Standard Adomian Decomposition Research Articles

Adomian Modification Methods via Orthogonal Polynomials: A Comparative Study

The present manuscript proposes different modification procedures for the standard Adomian Decomposition Method (ADM). These procedures are based on the application of orthogonal polynomials that play vital parts in approximation theories. Moreover, the study also scrutinizes four test nonlinear inhomogeneous initial-value problems, and distinctively examines their respective absolute error differences. Remarkably, different computational benefits of the proposed modification are noted with regard high-level of accuracy and fewer computational steps.

An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)

<p>The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.</p>

The dynamics of fractional KdV type equations occurring in magneto-acoustic waves through non-singular kernel derivatives

To study magneto-acoustic waves in plasma, we will use a numerical method based on the Natural Transform Decomposition Method (NTDM) to find the approximative solutions of nonlinear fifth-order KdV equations. The method combines the familiar Natural transform (NT) with the standard Adomian decomposition method. The fractional derivatives considered are the Caputo–Fabrizio and the Atangana–Baleanu derivatives in the sense of Caputo derivatives. Adomian polynomials may be employed to tackle nonlinear terms. In this method, the solution is calculated as a convergent series, and it is demonstrated that the NTDM solutions converge to the exact solutions. A range of two- and three-dimensional figures have been used to illustrate the dynamic behavior of the derived solutions. The tables provide a visual representation of numerical data. The physical behavior of the derived solutions about fractional order is further demonstrated by several simulations. When addressing nonlinear wave equations in science and engineering, the NTDM offers a broad range of applications. Several examples are given to highlight the importance of this work and to demonstrate the simplicity and trustworthiness of the method.

Approximate Series Solution For Two-Point Fuzzy Boundary Value Problems

In this work, we have used double decomposition method to find approximate analytical solutions for the two-point fuzzy boundary value problems. This method is based on the standard Adomian decomposition method, which is a highly efficient method for solving fuzzy and non-fuzzy differential equations. This method allows for the solution to be calculated in the form of convergent series, in which the obtained are accurate solutions and very close to the exact analytical solutions. Some numerical results have been given to illustrate the efficiency of the used method.

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

Purpose This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique. Design/methodology/approach First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs. Findings The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods. Originality/value Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.

The Spectral Adomian Decomposition Method for the Solution of MHD Jeffery–Hamel Problem

In this study, the effects of magnetic field on the Jeffery–Hamel problem is studied using a powerful numerical method called the spectral Adomian decomposition method (SADM). The traditional Navier–Stokes equation of fluid mechanics and Maxwell’s electromagnetism governing equations are reduced to nonlinear ordinary differential equations to model the problem. Comparisons with the numerical solutions are made to demonstrate the validity and high accuracy of the present approach. The velocity profile of the inner part of the divergent channel is studied for various values of magnetic field parameter and angle of channel. It was found that an increase in the magnetic field parameter leads to increase in the velocity profile. The results indicated that this technique is more efficient and converges faster than the standard Adomian decomposition method.

Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable.

A New Decomposition Method for Integro-Differential Equations

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 terms in the series were then selected in pairs to form the initials for the prevailing approximation. The newly modified Adomian decomposition method (MADM) accelerates the convergence of the solution faster than the Standard Adomian Decomposition Method (SADM). This study recommends the use of the MADM for solving integro-differential equations.

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.

An Efficient Computational Method for Differential Equations of Fractional Type

An effective solution method of fractional ordinary and partial differential equations is proposed in the present paper. The standard Adomian Decomposition Method (ADM) is modified via introducing a functional term involving both a variable and a parameter. A residual approach is then adopted to identify the optimal value of the embedded parameter within the frame of L2 norm. Numerical experiments on sample problems of open literature prove that the presented algorithm is quite accurate, more advantageous over the traditional ADM and straightforward to implement for the fractional ordinary and partial differential equations of the recent focus of mathematical models. Better performance of the method is further evidenced against some compared commonly used numerical techniques.

Resolution of system of Volterra integral equations of the first kind by derivation technique and modified decomposition methods

Abstract A solution method for various systems of integral equations of the first kind is presented in this paper. The method starts off by transforming the systems via the application of the Leibnitz’s derivation technique and then employs three different decomposition methods based on the Standard Adomian decomposition method (SADM) for solutions. To demonstrate the efficiency of the proposed method, some illustrative examples are considered and the obtained results indicate that the approach is indeed of practical interest.

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.

Solving Coupled Lane-Emden Equations by Green’s Function and Decomposition Technique

In this paper, the Green's function and decomposition technique is proposed for solving the coupled Lane-Emden equations. This approach depends on constructing Green's function before establishing the recursive scheme for the series solution. Unlike, standard Adomian decomposition method, the present method avoids solving a sequence of transcendental equations for the undetermined coefficients. Convergence and error estimation is provided. Three examples of coupled Lane-Emden equations are considered to demonstrate the accuracy of the current algorithm.

