Multi-step Differential Transform Method Research Articles

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

Cloud droplets scavenging of gaseous pollutant from the atmosphere: Nonlinear modelling and analyses

The principle of how hazardous pollutants are annulled by rain, dew or precipitations has been a subject of interest in addressing environmental challenges and climate change. This scavenging process when studied generates nonlinear dynamic models that require robust techniques for obtaining their solutions before use for predictions. Consequently, this work presents analytical solutions to such unsteady processes using PADE-Homotopy perturbation method (PADE-HPM) and Multi-step differential transform method (MSDTM). The obtained solutions are verified via a numerical finite difference method (FDM). These methods demonstrate adequate aptitude for predicting the scavenging processes. Furthermore, the study of the controlling parameters shows that reduction in the formation rate of the annulling agents causes a corresponding decrease in the density of the available raindrop for scavenging. Meanwhile, augmentation in formation rate rapidly neutralizes pollutants from the atmosphere. A synonymous impact is realized for growth rate parameter. In addition, it is established that as the transformation rates of hazardous gases reduce while scavenging processes are augmented, the threat posing pollutants would be completely annulled. It is envisaged that the present study will find usage in overcoming one of the dares facing the world at large on scavenging of hazardous pollutants.

Radiative MHD flow of Rivlin–Ericksen nanofluid of grade three through porous medium with uniform heat source

BackgroundThe paper investigates the analytical and numerical solution of the radiation effect on MHD flow of Rivlin–Ericksen nanofluid of grade three through a porous medium with a uniform heat source between two vertical flat plates. The governing equations are solved analytically using multi-step differential transform method (MDTM) and numerically using finite difference method (FDM) and shooting method by designing MATLAB and Mathematica algorithms. The study discovered that the MDTM, FDM, and shooting methods are effective for solving nonlinear differential equations like this one.ResultsGraphs and tables show the influence of different parameters on velocity and temperature. Figures and tables show the comparisons between current outcomes and previous results that are accessible.ConclusionsThe present results showed that the analytical and numerical solutions agree well with previously published outcomes.

An Approximate Solution to Predator-prey Models Using The Differential Transform Method and Multi-step Differential Transform Method, in Comparison with Results of The Classical Runge-kutta Method

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.

The kinematics behaviour of coupled pendulum using differential transformation method

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

NUMERICAL SIMULATION OF DENGUE FEVER (SIR MODEL) USING DIFFERENTIAL TRANSFORM METHOD, MULTI-STEP DIFFERENTIAL TRANSFORM METHOD AND RK4 METHOD

This study investigates the application of the differential transformation method(DTM), multi-step differential transform method(MsDTM) with step-size and RK4 method (Mathematica) for finding the numerical solution of the SIR model of dengue fever in epidemiology. This model is a system of non-linear ordinary differential equations that have no analytic solution. Both the methods DTM and MsDTM are applied directly without any linearization, perturbation or discretization in the model equations to obtain semi-analytic solutions. The accuracy of the MSDTM is excellent and comparable to the RK4 method of Mathematica.

Nonlinear vibration analysis of axially moving string

In this paper, the nonlinear transverse vibration arising from axially moving string is investigated analytically. Translating string eigenfunctions are employed to reduce a partial-differential equation to a set of second degree of freedom nonlinear systems. The multi-step differential transform method (MsDTM) is proposed in order to find accurate solutions of time-varying length of an axially moving string. To illustrate the applicability and accuracy of MsDTM, the axial motion model is treated with two different sets of parameters. The relationship between transverse displacement, angular velocity and time is obtained and discussed. The effect of the string’s speed, damping and tension on the transverse displacement of the string are also taken into consideration.

