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Solution of different type of the partial differential equation by differential transform method and adomian's decomposition method

Abstract In this paper, the definitions and operations of the differential transform method [J.K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huarjung University Press, Wuuhahn, China, 1986] and Adomian’s decomposition method which is given by George Adomian for approximate solution of linear and non-linear differential equations are expressed [G. Adomian, Convergent series solution of nonlinear equation, Comput. Appl. Math. 11 (1984) 113–117] . Different partial differential equations are solved under the view of these methods and compared with the approximate solution and analytic solution. At the end, these solutions are illustrated by tables and figures.

The Use of Fractional Order Derivative to Predict the Groundwater Flow

The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis-time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes.

Two-dimensional differential transform method, Adomian's decomposition method, and variational iteration method for partial differential equations

The implementation of the two-dimensional differential transform method (DTM), Adomians decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.

Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations

In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.

Approximate Solution of Tuberculosis Disease Population Dynamics Model

We examine possible approximate solutions of both integer and noninteger systems of nonlinear differential equations describing tuberculosis disease population dynamics. The approximate solutions are obtained via the relatively new analytical technique, the homotopy decomposition method (HDM). The technique is described and illustrated with numerical example. The numerical simulations show that the approximate solutions are continuous functions of the noninteger-order derivative. The technique used for solving these problems is friendly, very easy, and less time consuming.

A Generalized Version of a Low Velocity Impact between a Rigid Sphere and a Transversely Isotropic Strain-Hardening Plate Supported by a Rigid Substrate Using the Concept of Noninteger Derivatives

A low velocity impact between a rigid sphere and transversely isotropic strain-hardening plate supported by a rigid substrate is generalized to the concept of noninteger derivatives order. A brief history of fractional derivatives order is presented. The fractional derivatives order adopted is in Caputo sense. The new equation is solved via the analytical technique, the Homotopy decomposition method (HDM). The technique is described and the numerical simulations are presented. Since it is very important to accurately predict the contact force and its time history, the three stages of the indentation process, including (1) the elastic indentation, (2) the plastic indentation, and (3) the elastic unloading stages, are investigated.

Comparison of Adomian Decomposition Method and Taylor Matrix Method in Solving Different Kinds of Partial Differential Equations

In this paper, we will present a comparison between the Adomian Decomposition Method (ADM) and Taylor Matrix Method by solving some well-known partial differential equations (PDEs). In order to illustrate the analysis we examined the Telegraph equation, which is considered one of the most significant partial differential equations, describe wave propagation of electric signals in a cable transmission line and Klein-Gordon equation which is encountered in several applied physics fields such as, quantum field theory , fluid dynamics , optoelectronic devices design and numerical analysis. Our study shows that the decomposition method is faster and easy to use from a computational viewpoint.

Implementation of taylor collocation and adomian decomposition method for systems of ordinary differential equations

The importance of ordinary differential equation and also systems of these equations in scientific world is a crystal-clear fact. Many problems in chemistry, physics, ecology, biology can be modeled by systems of ordinary differential equations. In solving these systems numerical methods are very important because most realistic systems of these equations do not have analytic solutions in applied sciences In this study, we apply Taylor collocation method and Adomian decomposition method to solve the systems of ordinary differential equations. In these both scheme, the solution takes the form of a convergent power series with easily computable components. So, we will be able to make a comparison between Adomian decomposition and Taylor collocation methods after getting these power series.

New Iteration Methods for Time-Fractional Modified Nonlinear Kawahara Equation

We put side by side the methodology of two comparatively new analytical techniques to get to the bottom of the system of nonlinear fractional modified Kawahara equation. The technique is described and exemplified with a numerical example. The dependability of both methods and the lessening in computations give these methods a wider applicability. In addition, the computations implicated are very simple and undemanding.

Application of fractional calculus in the dynamics of beams

This paper deals with a viscoelastic beam obeying a fractional differentiation constitutive law. The governing equation is derived from the viscoelastic material model. The equation of motion is solved by using the method of multiple scales. Additionally, principal parametric resonances are investigated in detail. The stability boundaries are also analytically determined from the solvability condition. It is concluded that the order and the coefficient of the fractional derivative have significant effect on the natural frequency and the amplitude of vibrations.