Vedat Suat Ertürk

Generalized differential transform method: Application to differential equations of fractional order

Abstract In this paper we propose a new generalization of the one-dimensional differential transform method that will extend the application of the method to differential equations of fractional order. The new generalization is based on generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the obtained results. The new generalization introduces a promising tool for many linear and nonlinear models containing fractional derivatives.

Solutions of non-linear oscillators by the modified differential transform method

A numerical method for solving nonlinear oscillators is proposed. The proposed scheme is based on the differential transform method (DTM), Laplace transform and Pade approximants. The modified differential transform method (MDTM) technique introduces an alternative framework designed to overcome the difficulty of capturing the periodic behavior of the solution, which is characteristic of oscillator equations, and give a good approximation to the true solution in a very large region. The numerical results demonstrate the validity and applicability of the new technique and a comparison is made with existing results.

Comparing numerical methods for solving fourth-order boundary value problems

In this study, we present a numerical comparison between differential transform method and the Adomian decomposition method for solving fourth-order boundary value problems. Three examples are given. The numerical results reveal that differential transform method is very efficient and accurate.

Solutions of a fractional oscillator by using differential transform method

Abstract In this paper, we present an efficient algorithm for solving a fractional oscillator using the differential transform method. The fractional derivatives are described in the Caputo sense. The application of differential transform method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of a fractional oscillator. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method.

An approximate solution of a fractional order differential equation model of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

In this paper, a fractional order differential system for modeling human T-cell lymphotropic virus I (HTLV-I) infection of CD4^+ T-cells is studied and its approximate solution is presented using a multi-step generalized differential transform method. The method is only a simple modification of the generalized differential transform method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. The solutions obtained are also presented graphically.

A numeric-analytic method for approximating a giving up smoking model containing fractional derivatives

Smoking is one of the main causes of health problems and continues to be one of the worlds most significant health challenges. In this paper, the dynamics of a giving up smoking model containing fractional derivatives is studied numerically. The multistep generalized differential transform method (for short MSGDTM) is employed to compute accurate approximate solutions to a giving up smoking model of fractional order. The unique positive solution for the fractional order model is presented. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The solutions obtained are also presented graphically.

Solutions to the problem of prey and predator and the epidemic model via differential transform method

Purpose – The purpose of this paper is to solve both the prey and predator problem and the problem of the spread of a non‐fatal disease in a population which is assumed to have constant size over the period of the epidemic.Design/methodology/approach – The differential transform method (DTM) is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations of these problems.Findings – Results obtained using the scheme presented here agree well with the results obtained by the Adomian decomposition and power series methods. Some plots are presented to show the reliability and simplicity of the method.Originality/value – This paper is believed to represent a new application for DTM on solving systems of nonlinear ordinary differential equations.

The differential transform method and Padé approximants for a fractional population growth model

– The purpose of this paper is to propose an approximate method for solving a fractional population growth model in a closed system. The fractional derivatives are described in the Caputo sense., – The approach is based on the differential transform method. The solutions of a fractional model equation are calculated in the form of convergent series with easily computable components., – The diagonal Pade approximants are effectively used in the analysis to capture the essential behavior of the solution., – Illustrative examples are included to demonstrate the validity and applicability of the technique.

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

In this paper, a reliable algorithm based on an adaptation of the standard differential transform method is presented, which is the multi-step differential transform method (MSDTM). The solutions of non-linear oscillators were obtained by MSDTM. Figurative comparisons between the MSDTM and the classical fourth-order Runge-Kutta method (RK4) reveal that the proposed technique is a promising tool to solve non-linear oscillators.

A reliable algorithm for solving tenth-order boundary value problems

In this paper we present an efficient numerical algorithm for solving linear and nonlinear boundary value problems with two-point boundary conditions of tenth-order. The differential transform method is applied to construct the numerical solutions. The proposed algorithm avoids the complexity provided by other numerical approaches. Several illustrative examples are given to demonstrate the effectiveness of the present algorithm.