Two-step Adomian Decomposition Method Research Articles

A reliable algorithm for a class of singular nonlinear two-point boundary value problems arising in physiology

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis–Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Padé approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP.

Ulam–Hyers stability and analytical approach for m-dimensional Caputo space-time variable fractional order advection–dispersion equation

This article introduces the computational analytical approach to solve the m-dimensional space-time variable Caputo fractional order advection–dispersion equation with the Dirichlet boundary using the two-step Adomian decomposition method and obtain the exact solution in just one iteration. Moreover, with the help of fixed point theory, we study the existence and uniqueness conditions for the positive solution and prove some new results. Also, obtain the Ulam–Hyers stabilities for the proposed problem. Two generalized examples are considered to show the method’s applicability and compared with other existing numerical methods. The present method performs exceptionally well in terms of efficiency and simplicity. Further, we solved both examples using the two most well-known numerical methods and compared them with the TSADM solution.

NEW EXISTENCE, UNIQUENESS RESULTS FOR MULTI-DIMENSIONAL MULTI-TERM CAPUTO TIME-FRACTIONAL MIXED SUB-DIFFUSION AND DIFFUSION-WAVE EQUATION ON CONVEX DOMAINS

In this paper, we investigate an efficient analytical method known as two step Adomian decomposition method (TSADM). This method does not require approximation/discretization, lengthy calculations and due to involvement of fractional operators and provides an exact solution. In this study, we generalize the multi-term time-fractional mixed sub-diffusion and diffusion-wave equation into multi dimensions with Caputo derivative for time fractional operators and obtain the exact solution. Furthermore, we establish the new results of existence and uniqueness of the solution using fixed point theory. To demonstrate the effectiveness of the proposed method, several generalized examples on the convex domain are considered.

An analytical solution of multi-dimensional space fractional diffusion equations with variable coefficients

In this paper, we have considered the multi-dimensional space fractional diffusion equations with variable coefficients. The fractional operators (derivative/integral) are used based on the Caputo definition. This study provides an analytical approach to determine the analytical solution of the considered problems with the help of the two-step Adomian decomposition method (TSADM). Moreover, new results have been obtained for the existence and uniqueness of a solution by using the Banach contraction principle and a fixed point theorem. We have extended the dimension of the space fractional diffusion equations with variable coefficients into multi-dimensions. Finally, the generalized problems with two different types of the forcing term have been included demonstrating the applicability and high efficiency of the TSADM in comparison to other existing numerical methods. The diffusion coefficients do not require to satisfy any certain conditions/restrictions for using the TSADM. There are no restrictions imposed on the problems for diffusion coefficients, and a similar procedures of the TSADM has followed to the obtained analytical solution for the multi-dimensional space fractional diffusion equations with variable coefficients.

An Analytical Solution of Linear/Nonlinear Fractional-Order Partial Differential Equations and with New Existence and Uniqueness Conditions

In this paper, we have adopted the two-step Adomian decomposition method (TSADM) for solving linear and nonlinear partial differential equations of fractional order and compared with the variational iteration method (VIM) and Adomian decomposition method (ADM). It is successfully applied to linear and nonlinear partial differential equations involving variable coefficients. It is shown that the TSADM is a more effective and promising method and provides an exact solution with one iteration of the problems without discretization and linearization.

Existence and uniqueness results and analytical solution of the multi-dimensional Riesz space distributed-order advection–diffusion equation via two-step Adomian decomposition method

In this article, we introduced for the first time the two-step Adomian decomposition method (TSADM) for solving the multi-dimensional Riesz space distributed-order advection–diffusion (RSDOAD) equation. The TSADM was successfully applied to obtain the analytical solution of the multi-dimensional (RSDOAD) equation. The analytical solution has been obtained without approximation/discretization of the Riesz fractional operator. Furthermore, new results for the existence are obtained with the help of some fixed point theorems, while the uniqueness of the solution was investigated employing the Banach contraction principle. Finally, we included a generalized example to demonstrate the validity and application of the proposed method. The obtained results conclude that the proposed method is powerful and efficient for the considered problem compared to the other existing methods.

