

In this paper, we discuss the numerical solution of Adomian decomposition method and Taylor’s expansion method in Volterra linear integral equation. And we apply modified Adomian decomposition method and Newton Raphson method in Volterra nonlinear integral equation with the help of example and estimated an error in MATLAB 13 versions.

Mathematical Modelling in Engineering with Integral Transforms via Modified Adomian Decomposition Method

In this work three integral transforms through modified Adomian decomposition method (ADM) are proposed to obtain the approximate analytical solution of different types of mathematical models arising in physical problems. These transformations are applied for both homogeneous and non-homogeneous linear differential equations. The efficiency and accuracy of the proposed methods are implemented through higher order non-homogeneous ordinary differential equations. Numerical tests are reported for applicability of the current scheme based on different transformations and compared with exact solutions.

Modified Adomian Decomposition Method for the Solution of Integro-Differential Equations

This paper is concerned with modification of the Adomian Decomposition Method for solving linear and non-linear Volterra and Volterra-Fredholm Integro-Differential equations. The Modified form of ADM was carried out by replacing the Adomian polynomials constructed in the conventional Adomian Decomposition Method with the constructed canonical polynomials. The modified Adomian Decomposition Method was applied to solve some existing example. The results obtained using the newly modified ADM proved superior when compared with the conventional ADM.

On an approximate solution of a boundary value problem for a nonlinear integro-differential equation

The aim of this work is to discuss the solvability of a boundary value problem for a nonlinear integro-differential equation. First, we derive an equivalent nonlinear Fredholm integral equation (NFIE) to this problem. Second, we prove the existence of a solution to the NFIE using the Krasnosel’skii fixed point theorem under verifying some sufficient conditions. Third, we solve the NFIE numerically and study the convergence rate via methods based upon applying the modified Adomian decomposition method and Liao’s homotopy analysis method. As applications, some examples are illustrated to support our work. The results in this work refer to both methods are efficient and converge rapidly, but the homotopy analysis method may converge faster when we succeed in choosing the optimal homotopy control parameter.

Numerical solutions for nonlinear Volterra-Fredholm integral equations of the second kind with a phase lag

<abstract><p>This study is focused on the numerical solutions of the nonlinear Volterra-Fredholm integral equations (NV-FIEs) of the second kind, which have several applications in physical mathematics and contact problems. Herein, we develop a new technique that combines the modified Adomian decomposition method and the quadrature (trapezoidal and Weddle) rules that used when the definite integral could be extremely difficult, for approximating the solutions of the NV-FIEs of second kind with a phase lag. Foremost, Picard's method and Banach's fixed point theorem are implemented to discuss the existence and uniqueness of the solution. Furthermore, numerical examples are presented to highlight the proposed method's effectiveness, wherein the results are displayed in group of tables and figures to illustrate the applicability of the theoretical results.</p></abstract>

Optimal Parameters for Nonlinear Hirota-Satsuma Coupled KdV System by Using Hybrid Firefly Algorithm with Modified Adomian Decomposition

In this paper, several parameters of the non-linear Hirota-Satsuma coupled KdV system were estimated using a hybrid between the Firefly Algorithm (FFA) and the Modified Adomian decomposition method (MADM). It turns out that optimal parameters can significantly improve the solutions when using a suitably selected fitness function for this problem. The results obtained show that the approximate solutions are highly compatible with the exact solutions and that the hybrid method FFA_MADM gives higher efficiency and accuracy compared to the classic MADM method.

On analysis of the fuzzy fractional order Volterra-Fredholm integro-differential equation

In this paper, we study the fractional integro-differential equation under the notion of fuzziness. We consider the fuzzy fractional Volterra-Fredholm integro-differential equation in the Caputo sense and show its existence and uniqueness by using fixed point theory. Further, we use the modified Adomian decomposition method (MADM) to determine the solution of the proposed problem. We provide some examples to support our method. We also provide graphical representation to visualize the behavior of the solutions. At the end of the paper, we present the conclusion part.

