Adomian Decomposition Method Method Research Articles

Advanced medical image encryption techniques using the fractional-order Halvorsen circulant systems: dynamics, control, synchronization and security applications

In this paper, we describe the Halvorsen circulant system (HCS) with a fractional-order Caputo derivative and its qualitative properties. The numerical solution of the fractional order Halvorsen circulant system (FO-HCS) is proposed based on the Adomian decomposition method (ADM). The ADM method is used to solve fractional-order systems. Then, dynamics is analyzed using different methods including Lyapunov exponents, bifurcation diagrams, complexity, and phase diagrams. This paper also investigates the stabilization and synchronization of identical FO-HCS, and stability theory proves adaptive feedback control and synchronization. In addition, using the fractional-order system’s remarkable properties to develop the image encryption technique using the extended fractional sequences. The proposed method uses a keystream generator for high security based on the enhanced fractional Halvorsen circulant chaotic behavior. The simulation results confirm that it can resist various attacks, including statistical analysis, differential attacks, brute-force attacks, known plaintext attacks, and chosen plaintext attacks, with high security, and low computational complexity. Finally, the results of the simulation and its performance prove that it's effective and secure in image data.

Comparison the solutions for some kinds of differential equations using iterative methods

This manuscript presents several applications for solving special kinds of ordinary and partial differential equations using iteration methods such as Adomian decomposition method (ADM), Variation iterative method (VIM) and Taylor series method. These methods can be applied as well as to solve nonperturbed problems and 3rd order parabolic PDEs with variable coefficient. Moreover, we compare the results using ADM, VIM and Taylor series method. These methods are a commination of the two initial conditions.

On the expedient solution of the magneto-hydrodynamic Jeffery-Hamel flow of Casson fluid

The equation of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel is derived and solved using a new modified Adomian decomposition method (ADM). So far in all problems where semi-analytical methods are used the boundary conditions are not satisfied completely. In the present research, a hybrid of the Fourier transform and the Adomian decomposition method (FTADM), is presented in order to incorporate all boundary conditions into our solution of magneto-hydrodynamic Jeffery-Hamel flow of non-Newtonian Casson fluid in a stretching/shrinking convergent/divergent channel flow. The effects of various emerging parameters such as channel angle, stretching/shrinking parameter, Casson fluid parameter, Reynolds number and Hartmann number on velocity profile are considered. The results using the FTADM are compared with the results of ADM and numerical Range-Kutta fourth-order method. The comparison reveals that, for the same number of components of the recursive sequences over a wide range of spatial domain, the relative errors associated with the new method, FTADM, are much less than the ADM. The results of the new method show that the method is an accurate and expedient approximate analytic method in solving the third-order nonlinear equation of Jeffery-Hamel flow of non-Newtonian Casson fluid.

Application of the Adomian Decomposition Method (ADM) for Solving the Singular Fourth-Order Parabolic Partial Differential Equation

In this paper, the ADM method is used to construct the solution of the singular fourth-order partial differential equation.

Solving some System of Linear Fuzzy Fractional Differential Equations by Adomian Decomposition Method

In this paper, we present the numerical method relatively new for solving fuzzy fractional differential equations using Adomian decomposition method (ADM) where the fractional derivative is assumed in terms Riemann-Livoullie sense. The solutions here are expressed in terms of Mittag-Leffler functions. ADM method is an interesting method to solve some fractional order differential equation numerically. Our aim in this paper is to find solution of fractional differential equation numerically with fuzzy valued initial condition by applying ADM.

Solutions of the modified Kawahara equation with time-and space-fractional derivatives

In this paper, we consider fractional differential equations (FDEs), specially modified Kawahara equation with time and space fractional derivatives, also we use Adomian decomposition method (ADM) to approximate the exact solutions of this equation. The ADM method converts the FKEs to an iterated formula that approximate solution is computable. The numerical examples illustrate efficiency and accuracy of the proposed method.

Two reliable methods for solving the Volterra integral equation with a weakly singular kernel

Two reliable methods, namely the Adomian decomposition method (ADM) and the variational iteration method (VIM), are used for solving the Volterra integral equation with a weakly singular kernel in the reproducing kernel space. Both methods provide convergent series solutions for this equation. The ADM method gives a sequence of components of the solution, which composes a sequence of approximations, whereas the VIM more directly provides a sequence of approximations; both exhibit high accuracy. Four numerical examples are examined to confirm the validity and the power of these two methods.

