A generalized differential transform method for linear partial differential equations of fractional order

In this letter we develop a new generalization of the two-dimensional differential transform method that will extend the application of the method to linear partial differential equations with space- and time-fractional derivatives. The new generalization is based on the two-dimensional differential transform method, generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the present method. The results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order

A Handy Approximation Technique for Closedform and Approximate Solutions of Time- Fractional Heat and Heat-Like Equations with Variable Coefficients

In this paper, we propose a handy approximation technique (HAT) for obtaining both closed-form and approximate solutions of time-fractional heat and heat-like equations with variable coefficients. The method is relatively recent, proposed via the modification of the classical Differential Transformation Method (DTM). It devises a simple scheme for solving the illustrative examples, and some similar PDEs. Besides being handy, the results obtained converge faster to their exact forms. This shows that this modified DTM (MDTM) is very efficient and reliable. It involves less computational work, even without given up accuracy. Therefore, we strongly recommend it for solving both linear and nonlinear time-fractional partial differential equations (PDEs) with applications in other aspects of pure and applied sciences, management, and finance

The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics

AbstractA non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald–Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order

Numerical solution for the time-Fractional Diffusion-wave Equations by using Sinc-Legendre Collocation Method

In this paper the numerical solution of fractional diffusion wave equation is proposed. The fractional derivative will be in the Caputo sense. The proposed method will be based on shifted Legendre collocation scheme and sinc function approximation for time and space respectively. The problem is reduced to the problem into a system of algebraic equations after implementing this method. For demonstrating the validity and applicability of the proposed numerical scheme some examples are presented. Keywords: Fractional diffusion equation, Sinc functions, shifted Legendre  polynomials, Collocation method

Solution of the SIR models of epidemics using MSGDTM

Stochastic compartmental (e.g., SIR) models have proven useful for studying the epidemics of childhood diseases while taking into account the variability of the epidemic dynamics. Here, we use the multi-step generalized differential transform method (MSGDTM) to approximate the numerical solution of the SIR model and numerical simulations are presented graphically

SHIFTED LEGENDRE POLYNOMIAL BASED GALERKIN AND COLLOCATION METHODS FOR SOLVING FRACTIONAL ORDER DELAY DIFFERENTIAL EQUATIONS

In this article, effective numerical methods for the solution of fractional order delay differential equations (FODDEs) are presented. The fractional derivative (FD) is defined in Caputo sense. Shifted Legendre polynomials are used in the Collocation and Galerkin methods to convert FDDEs to the linear and/or nonlinear system in algebraic form of equations. Example problems are addressed to show the powerfulness and efficacy of the methods

Symbolic computation of exact solutions for fractional differential-difference equation models

The aim of the present study is to extend the G'/G-expansion method to fractional differential-difference equations of rational type. Particular time-fractional models are considered to show the strength of the method. Three types of exact solutions are observed: hyperbolic, trigonometric and rational. Exact solutions in terms of topological solitons and singular periodic functions are also obtained. As far as we are aware, our results have not been published elsewhere previously

A new analytic numeric method solution for fractional modified epidemiological model for computer viruses

Computer viruses are an extremely important aspect of computer security, and understanding their spread and extent is an important component of any defensive strategy. Epidemiological models have been proposed to deal with this issue, and we present one such here. We consider the modified epidemiological model for computer viruses (SAIR) proposed by J. R. C. Piqueira and V. O. Araujo. This model includes an antidotal population compartment (A) representing nodes of the network equipped with fully effective anti-virus programs. The multi-step generalized differential transform method (MSGDTM) is employed to compute an approximation to the solution of the model of fractional order. The fractional derivatives are described in the Caputo sense. Figurative comparisons between the MSGDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that this method is very effective. Mathematica 9 is used to carry out the computations. Graphical results are presented and discussed quantitatively to illustrate the solution