Differential Transformation Method Results Research Articles

Vibration analysis of buried pipelines and piles using differential transformation method

Investigating the interaction between structures and their elastic foundations is essential in structures with elastic support conditions, such as buried pipelines and piles. Research investigation was carried out to find the frequency of a buried structure simulated by Euler–Bernoulli beam equations, with distributed damping and general boundary conditions on an elastic foundation. The elastic bed was modelled by a linear spring, and boundary conditions were changeable at two ends of the beam's support with variations in the stiffness of transitional and rotational springs. The differential transformation method (DTM) was then used to solve the equations by correlating the function and its derivatives. The frequency of the structure was obtained for various damping and stiffness values. The increase in the stiffness of the elastic support and the decrease in damping led to a higher frequency. The results of DTM agreed with the results of the exact method.

Differential Transform Method as an Effective Tool for Investigating Fractional Dynamical Systems

This study investigates the use of the differential transform method (DTM) for integrating the Rössler system of the fractional order. Preliminary studies of the integer-order Rössler system, with reference to other well-established integration methods, made it possible to assess the quality of the method and to determine optimal parameter values that should be used when integrating a system with different dynamic characteristics. Bifurcation diagrams obtained for the Rössler fractional system show that, compared to the RK4 scheme-based integration, the DTM results are more resistant to changes in the fractionality of the system.

Differential transformation method for circular membrane vibrations

The purpose of this research is to present the steps of one-dimensional differential transformation method (DTM) to find the series solutions for the vibrations of a circular membrane under the specified initial and boundary conditions. The problems will be studied in the both cases of vibrations depending only on radius and of the vibrations depending on both radius and angle. We illustrate four examples of problems which the exact solutions can be solve analytically and compare them to the DTM results, to show that the DTM is reliable and of high accuracy. This work shows that the DTM is easier to use than the analytical method from the point of view of programming.

Elasto-Plastic Analysis of Cylindrical Vessel with Arbitrary Material Gradation Subjected to Thermo-Mechanical Loading Via DTM

For the first time, the response of the functionally graded (FG) hollow cylinder vessel with arbitrary material gradation in radial direction subjected to thermo-mechanical loads is investigated by employing differential transformation method (DTM). The coordinate transformation and proper scale factor are applied to eliminate redundant terms in DTM; as a result, fast convergence with less computational effort is observed. The elasticity modulus, thermal expansion coefficient, thermal conductivity and yield stress are assumed to be arbitrary functions of radius. The thick-walled FG reservoir is in steady-state condition, and strain hardening in plastic zone is neglected by using elastic–perfectly plastic behavior assumptions. The incompressible plastic deformation and plane-strain condition are considered to obtain radial displacement in the plastic zones. The effects of various parameters on commencement and propagation of possible plastic zones, yield pressure and yield temperature gradient are investigated. The analytical solutions for FG cylindrical container with conventional power law gradation are presented. In the case of arbitrary material gradation, the validity of the numerical procedure is proved by observing convergence and good agreement between results of DTM and results of finite element software Abaqus.

Heat Transfer in Hydro-Magnetic Nano-Fluid Flow between Non-Parallel Plates Using DTM

This study presents a computational investigation on heat and flow behaviors between non parallel plates with the influence of a transverse magnetic field when the medium is filled with solid nanoparticles. The nonlinear governing equations are treated analytically via Differential Transform Method (DTM). Thereafter, obtained DTM results are validate with the help of numerical fourth order Runge-Kutta (RK4) solution. The main aim of this research work is to analyze the influence of varying physical parameters, in particular Reynolds number, nanofluid volume fraction, and Hartmann number. It was found that the presence of solid nanoparticles in a water base liquid has a notable effect on the heat transfer improvement within convergent-divergent channels. The comparison of DTM results with numerical RK4 solution also shows the validity of the analytical DTM technique. In fact, results demonstrate that the DTM data match perfectly with numerical ones and those available in literature.