A semianalytical technique for MHD peristalsis of pseudoplastic nanofluid with temperature‐dependent viscosity: Application in drug delivery system

AbstractAn analysis has been implemented to study the influences of nonconstant viscosity and magnetohydrodynamics on pseudoplastic nanofluid through a porous medium. Ohmic dissipation, chemical reaction, and heat generation are taken into consideration. The current problem is debated under the molds of tiny or zero and approximation. Two models of nonconstant viscosity are deliberated. Model (I)—all parameters are nondimensional and have been measured as constants inside the flow. Model (II)—all these stated nondimensional parameters have been considered to differ with the temperature. Comparison among the solutions achieved by utilizing numerical results and multi‐step differential transform method (Ms‐DTM) is displayed in excellent agreement. Attention is dedicated to , ‐, and parameters. The governing equations for each case have been explained by an easy and highly perfect series established seminumerical Ms‐DTM utilizing Mathematica 11, which uses mathematical software package. Nanofluids are active for drug carrying and drug delivery systems because of the control with the velocity of fluid.

Numerical Solution of Tumor-immune Model including small Molecule Drug by Multi-step Differential Transform Method

Numerical Solution of Tumor-immune Model including small Molecule Drug by Multi-step Differential Transform Method

Temperature-Dependent Electrical Conductivity and Thermal Radiation Effects on MHD Peristaltic Motion of Williamson Nanofluids in a Tapered Asymmetric Channel

ABSTRACTThis paper is intended for dealing with the peristaltic flow of an electrically conducting Williamson nanofluid in a tapered asymmetric channel through a porous medium with heat and mass transfer. In the current paper, temperature-dependent electrical conductivity formulation was introduced for the first time in peristaltic literature. The flow is pervaded by an oblique uniform magnetic field. The present investigation includes the influences of thermal radiation, Joule heat, viscous dissipation, Hall Current, 1st order chemical reaction, and Dofour and Soret numbers. Current problem is reformulated under the molds of low Reynolds number and long wavelength approximation. Afterwards, semi analytical solutions have been evaluated for the distributions of velocity, temperature, nanoparticle concentrations as well as longitudinal pressure gradient. Solutions can be obtained by using multi-step differential transform method (MS-DTM), a reliable and powerful technique that improve accuracy and overcome drawbacks raised in using the standard differential transform method (DTM). Detailed comparisons have been made at different values of 𝑥 through graphs by Ms-DTM. The graphically results were prepared to visualize the effects of various physical parameters of interest. The semi-analytical results had shown that, as the thermal radiation increases, the nanoparticles diameter and concentration of fluid increase (thermal radiation is a decreasing function in temperature when the temperature decreases the diameter of the nanoparticles increases i.e. the volume of nanoparticle and its concentration increases and become more effective near to tumor tissues). Consequently, it can be used as agents for radiation therapy, generate localized raises in radiation doses and selectively target tumor cells for localized damage (Radiotherapy of oncology).

Application of multi-step differential transform method to the nonlinear behaviour of cloud droplets on gaseous atmospheric pollutant removal

The determination of the concentration of gaseous atmospheric pollutants and their neutralization by rain or dew have been the subjects of research interests over the years. This paper presents close-form solutions to the transient behaviour and determination of the concentration of gaseous atmospheric pollutants using Multi-Step Differential Transform Method (MsDTM). Also, parametric studies are carried out to properly understand the chemistry of the process and the associated density of droplet required for complete neutralization. The generated analytical solution obtained by MsDTM are verified with an efficient numerical (fourth order Runge-Kutta, RK45) scheme. Results show that MsDTM is an efficient scheme for the problem. Also, the parametric study performed in this work shows that an increase in the cloud water droplet formation rate increases the atmospheric rain drop density which consequently reduces the cumulative concentration of gaseous pollutants in the atmosphere. The same effect is established for the growth rate constant of the atmospheric rain drop parameter. This work will be useful in solving to a large extent one of the challenges facing the world at large on gaseous atmospheric pollutants as well as in the determination of different pollutants neutralization period under different atmospheric conditions.