Adomian Decomposition Method (ADM) Versus Nonlinear Solution of Natural Gas Compressibility Factor Correlations

Many mathematical correlations were proposed for the calculation of the natural gas compressibility or z factor. Those correlations are implicit in z which means that they are iteratively solved using nonlinear solvers. In some normal cases, however, nonlinear solvers may diverge from the solution. This behavior has led some researchers to think about simpler and explicit methods for solving the z factor correlations. In a recent paper by Fatoorehchi et al. (J Petrol Sci Eng. doi:10.1016/j.petrol.2014.03.004, 2004), the authors have implemented the Adomian Decomposition Method (ADM) which was introduced by George Adomian in the 1980’s. They used the ADM method to tackle the nonlinearity of natural gas z-factor of the Hankinson–Thomas–Phillips correlation. The authors presented examples of gas gravities in the range 0.6–0.8 and pressure ranges no more than 6,000 psi. The contribution of this paper, however, is that it presents the discretization and solution method of the widely accepted natural gas z-factor correlations using the ADM method. It extensively covers the procedure and methodology of evaluating the Adomian polynomials along with a listing of the first ten polynomials and nonlinearities in a very compact form. In addition, it compares the nonlinear and ADM solutions via extensive examples for gas gravities 0.6–1.5 and for pressures 100–9,000 psi.

Application of Adomian decomposition method to solve hybrid fuzzy differential equations

In this paper, we study the numerical solution of hybrid fuzzy differential equations by using Adomian decomposition method (ADM). This is powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. Several numerical examples are given and by comparing the numerical results obtained from ADM and predictor corrector method (PCM), we have studied their accuracy.

A comparison of Adomian and generalized Adomian methods in solving the nonlinear problem of flow in convergent-divergent channels

The principal aim of this paper is to compare two analytical methods of decomposition employed for solving nonlinear problem of flow in converging-diverging channels, namely the Adomian Decomposition Method (ADM) and Generalized Decomposition Method (GDM). The comparison between ADM and GDM is conducted by using the numerical results of the studied problem as a guide. The results show that GDM method is more accurate and very efficient than ADM method. Indeed, ADM is a good method in solving the nonlinear problems, but we notice in solving the nonlinear problem of the Jeffery-Hamel flow that GDM is powerful, superior than ADM and clearly constitute an excellent choice for solving nonlinear problems.

Numerical analysis of different second order systems using Adomian decomposition method

In this article, the Adomian Decomposition Method (ADM) is used to obtain the numerical solution the different types of second order systems like stable, unstable, stiff systems and system with singular-A. The obtained discrete solutions using ADM and fourth order Runge-Kutta Method (RK) are compared with the exact solutions of the problems and are found to be very accurate. Tables for discrete and exact solutions are presented to show the efficiency of ADM. This ADM can be easily implemented in a digital computer and the solution can be obtained for any length of time.

A Comparison between Adomian Decomposition and Tau Methods

We present a comparison between Adomian decomposition method (ADM) and Tau method (TM) for the integro-differential equations with the initial or the boundary conditions. The problem is solved quickly, easily, and elegantly by ADM. The numerical results on the examples are shown to validate the proposed ADM as an effective numerical method to solve the integro-differential equations. The numerical results show that ADM method is very effective and convenient for solving differential equations than Tao method.

The ADM method for solving the modified kawahara equation with fractional spatial and temporal derivatives

In this paper, by introducing the fractional derivative in the sense of Caputo, we apply the Adomian decomposition method (ADM) to solve the modified Kawahara equation with fractional derivatives. Two cases of special interest such as the temporal-fractional modified Kawahara eqaution and the spatial-fractional modified Kawahara equation are discussed in details. Further, we compare the obtained results with those obtained for integer derivatives

Free vibration analysis of multiple-stepped beams by using Adomian decomposition method

The Adomian decomposition method (ADM) is employed in this paper to investigate the free vibrations of the Euler–Bernoulli beams with multiple cross-section steps. The proposed ADM method can be used to analyze the vibration of beams consisting of an arbitrary number of steps through a recursive way. The solution can be obtained by solving a set of algebraic equations with only three unknown parameters. Furthermore, the method can be extended to obtain an approximate solution to vibration problems of any type of non-uniform beams. Several numerical examples are presented and compared to those given in the paper. It is shown that the ADM offers an accurate and effective method of free vibration analysis of multiple-stepped beams with arbitrary boundary conditions.

A numerical method for solving Jaulent–Miodek equation

This Letter presents a numerical method, based on the Adomian's decomposition method (ADM), is presented of the Jaulent–Miodek (JM) equation. We obtain the numerical and exact travelling wave solutions of the JM equation for the initial conditions. The ADM method provides a better approximation to the exact solution than the classical methods. In order to show the efficiency of this method, exhaustive numerical examples are presented.