Implementation of multi-step differentialtransformation method for hyperchaotic Rossler system

This paper investigates the accuracy of the differential transformation method (DTM) for solving the three-species food chain models which is described as three-dimensional system of ODES with quadratic and rational nonlinearities. Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. The direct symbolic-numeric scheme is indicated to be efficient and accurate. Key words: Three-species food chain models, multi-step differential transformation method, attractors.

A compliant beam subjected to large deformation is governed by a multifaceted nonlinear differential equation. In the context of theoretical mechanics, solution for such equations plays an important role. Since it is hard to find closed-form solutions for this nonlinear problem and attempt at direct solution results in linearising the model. This paper investigates the aforementioned problem via the multi-step differential transformation method (MsDTM), which is well-known approximate analytical solutions. The nonlinear governing equation is established based on a large radius of curvature that gives rise to curvature-moment nonlinearity. Based on established boundary conditions, solutions are sort to address the free vibration and static response of the deforming flexible beam. The geometrically linear and nonlinear theory approaches are related. The efficacy of the MsDTM is verified by a couple of physically related parameters for this investigation. The findings demonstrate that this approach is highly efficient and easy to determine the solution of such problems. In new engineering subjects, it is forecast that MsDTM will find wide use.

Investigation of third-grade non-Newtonian blood flow in arteries under periodic body acceleration using multi-step differential transformation method

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.

In recent years, there has been a tendency towards renewable energy sources considering the damages caused by non-renewable energy resources to nature and humans. One of the renewable energy sources is wind and energy is obtained with the help of wind turbines. To determine the behavior of wind turbines under earthquake loads, dynamic characteristics are required. In this study, the differential transformation method is proposed to determine the free vibration analysis of wind turbines with a variable cross-section. The wind turbine is modeled as an equivalent variable continuous flexural beam and blade weight is considered as a point mass at the top of the structures. The differential equation representing the free vibration of the wind turbine is transformed into an algebraic equation with the help of differential transformation method and the angular frequencies and the mode shapes of the wind turbine are obtained by the help of the differential transformation method. In the study, a sample taken from the literature was solved with the presented method and the suitability of the method was investigated. The same wind turbine example also modeled by finite element modelling software, ABAQUS. Results of the finite element model and differential transformation method are compared with each other and the results are in good agreement.

Analyzing the free vibrations of a plate using finite difference and differential transformation method

The application of differential transformation method to nonlinear equations arising in heat transfer

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.

Application of the differential transformation method to vibration analysis of pipes conveying fluid

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.

Problem statement: In this study, a continuous solution for Genesio system was considered using Differential Transformation Method (DTM). Approach: Numerical results were compared to those obtained by the Runge-Kutta method to evaluate the performance of the suggested method. Results: The accuracy of the DTM was tested as the chaotic Genesio system. Conclusion/Recommendations: It was shown that the DTM was robust, accurate and easy to apply and gave analytical solution on each subinterval, which was not possible in the purely numerical method.

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.

Tapered composite beams are widely used in various engineering applications, the dynamic analysis of this structure appear more and more important in its employment areas. The purpose of this paper is to carry on a research of dynamic characteristic of a steel beam with taped damping layer by Differential Transformation Method(DTM).Since parameters of material used in the damping layer depend on frequency, the method which compares the natural frequency of the composite beam with material frequency is added to research of the free vibration characteristics of this structure.A comparison on calculation cost time between ANSYS and DTM which are applied to analyze same structure demonstrates that DTM has the advantages of far less computation time than ANSYS.

A numeric–analytic method for approximating the chaotic Chen system

Vibration analysis of a circular arch with variable cross-section using differential transformation and generalized differential quadrature