Semi analytical analysis for transient Eyring-Powell squeezing flow in a stretching channel due to magnetic field using DTM

This paper deals with heat and mass transfer in unsteady two-dimensional MHD squeezing flow of Eyring-Powell fluid between two parallel infinite plates. Heat generation/absorption, thermal radiation and Joule heating influences are taken into consideration. The governing nonlinear PDEs are converted to nonlinear ODEs using similarity transformations. Analytical solution for the velocity, temperature and concentration distribution has been obtained by means of Differential Transformation Method (DTM).The comparisons between the results of DTM and numerical method (Runge-Kutta Fehlbrg), proved high precision of DTM. The effects of various parameters, including squeeze number, radiation parameter, heat generation/absorption parameter and Hartmann number on the flow field, skin friction coefficient and heat transfer have been studied. The achieved results reveal that Lorentz force generated by magnetic field parameter (Ha), reduces velocity distribution. Also, it is found that the temperature profile reduces with the increase of radiation parameter (R).

Analysis of Computer Virus Propagation Based on Compartmental Model

Computer viruses pose a considerable problem for users of personal computers. In order to effectively defend against a virus, this paper proposes a compartmental model SAEIQRS (Susceptible – Antidotal – Exposed - Infected – Quarantine - Recovered - Susceptible) of virus transmission in a computer network. The differential transformation method (DTM) is applied to obtain an improved solution of each compartment. We have achieved an accuracy of order O(h6) and validated the results of DTM with fourth-order Runge-Kutta (RK4) method. Based on the basic reproduction number, we analyzed the local stability of the model for virus free and endemic equilibria. Using a Lyapunov function, we demonstrated the global stability of virus free equilibria. Numerically the eigenvalues are computed using two different sets of parameter values and the corresponding dominant eigenvalues are determined by means of power method. Finally, we simulate the system in MATLAB. Based on the analysis, aspects of different compartments are investigated.

Free vibration investigation of nano mass sensor using differential transformation method

In the present study, transverse vibration of nano-cantilever beam with attached mass and two rotational and transverse springs at its end is studied. Resonance frequency of vibrating system is influenced by changing mass particle and stiffness coefficients. Euler–Bernoulli beam theory, nonlocal constitutive equations of Eringen, and Hamilton’s principle are used to develop equations of motion. Differential transformation method (DTM) is applied to solve the governing equations of the nanobeam with attached mass particle. Accurate results with minimum mathematical calculation are the advantages of DTM. A detailed parametric study is conducted to investigate the influences of nonlocal parameter. The results can be used in designing of nanoelectromechanical systems. To verify the results, some comparisons are presented between differential transform method results and open literature to show the accuracy of this new approach.

A numerical study on slip flow in packed hyper-porous media inside micro-channels using differential transformation method

Velocity slip boundary condition in parallel-plate micro-channels filled with hyper-porous media is studied using differential transformation method (DTM) in order to find an analytical approximate solution. The results focus on slip flow regime (i.e., for Knudsen numbers in the range 10−3 < Kn < 10−1). The Darcy-Brinkman-Forchheimer model is applied to study the effect of nonlinear drag term further boundary-friction effects on hydrodynamic of gas flow in micro-channels. The results show that DTM results are in good agreement with numerical ones. Also, it is observed that decreasing the value of Darcy number flattens the velocity profile while this trend is opposite for decreasing the Forchheimer number. Also, increasing the value of causes to increase the velocity slip at the wall.

A Semi Analytical Method For Electro-Thermo-Mechanical Nonlinear Vibration Analysis of Nanobeam Resting On the Winkler-Pasternak Foundations With General Elastic Boundary Conditions

This study presents an examination of nonlinear free vibration of a nanobeam under electro-thermo-mechanical loading with elastic medium and various boundary conditions, especially the elastic boundary condition. The nanobeam is modeled as an Euler–Bernoulli beam. The von Kármán strain-displacement relationship together with Hamilton’s principle and Eringen’s theory are employed to derive equations of motion. The nonlinear free vibration frequency is obtained for simply supported (S-S) and elastic supported (E-E) boundary conditions. E-E boundary condition is a general and actual form of boundary conditions and it is chosen because of more realistic behavior. By applying the differential transform method (DTM), the nanobeam’s natural frequencies can be easily obtained for the two different boundary conditions mentioned above. Performing a precise study led to investigation of the influences of nonlocal parameter, temperature change, spring constants (either for elastic medium or boundary condition) and imposed electric potential on the nonlinear free vibration characteristics of nanobeam. The results for S-S and E-E nanobeams are compared with each other. In order to validate the results, some comparisons are presented between DTM results and open literature to show the accuracy of this new approach. It has been discovered that DTM solves the equations with minimum calculation cost.