Modified Adomian decomposition method for solving the problem of boundary layer convective heat transfer

In this paper, we apply a new modification of the Adomian decomposition method for solving the problem of boundary layer convective heat transfer with viscous dissipation and low pressure gradient over a at plate. The technique is based on the standard Adomian decomposition method and the Chebyshev pseudospectral method. Comparisons are made between the pro-posed technique, the standard Adomian decomposition method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the present approach. The results demonstrate that the new modification is more efficient and converges faster than the Adomian decomposition method.

Radiative nanofluid flow and heat transfer between parallel disks with penetrable and stretchable walls considering Cattaneo–Christov heat flux model

AbstractIn this work, we explore the unsteady squeezing flow and heat transfer of nanofluid between two parallel disks in which one of the disks is penetrable and the other is stretchable/shrinkable, in the presence of thermal radiation and heat source impacts, and considering the Cattaneo–Christov heat flux model instead of the more conventional Fourier's law of heat conduction. A similarity transformation is utilized to transmute the governing momentum and energy equations into nonlinear ordinary differential equations with the proper boundary conditions. The achieved nonlinear ordinary differential equations are solved by the Duan–Rach Approach (DRA). This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The impacts of diverse active parameters, such as the suction/injection parameter, the solid volume fraction, the heat source parameter, the thermal relaxation parameter, and the radiation parameter on flow and heat transfer traits are examined. In addition, the value of the Nusselt number is calculated and portrayed through figures.

A novel fast modification of the Adomian decomposition method to solve integral equations of the first kind with hypersingular kernels

In this paper, using the modified Adomian decomposition method, a novel fast scheme, has been presented, being interactive, to be used for solving a class of hypersingular integral equations of the first kind. Compared to the standard Adomian decomposition method, this method expedites the quick accumulation of a series of solutions. Different problems, for exemplification, have been solved in order to display the accuracy and validity of the aforementioned method. According to the results, it gets known that the novel fast modified Adomian decomposition method is so efficient and simple which gives the exact solution, and also shuns the complicated function of long theoretical calculations of collocation methods based on polynomial producing the exact solution.

On series solution for second order semilinear parabolic IBVPs

This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order ( N ), the number of eigenfunctions ( M ) and the diffusion coefficient is rather complex.

Impact of Cattaneo–Christov heat flux on MHD nanofluid flow and heat transfer between parallel plates considering thermal radiation effect

Abstract In this work, the unsteady squeezing MHD nanofluid flow and heat transfer between two parallel plates in the attendance of thermal radiation impact and considering Cattaneo–Christov heat flux model instead of conventional Fourier's law of heat conduction are examined. A similarity transformation is used to transmute the governing momentum and energy equations into non-linear ordinary differential equations with the appropriate boundary conditions. The gained non-linear ordinary differential equations are solved by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The impacts of diverse active parameters such as the magnetic parameter, the squeeze number, the volume fraction of nanofluid, the heat source parameter, the thermal relaxation parameter and the radiation parameter on the velocity and temperature profiles are examined. In addition, the value of the Nusselt number is calculated and presented through figures. The outcomes indicate that the temperature distribution is fewer in the case of Cattaneo–Christov heat flux model as compared to Fourier's law. Furthermore, Nusselt number is an incrementing function of heat source parameter, while it is a diminishing function of the thermal relaxation parameter.

Analytical solution and heat transfer of two-phase nanofluid flow between non-parallel walls considering Joule heating effect

In this paper, MHD nanofluid flow, heat and mass transfer between non-parallel walls is investigated. Brownian diffusion and thermophoresis effects which are two momentous slip mechanisms in nanofluids have been considered in the nanofluid model. Also, Joule heating effect is taken into account. The governing radial momentum, energy and mass equations are solved by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The effects of active parameters such as the Reynolds number, the opening angle parameter, the Schmidt number, thermophoretic parameter and Brownian parameter are investigated on the velocity, temperature and concentration. Also, the value of the Nusselt number is calculated and presented through figures. The results indicate that the temperature and concentration profiles and Nusselt number increase with the increasing Schmidt number. In addition, the limiting cases are gained and found to be in an excellent agreement with the previous works.

Investigation of MHD Go-water nanofluid flow and heat transfer in a porous channel in the presence of thermal radiation effect

In this paper, flow and heat transfer of MHD Go-water nanofluid between two parallel flat plates in the presence of thermal radiation are studied. One of plates is externally heated and cooled by coolant injection through the other plate, which also expands or contracts with time. A similarity transformation is used to transmute the governing momentum and energy equations into non-linear ordinary differential equations with the appropriate boundary conditions. The obtained non-linear ordinary differential equations are solved by Duan–Rach Approach (DRA). This method allows us to find a solution without using numerical methods to evaluate the undetermined coefficients. This method modifies the standard Adomian Decomposition Method by evaluating the inverse operators at the boundary conditions directly. The impacts of various parameters such as the Reynolds number, the expansion ratio, the magnetic parameter, the power law index, the solid volume fraction and the radiation parameter are investigated on the velocity and temperature. Furthermore, the value of the Nusselt number is calculated and presented through figures. The results indicate that the temperature profile and the Nusselt number have a direct relationship with the solid volume fraction and have an inverse relationship with the radiation parameter. In addition, the limiting cases are gained and found to be in an excellent agreement with the previous works.