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Adomian Decomposition Method for Solving Abelian Differential Equations  [PDF]
Omar K. Jaradat
Journal of Applied Sciences , 2008,
Abstract: In this study, we implement a relatively new analytical technique, the Adomian Decomposition Method (ADM), for solving Abelian differential equations. The analytical and numerical results of the equations have been obtained in terms of convergent series with easily computable components. The method is applied to solve two problems. The current results are compared with these derived from the established Runge-Kutta method in order to verify the accuracy of the ADM. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the ADM is powerful and efficient tool for solving Abelian differential equations.
Spectral decomposition of 3D Fokker - Planck differential operator  [PDF]
Igor A. Tanski
Physics , 2006,
Abstract: We construct spectral decomposition of 3D Fokker - Planck differential operator in this paper. We use the decomposition to obtain solution of Cauchy problem - and especially the fundamental solution. Then we use the decomposition to calculate macroscopic parameters of Fokker - Planck flow.
On Adomian's Decomposition Method for Solving Differential Equations  [PDF]
Petre Dita,Nicolae Grama
Physics , 1997,
Abstract: We show that with a few modifications the Adomian's method for solving second order differential equations can be used to obtain the known results of the special functions of mathematical physics. The modifications are necessary in order to take correctly into account the behaviour of the solutions in the neighborhood of the singular points.
Laplace Discrete Adomian Decomposition Method for Solving Nonlinear Integro Differential Equations  [PDF]
H. O. Bakodah, M. Al-Mazmumy, S. O. Almuhalbedi, Lazim Abdullah
Journal of Applied Mathematics and Physics (JAMP) , 2019, DOI: 10.4236/jamp.2019.76093
Abstract:
This paper proposes the Laplace Discrete Adomian Decomposition Method and its application for solving nonlinear integro-differential equations. This method is based upon the Laplace Adomian decomposition method coupled with some quadrature rules of numerical integration. Four numerical examples of integro-differential equations in both Volterra and Fredholm integrals are used to be solved by the proposed method. The performance of the proposed method is verified through absolute error measures between the approximated solutions and exact solutions. The series of experimental numerical results show that our proposed method performs in high accuracy and efficiency. The study clearly highlights that the proposed method could be used to overcome the analytical approaches in solving nonlinear integro-differential equations.
Spectral decomposition of 1D Fokker - Planck differential operator  [PDF]
Igor A. Tanski
Physics , 2006,
Abstract: We construct spectral decomposition of 1D Fokker - Planck differential operator. This reveal solution of Cauchy problem. We develop fundamental solution of Cauchy problem and compare it with one obtained by other means in our former work [5].
A Comparison Between Adomian’s Decomposition Method and the Homotopy Perturbation Method for Solving Nonlinear Differential Equations  [PDF]
E. Babolian,A.R. Vahidi,Z. Azimzadeh
Journal of Applied Sciences , 2012,
Abstract: In this study, we have compared the performance of the Adomian Decomposition Method (ADM) and the Homotopy Perturbation Method (HPM) for solving nonlinear differential equations. By comparative theoretical analysis of these methods, we show that the ADM is equivalent to the HPM with a specific convex homotopy for nonlinear differential equations.
The Adomian Decomposition Method for a Type of Fractional Differential Equations  [PDF]
Peng Guo
Journal of Applied Mathematics and Physics (JAMP) , 2019, DOI: 10.4236/jamp.2019.710166
Abstract: Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.
The introduction to the operator method for solving differential equations.First-order DE  [PDF]
Yu. N. Kosovtsov
Mathematics , 2002,
Abstract: We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of the equations and then to transform them to more convenient form with help of developed family of operator identities. On example of non-linear first-order DEs we analyse analytical and algorithmical possibilities for solutions obtaining. Different forms of solutions for first-order DEs are given, including for some integro-differential equations and equations with variational derivatives. We describe new algorithms for direct computing the solutions with help of computer algebra system (CAS). We also discuss recipe for finding new solvability conditions, which allow to enlarge DE solving abilities of existent CAS.
A Superexponentially Convergent Functional-Discrete Method for Solving the Cauchy Problem for Systems of Ordinary Differential Equations  [PDF]
Makarov Volodymyr,Dragunov Denis
Mathematics , 2010,
Abstract: In the paper a new numerical-analytical method for solving the Cauchy problem for systems of ordinary differential equations of special form is presented. The method is based on the idea of the FD-method for solving the operator equations of general form, which was proposed by V.L. Makarov. The sufficient conditions for the method converges with a superexponential convergence rate were obtained. We have generalized the known statement about the local properties of Adomian polynomials for scalar functions on the operator case. Using the numerical examples we make the comparison between the proposed method and the Adomian Decomposition Method.
Modified Decomposition Method with New Inverse Differential Operators for Solving Singular Nonlinear IVPs in First- and Second-Order PDEs Arising in Fluid Mechanics  [PDF]
Nemat Dalir
International Journal of Mathematics and Mathematical Sciences , 2014, DOI: 10.1155/2014/793685
Abstract: Singular nonlinear initial-value problems (IVPs) in first-order and second-order partial differential equations (PDEs) arising in fluid mechanics are semianalytically solved. To achieve this, the modified decomposition method (MDM) is used in conjunction with some new inverse differential operators. In other words, new inverse differential operators are developed for the MDM and used with the MDM to solve first- and second-order singular nonlinear PDEs. The results of the solutions by the MDM together with new inverse operators are compared with the existing exact analytical solutions. The comparisons show excellent agreement. 1. Introduction Singular nonlinear partial differential equations (PDEs) arise in various physical phenomena in applied sciences and engineering from such areas as fluid mechanics and heat transfer, Riemannian geometry, applied probability, mathematical physics, and biology. The Adomian decomposition method (ADM) and modified decomposition method (MDM) are semianalytical methods that give approximate analytical solutions for the differential equations. MDM was first developed by Wazwaz and El-Seyed [1] who applied it to solve the ordinary differential equations (ODEs). Since then the MDM has been used for solving various equations in mathematics and physics [2–4], boundary value problems [5–9], various problems in engineering [10–13], and initial-value problems [14–17]. Adomian et al. [14] solved the Lane-Emden equation using the MDM. Wazwaz [15] investigated singular initial-value problems, linear and nonlinear, homogeneous and nonhomogeneous, by using the ADM. Hasan and Zhu [16] reported the solution of singular nonlinear initial-value problems in ordinary differential equations (ODEs) by the ADM. Wu [17] extended the ADM for the calculations of the nondifferentiable functions in nonsmooth initial-value problems. His iteration procedure was based on Jumarie Taylor series. Abassy [18] introduced a qualitative improvement in the ADM for solving nonlinear nonhomogenous initial-value problems. Lin et al. [19], based on a new definition of the Adomian polynomials and the two-step Adomian decomposition method combined with the Pade technique, proposed a new algorithm to construct accurate analytical approximations of nonlinear differential equations with initial conditions. Wazwaz et al. [20] used the ADM to handle the integral form of the Lane-Emden equations with initial values and boundary conditions. To the best knowledge of author, till now, no one has attempted the modified decomposition method on solving singular nonlinear
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