Electro-thermo-mechanical nonlinear free vibration of nanobeam resting on the winkler-pasternak foundations based on nonlocal elasticity using differential transform method

This study presents electro-thermo-mechanical nonlinear vibration of nanobeam resting on the Winkler-Pasternak foundations. The nanobeam is considered as an Euler–Bernoulli beam together with von Karman assumptions are employed to model the geometrical nonlinearity where embedded on an elastic foundation which is simulated by spring constant of the Winkler-model and the shear constant of the Pasternak-model. The corresponding governing equations are obtained using Hamilton’s principle considering nonlinear strains. In order to obtain the nonlinear frequency for simply supported mechanical boundary condition at two ends of the nanobeam, differential transform method (DTM) is used in conjunction with a program. By applying this technique, the nanobeam’s natural frequencies can be easily obtained and a rapidly convergent sequence is obtained during the solution. A detailed parametric study is conducted to investigate the influences of nonlocal parameter, temperature change, spring constants and imposed electric potential on the nonlinear free vibration characteristics of nanobeam. To verify the results some comparisons are presented between differential transform method results and open literature to show the accuracy of this new approach. It has been discovered that the DTM does not necessitate small perturbation and is also suitably precise to both linear and nonlinear problems in physics and engineering.

Differential Transformation Method for Newtonian and non-Newtonian nanofluids flow analysis: Compared to numerical solution

In this study, a simple and high accurate series-based method called Differential Transformation Method (DTM) is used for solving the coupled nonlinear differential equations in fluids mechanic problems. The concept of the DTM is briefly introduced, and its application on two different cases, natural convection of a non-Newtonian nanofluid between two vertical plates and Newtonian nanofluid flow between two horizontal plates, has been studied. DTM results are compared with those obtained by a numerical solution (Fourth-order Runge–Kutta) to show the accuracy of the proposed method. Results reveal that DTM is very effective and convenient which can achieve more reliable results compared to other analytical methods in solving some engineering and sciences problems.

Differential transformation method for Newtonian and Non-Newtonian fluids flow analysis: comparison with HPM and numerical solution

In this study, a simple and high accurate series-based method called differential transformation method (DTM) is used for solving the coupled nonlinear differential equations in fluids’ mechanic problems. The concept of the DTM is briefly introduced, and its application for two different cases, natural convection of a non-Newtonian fluid between two vertical plates and Newtonian fluid flow between two horizontal plates, has been studied. DTM results are compared with those obtained by another series-based analytical technique namely homotopy perturbation method (HPM) and a numerical solution (Fourth-order Runge–Kutta) to show the accuracy of the proposed method. Results reveal that DTM is very effective and convenient and can achieve more suitable results compared to HPM in some areas of equations in engineering and science problems.

Analytical investigation of convective heat transfer of a longitudinal fin with temperature-dependent thermal conductivity, heat transfer coefficient and heat generation

In this article, the heat transfer through a longitudinal fin is studied. The heat transfer coefficient, thermal conductivity and heat generation are variables and supposed to be temperature-dependent. The temperature distribution in fin with longitudinal rectangular profile was carried out by using the differential transformation method (DTM) which is an analytical solution technique. For validation of the analytical solution, the heat equation is solved numerically. The temperature distribution is shown for different values of the embedding parameters. The DTM results indicate that the fin tip temperature increases with an increase in the heat generation gradient. Results reveal that DTM is very effective and convenient. Comparison of the results (DTM and numerical) was shown that the analytical method and numerical data are in a good agreement with each other. Key words: Fins, temperature dependent thermal properties, heat generation, analytical solutions, differential transformation method (DTM).

Approximate solution of the nonlinear heat transfer equation of a fin with the power-law temperature-dependent thermal conductivity and heat transfer coefficient

In this paper, differential transform method (DTM) is used to solve the nonlinear heat transfer equation of a fin with the power-law temperature-dependent both thermal conductivity and heat transfer coefficient. Using DTM, the differential equation and the related boundary conditions transformed into a recurrence set of equations and finally, the coefficients of power series are obtained based on the solution of this set of equations. DTM overcame on nonlinearity without using restrictive assumptions or linearization. Results are presented for the dimensionless temperature distribution and fin efficiency for different values of the problem parameters. DTM results are compared with special case of the problem that has an exact closed-form solution, and an excellent accuracy is observed.

