A multi-step differential transform method and application to non-chaotic or chaotic systems

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.

Predator-prey models are the building blocks of the ecosystems as biomasses are grown out of their resource masses. Different relationships exist between these models as different interacting species compete, metamorphosis occurs and migrate strategically aiming for resources to sustain their struggle to exist. To numerically investigate these assumptions, ordinary differential equations are formulated, and a variety of methods are used to obtain and compare approximate solutions against exact solutions, although most numerical methods often require heavy computations that are time-consuming. In this paper, the traditional differential transform (DTM) method is implemented to obtain a numerical approximate solution to prey-predator models. The solution obtained with DTM is convergent locally within a small domain. The multi-step differential transform method (MSDTM) is a technique that improves DTM in the sense that it increases its interval of convergence of the series expansion. One predator-one prey and two-predator-one prey models are considered with a quadratic term which signifies other food sources for its feeding. The result obtained numerically and graphically shows point DTM diverges. The advantage of the new algorithm is that the obtained series solution converges for wide time regions and the solutions obtained from DTM and MSDTM are compared with solutions obtained using the classical Runge-Kutta method of order four. The results demonstrated is that MSDTM computes fast, is reliable and gives good results compared to the solutions obtained using the classical Runge-Kutta method.

A numeric–analytic method for approximating the chaotic Chen system

This paper presents an analysis of PSIuIeTR type model, which are used to study the transmission dynamics of typhoid fever diseases in a population. Basic idea of typhoid fever disease transmission using compartmental modeling is discussed. Differential Transformation Method (DTM) is discussed in detail, which is used to compute the series solution of the non-linear system of differential equation governing the model equations. The validity of the (DTM) in solving the proposed model is established by classical fourth-order Runge-Kutta method which is implemented in Maple 18. Graphical results confirm that (DTM) is in good agreement with RK-4 and this produced correctly same behaviour of the model, thus validating the efficiency and accuracy of (DTM) in finding the series solution of an epidemic model.

This paper proposed a new analytic method to calculate the Lyapunov exponent based on the Differential Transform Method (DTM). In this work, the DTM method is applied for finding approximate variable coefficients. In this method, first the time series of the dynamical system is estimated based on DTM method, then the Lyapunov exponent is calculated analytically. The Lorenz system and the Colpitts oscillator are used as a practical case study of this research to validate the results. Simulation results are used to show the main points of this paper. We have calculated of the Lyapunov exponent based on DTM method for the first time.

In this article, we have described a mathematical model for tumor-immune system with impact of small molecules drug, and solved by multi step differential transform method.The mathematical model has four compartments; population of tumor cell, CD8 T-killer cell, CD4 T-helper cell and amount of drug in the blood stream.Calculations are presented to verify the theoretical results so obtained for tumor free equilibrium point.The model is fit for removing tumor with passage of time irrespective of the initial population or size of the tumor.

In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. The method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems.

In this paper the Modified Equations of Emden type (MEE), χ+αχχ+βχ 3 is solved numerically by the differential transform method. This technique doesn’t require any discretization, linearization or small perturbations and therefore it reduces significantly the numerical computation. The current results of this paper are in excellent agreement with those provided by Chandrasekar et al. [1] and thereby illustrate the reliability and the performance of the differential transform method. We have also compared the results with the classical Runge-Kutta 4 (RK4) Method.

Solving a fractional order model of HIV infection of CD4 + T cells

This paper employs differential transformation method (DTM) to study the dynamic behavior of a gyroscope control system. The analytical results reveal a complex dynamic behavior comprising periodic, sub-harmonic, quasi-periodic, and chaotic responses of the center of gravity. Furthermore, the results reveal the changes which take place in the dynamic behavior of the gyroscope control system as the external force is increased. The current analytical results by DTM are found to be in good agreement with those of Runge-Kutta method (RKM). Therefore, the proposed method provides an effective means of gaining insights into the nonlinear dynamics of gyroscope systems.

The stability of a linear generalized synchronization approach between different chaot- ic systems with unknown parameters is investigated in this paper. An adaptive controller and some parameter update laws are proposed to achieve stable linear generalized synchronization and parame- ter identification of different chaotic systems. This novel synchronization method effectively unifies complete synchronization, anti-synchronization, projective synchronization, rotation synchronization and function synchronization. The main results are illustrated by two examples: the rotation synchro- nization between Lorenz and Chen system; the modified projective synchronization between Lorenz system and Lu system.

This paper presents the solution of coupled equations which are of fractional order using differential transform method. In this paper we extend the scope of differential transform method to system of fractional differential equations so that we get the analytical solutions. The coupled fractional differential equations of a physical system, namely, coupled fractional oscillator with some applications is given via differential transform method. Here we introduce the solution of coupled oscillation of equal fractional order which can be enhanced to unequal fractional order.

In this paper, differential transform method (DTM) is described and is applied to solve systems of nonlinear ordinary differential equations which is arising in HIV infections of cell. Intervals of validity of the solution will be extended by using Pade approximation. The results also will be compared with those results obtained by Runge-Kutta method. The technique is described and is illustrated with one numerical example. The numerical results shown that the reliability and efficiency of the method.

SOLUTION OF HIGHER-ORDER ABEL EQUATIONS BY DIFFERENTIAL TRANSFORM METHOD

The main objective of this work is to use and compare some techniques, namely the variational Iteration method (VIM), Decomposition Method (ADM), Differential Transform Method (DTM), and Temimi and Ansari Method (TAM) for differential equations of the first and second orders that appeared in physics and engineering. Notably, the results indicate that the VIM gives a solution in series form, which converges to an exact solution;. At the same time, DTM is easy to apply but requires transformation, ADM does not need any transformation except the calculation of Adomain polynomials. TAM solved nonlinear problems in much easier iteration steps, which converge to the true solution without applying any assumption. The results obtained converge rapidly to the exact solution.