Modified Variational Iteration Technique for the Numerical Solution of Fifth Order KdV-type Equations

In this article, a simple and new algorithm is proposed, namely the modified variational iteration algorithm-I (mVIA-I), for obtaining numerical solutions to different types of fifth-order Korteweg de-Vries (KdV) equations. In order to verify the precision, accuracy and stability of the mVIA-I method, generated numerical results are compared with the Laplace decomposition method, Adomian decomposition method, Homotopy perturbation transform method and the modified Adomian decomposition method. Comparison with the mentioned methods reveals that the mVIA-I is computationally attractive, exceptionally productive and achieves better accuracy than the others.

Development of a Model for the Process of Anaerobic Digestion and Its Solution by the Modified Adomian Decomposition Method

Modeling the transformation of biomass into biogas is complex, because it involves a nonlinear and coupled set of ordinary differential equations. Thus, obtaining an analytical-numerical solution becomes attractive for this problem. In this paper, five chemical reactions are used to model the chemical kinetics of the anaerobic digestion process. The rate of production of each reaction is estimated by Gibbs free energy value. The equation system of the model is solved by the Modified Adomian Decomposition Method, applied to the time variable. The results obtained agree with the expected solution.

Application of Adomian Decomposition Method to Solving Higher Order Singular Value Problems for Ordinary Differential Equations

This paper is an attempt to solve singular value problems for higher order ordinary differential equation by using new modification of Adomian Decomposition Method (ADM). Convergent series solution of considered problem have been obtained. Three numerical examples are discussed to validate the strength and ease of the method used.

Inverse and efficiency of heat transfer convex fin with multiple nonlinearities

AbstractIn this article, we first propose the novel semi‐analytical technique—modified Adomian decomposition method (MADM)—for a closed‐form solution of the nonlinear heat transfer equation of convex profile with singularity where all thermal parameters are functions of temperature. The longitudinal convex fin is subjected to different boiling regimes, which are defined by particular values of n (power index) of heat transfer coefficient. The energy balance equation of the convex fin with several temperature‐dependent properties are solved separately using the MADM and the spectral quasi‐linearization method. Using the values obtained from the direct heat transfer method, the unknown parameters of the profile, such as thermal conductivity, surface emissivity, heat generation number, conduction‐convection parameter, and radiation‐conduction parameter are inversely predicted by an inverse heat transfer analysis using the simplex search method. The effect of the measurement error and the number of measurement points has been presented. It is found that present measurement points and reconstruction of the exact temperature distribution of the convex fin are fairly in good agreement.

Analytical solution with existence and uniqueness conditions of non-linear initial value multi-order fractional differential equations using Caputo derivative

The main focus of this study is to apply the two-step Adomian decomposition method (TSADM) for finding a solution of a fractional-order non-linear differential equation by using the Caputo derivative. We are interested in obtaining an analytical solution with two main constraints, that are, without converting the non-linear fractional differential equation to a system of linear algebraic fractional equation, and secondly, with less number of iterations. Moreover, we have investigated conditions for the existence and uniqueness of a solution with the help of some fixed point theorems. Furthermore, the method is demonstrated with the help of some examples. We also compare the results with the Adomian decomposition method (ADM), the modified Adomian decomposition method, and the combination of the ADM and a spectral method. It is concluded that the TSADM provides an analytical solution of fractional-order non-linear differential equation, while the other methods furnish an approximate solution.

Numerical solution of Lane-Emden type equations using Adomian decomposition method with unequal step-size partitions

PurposeThe purpose of this paper is to obtain the highly accurate numerical solution of Lane–Emden-type equations using modified Adomian decomposition method (MADM) for unequal step-size partitions.Design/methodology/approachFirst, the authors describe the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. After that, for the fast calculation of the Adomian polynomials, an algorithm is presented based on Duan’s corollary and Rach’s rule. Then, MADM is discussed for the unequal step-size partitions of the domain, to obtain the numerical solution of Lane–Emden-type equations. Moreover, convergence analysis and an error bound for the approximate solution are discussed.FindingsThe proposed method removes the singular behaviour of the problems and provides the high precision numerical solution in the large effective region of convergence in comparison to the other existing methods, as shown in the tested examples.Originality/valueUnlike the other methods, the proposed method does not require linearization or perturbation to obtain an analytical and numerical solution of singular differential equations, and the obtained results are more physically realistic.