Two dominant analytical methods for thermal analysis of convective step fin with variable thermal conductivity

Heat transfer in a straight fin with a step change in thickness and variable thermal conductivity which is losing heat by convection to its surroundings is developed via differential transformation method (DTM) and variational iteration method (VIM). In this study, we compare DTM and VIM results, with those of homotopy perturbation method (HPM) and an accurate numerical solution to verify the accuracy of the proposed methods. As an important result, it is depicted that the DTM results are more accurate in comparison with those obtained by VIM and HPM. After these verifications the effects of parameters such as thickness ratio, ?, dimensionless fin semi thickness,?, length ratio, ?, thermal conductivity parameter, ?, Biot number, Bi, on the temperature distribution are illustrated and explained.

Convection-radiation thermal analysis of triangular porous fins with temperature-dependent thermal conductivity by DTM

Increasing performance of porous fins is one of the common priorities nowadays. Hence attention here is given to the convection and radiation effects in the analysis of performance of a porous triangular fin with temperature-dependent thermal conductivity. Mathematical problem is solved by differential transformation method (DTM). The DTM results are compared with the numerical results obtained by fourth-order Runge–Kutta method in order to confirm the accuracy of the analytical solution. The dimensionless temperature distributions, fin efficiency and fin effectiveness are studied with respect to emerging parameters involved in the analysis. It is noted that there is an increase in fin performance in view of porous constraint.

Taylor Series Solution of the Electrohydrodynamic Flow Equation

In this work, the electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is considered. The differential equation of the problem is a highly nonlinear and singular boundary value problem (BVP). An analytical solution of the problem is presented using the differential transform method (DTM). In this method, the differential equation and related boundary conditions transformed into a recurrence set of equations and at the end, the coefficients of Taylor series are calculated based on the solution of this set of equations. DTM results are compared with the numerical solution of the problem and an excellent accuracy is observed. It showed that the proposed method overcame the nonlinearity and singularity of the problem without any need to restrict assumptions or linearization. Most of the scientific problems and engineering phenomena are usually defined based on the nonlinear differential equations of both initial value problems (IVP) and boundary value problems (BVP). It is important to remember that except a limited number of these problems, finding exact analytical solution is difficult. Therefore, researchers use semi-analytical techniques and numerical methods. The semi-analytical methods such as the perturbation method (PM), Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and differential transform method (DTM) are useful mathematical ones, which present the analytical solutions for nonlinear differential equations. In this paper, solution of electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is presented using DTM. Already, the electrohydrodynamic flow of a fluid and its governing equations is considered (1). The electrohydrodynamic flow is important in analysis of the flow meters, accelerators, pumps and magnetohydrodynamic generators. The differential equation of the problem is the nonlinear singular boundary value problem. 2 2

Solution of the Boundary Layer Equation of the Power-Law Pseudoplastic Fluid Using Differential Transform Method

The boundary layer equation of the pseudoplastic fluid over a flat plate is considered. This equation is a boundary value problem (BVP) with the high nonlinearity and a boundary condition at infinity. To solve such problems, powerful numerical techniques are usually used. Here, through using differential transform method (DTM), the BVP is replaced by two initial value problems (IVP) and then multi-step differential transform method (MDTM) is applied to solve them. The differential equation and its boundary conditions are transformed to a set of algebraic equations, and the Taylor series of solution is calculated in every sub domain. In this solution, there is no need for restrictive assumptions or linearization. Finally, DTM results are compared with the numerical solution of the problem, and a good accuracy of the proposed method is observed.

Erratum to: “Application of the differential transformation method and variational iteration method to large deformation of cantilever beams under point load”

Large deflection of a cantilever beam subjected to a tip-concentrated load is governed by a non-linear differential equation. Since it is hard to find exact or closed-form solutions for this non-linear problem, this paper investigates the aforementioned problem via the differential transformation method (DTM) and the variational iteration method (VIM), which are well-known approximate analytical solutions. The mathematical formulation is yielded to a non-linear two-point boundary value problem. In this study, we compare the DTM and VIM results, with those of Adomian decomposition method (ADM) and the established numerical solution obtained by the Richardson extrapolation in order to verify the accuracy of the proposed methods. As an important result, it is depicted from tabulated data that the DTM results are more accurate in comparison with those obtained by the VIM and ADM, which is one of the objectives of this article. Moreover, the effects of dimensionless end point load, α, on the slope of any point along the arc length and the dimensionless vertical and horizontal displacements are illustrated and explained. The results reveal that these methods are very effective and convenient in predicting the solution of such problems, and it is predicted that the DTM and VIM can find a wide application in new engineering problems.