An analytical solution with existence and uniqueness conditions for fractional integro-differential equations

This study aims to apply the two-step Adomian decomposition method (TSADM) to find an analytical solution of integro-differential equations for fractional order without discretization/approximation with less number of iterations and reduce the computational efforts. Moreover, we have established the results for the existence and uniqueness of a solution with the help of some fixed point theorems and the Banach contraction principle. Furthermore, the method is demonstrated on different test examples arising in real life situations. It is concluded that the TSADM provides exact solution of the fractional integro-differential equations in one iteration. At the same time, the other existing methods furnish an approximate solution and require lots of computation to solve the problem applying discretization/approximation on fractional operators.

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.

Exact solution with existence and uniqueness conditions for multi-dimensional time-space tempered fractional diffusion-wave equation

In this paper, we investigate the existence and uniqueness of the multi-dimensional time-space tempered fractional diffusion-wave (TSTFDW) equation by using the fixed point theory. We also found an exact solution of the multi-dimensional TSTFDW equation. The two-step Adomian decomposition method is proposed to achieve the exact solution of this problem. The main advantages of this method are its easy implementation and higher efficiency than other existing numerical methods.

Invariant subspace and approximate analytic solutions of a fractional model of convective longitudinal fins in thermal conductivity

In this work, a fractional model of convective longitudinal fins in thermal conductivity are studied by using the invariant subspace method (ISM). The method determines an invariant subspace and construct the exact solutions of the partial differential equations (PDEs) by reducing them to ordinary differential equations (ODEs). As a result of the calculations, exponential solutions of the model are derived. Furthermore, the two step Adomian decomposition method (TSADM) and the Pade approximation techniques are used to derive the Pade approximate solutions of the model. The are derived along with interesting figures showing both the exact and approximate solutions. Ultimately, for illustrating the acquired results, some numerical simulations are performed.

ADMP: A Maple Package for Symbolic Computation and Error Estimating to Singular Two-Point Boundary Value Problems with Initial Conditions

A two step Adomian decomposition method (TSADM) coupled with pade approximations is applied for solving various two-point boundary value problems with initial conditions. This technique is based on a new definition of the Adomian polynomials and the TSADM. A Maple package ADMP is used to implement the proposed algorithm. The algorithm does not require linearization, perturbation, predicting the initial terms or a set of basis function or other restrictive assumptions and approximates the analytical solution. The validity of the applied package is given by two examples.

Solution of Nonlinear Integro Differential Equations by Two-Step Adomian Decomposition Method (TSAM)

The Adomian decomposition method (ADM) can be used to solve a wide range of problems and usually gets the solution in a series form. In this paper, we propose two-step Adomian Decomposition Method (TSAM) for nonlinear integro-differential equations that will facilitate the calculations. In this modification, compared to the standard Adomian decomposition method, the size of calculations was reduced. This modification also avoids computing Adomian polynomials. Numerical results are given to show the efficiency and performance of this method.

Symbolic computation of analytic approximate solutions for nonlinear differential equations with initial conditions

The Adomian decomposition method (ADM) is one of the most effective methods for constructing analytic approximate solutions of nonlinear differential equations. In this paper, based on the new definition of the Adomian polynomials, and the two-step Adomian decomposition method (TSADM) combined with the Padé technique, a new algorithm is proposed to construct accurate analytic approximations of nonlinear differential equations with initial conditions. Furthermore, a MAPLE package is developed, which is user-friendly and efficient. One only needs to input a system, initial conditions and several necessary parameters, then our package will automatically deliver analytic approximate solutions within a few seconds. Several different types of examples are given to illustrate the validity of the package. Our program provides a helpful and easy-to-use tool in science and engineering to deal with initial value problems. Program summary Program title: NAPA Catalogue identifier: AEJZ_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEJZ_v1_0.html Program obtainable from: CPC Program Library, Queenʼs University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 4060 No. of bytes in distributed program, including test data, etc.: 113 498 Distribution format: tar.gz Programming language: MAPLE R13 Computer: PC Operating system: Windows XP/7 RAM: 2 Gbytes Classification: 4.3 Nature of problem: Solve nonlinear differential equations with initial conditions. Solution method: Adomian decomposition method and Padé technique. Running time: Seconds at most in routine uses of the program. Special tasks may take up to some minutes.

A two-step Laplace decomposition method for solving nonlinear partial differential equations

The Adomian decomposition method (ADM) is an analytical method to solve linear and nonlinear equations and gives the solution a series form. Two-step Adomian decomposition method (TSADM) is a modification on ADM and makes the calculations much simpler. In this paper we combine Laplace transform and TSADM and present a new approach for solving partial differential equations. Key words: Two-step Adomian decomposition method, Laplace decomposition method, Adomian decomposition method. Mathematics subject classification: 47J30, 49S05.

Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method

Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical superdiffusive problems in fluid flow, finance and other areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by two-step Adomian Decomposition Method (TSADM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form and then their solutions have been represented graphically. Two examples, the first one is one-dimensional and the second one is two-dimensional fractional diffusion equation, are presented to show the application of the present technique. The solutions obtained by the standard decomposition method have been numerically evaluated and presented in the form of tables and then compared with those obtained by TSADM. The present TSADM performs extremely well in terms of efficiency and simplicity.

Solution of the heat equation with variable properties by two-step Adomian decomposition method

Adomian Decomposition Method [G. Adomian, R. Rach, Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations, Appl. Math. Comput. 19 (1990) 9–12; G. Adomian, R. Rach, Analytical solution of non linear boundary value problem in several dimension by decomposition, J. Appl. Math. 174 (1993) 118–137; G. Adomian, R. Rach, Modified Adomian polynomials, Math. Comput. Modelling 24 (1996) 39–46] is useful to find the solution of linear and nonlinear equations. There is a renewed interest in the method [A.M. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1999) 77–86; A.M. Wazwaz, The modified decomposition method for analytic treatment of differential equations, Appl. Math. Comput. 173 (2006) 165–176; X.G. Luo, A two-step Adomian decomposition method, Appl. Math. Comput. 170 (2005) 570–583; B.Q. Zhang, X.G. Luo, Q.B. Wu, Experimentation with two-step Adomian decomposition method to solve evolution models, Appl. Math. Comput. 175 (2006) 1495–1502; B.Q. Zhang, X.G. Luo, Q.B. Wu, Revisit on partial solutions in the Adomian decomposition method: Solving heat and wave equations, J. Math. Anal. Appl. 321 (2006) 353–363; B.Q. Zhang, X.G. Luo, Q.B. Wu, The restrictions and improvement of the Adomian decomposition method: Solving heat and wave equations, Appl. Math. Comput. 177 (1999) 99–104; Necdet Bildik, Hatice Bayramoglu, The solution of two dimensional nonlinear differential equation by the Adomian decomposition method, Appl. Math. Comput. 163 (1999) 551–567] and a lot of research is being conducted using this method. We attempt to enlarge the scope of its application by presenting the solution of the diffusion equation with variable properties. In this, we present two problems dealing with the heat conduction with variable properties. The compression of the first problem with eigenfunction expansion is also made. The two analytical solution agree exactly with each other. Although, the two methods arrive at the same result, nevertheless, it is of much value to obtain the solution by ADM which provides a powerful method of finding the solution of both linear and nonlinear problems. To apply this method, we have shown that generalized Fourier series is required to build up the solution instead of trigonometric Fourier series.

Experimentation with two-step Adomian decomposition method to solve evolution models

The two-step Adomian decomposition method (TSADM) is derived by separate the standard Adomian decomposition method into two steps. Although the TSADM needs only a slight variation from the standard Adomian decomposition method, experimentation shows that the TSADM is also promising for systems of inhomogeneous differential and integral equations, hyperbolic partial differential equations and singular initial value problems.

A two-step Adomian decomposition method

The Adomian decomposition method (ADM) is a nonnumerical method for solving a wide variety of functional equations and usually gets the solution in a series form. In this paper, we propose an efficient modification of ADM, namely two-step Adomian decomposition method (TSADM), that will facilitate the calculations. We then conduct a comparative study between the TSADM and previous methods with the help of several illustrative examples. The results obtained indicate that the TSADM is effective and promising.