Multi-Step Differential Transform Method for Solving the Influenza Virus Model with Disease Resistance

The spread of the influenza virus with disease resistance has been modeled by the SEIR epidemic model. The model is of the form a system of four nonlinear differential equations where its exact solution is difficult to be determined. In this paper, the model is solved by a multi-step differential transform method (MS-DTM), which is a semi-analytical method. In this case, the domain of computation is divided into a finite number of sub-intervals. At each sub-interval, we apply DTM and continuity condition to ensure the continuity solutions. To see the effectiveness of the MS-DTM, we perform some comparative study between the MS-DTM and the MATLAB ode45 routine. Our MS-DTM solutions have good agreement with those of ode45 routine. The accuracy of MS-DTM can be improved by taking a smaller step size or increase the number of terms in each subinterval.

Efficient Solution of Nonlinear Duffing Oscillator

In this paper, the efficient multi-step differential transform method (EMsDTM) is applied to get the accurate approximate solutions for strongly nonlinear duffing oscillator. The main improvement of EMsDTM which is to reduce the number of arithmetic operations, is thoroughly investigated and compared with the classic multi-step differential transform method (MsDTM). To illustrate the applicability and accuracy of the new method, six case studies of the free undamped and forced damped conditions are considered. The periodic response curves of both MsDTM and EMsDTM methods are obtained and contrasted with the exact solution or the numerical solution of Runge Kutta 4th order (RK4) method. This approach can be easily extended to other nonlinear systems and therefore is widely applicable in engineering and other sciences.

COMPUTATION OF TRANSIENT RADIATIVE REACTIVE THERMOSOLUTAL MAGNETOHYDRODYNAMIC CONVECTION IN INCLINED MHD HALL GENERATOR FLOW WITH DISSIPATION AND CROSS DIFFUSION

The present article investigates the collective influence of chemical reaction, viscous dissipation and Hall current magnetic effects on timedependent radiative magnetohydrodynamic flow, heat and mass transfer from an inclined wall embedded in a homogenous, isotropic highpermeability porous medium. The model developed is relevant to near wall magnetohydrodynamic energy generator transport phenomena in which chemical corrosion effects may arise during operations. The governing non-linear partial differential equations for mass, momentum, energy and species conservation are transformed into a system of coupled non-linear dimensionless partial differential equations with appropriate similarity variables. The normalized conservation equations are then solved with a robust finite element method (MATLABFEM) subject to corresponding initial and boundary conditions. Important dimensionless parameters emerging are Eckert number, thermal Grashof number, solutal Grashof number, magnetic body force parameter, Hall parameter, permeability parameter, Dufour number, Soret number, time, radiation-conduction parameter, chemical reaction parameter, heat absorption parameter, Prandtl number, Schmidt number and wall angle. Primary velocity is enhanced with Eckert number, thermal Grashof number, solutal Grashof number, Hall parameter, permeability parameter, Dufour number, Soret number, radiation-conduction parameter and time whereas it is reduced with first order chemical reaction parameter, heat absorption, magnetic body force parameter, Prandtl number, Schmidt number and wall inclination. Secondary velocity is elevated with Eckert number, solutal Grashof number, thermal Grashof number, magnetic body force parameter, Hall parameter, radiation-conduction parameter, Dufour number, Soret number and time whereas it is suppressed with reaction parameter, heat absorption, Prandtl number, Schmidt number and wall inclination. Temperature is enhanced with Eckert number, Dufour number, heat absorption, radiation-conduction parameter and time whereas it is depressed with Prandtl number. Species concentration is reduced with increasing chemical reaction parameter (destructive homogenous reaction) and Schmidt number whereas it is elevated with Soret number and time. Extensive discussion of the finite element formulation, convergence and validation is provided Skin friction, Nusselt number and Sherwood number distributions are also provided for selected parameter variation. Validation of solutions with published literature is also included for several special cases, namely non-reactive, non-dissipative flow in the absence of heat generation or absorption. Further validation is included using a multi-step differential transform method (MS-DTM). The present simulations provide an interesting insight into complex fluid/thermal/species diffusion characteristics in the boundary layer region of relevance to working MHD generator systems.

Approximate analytical solutions to nonlinear peristaltic flow with temperature-dependent viscosity parameters: Application of multi-step differential transform method (MsDTM)

This paper investigates the influence of temperature-dependent viscosity on peristaltic flow of a Newtonian fluid past a vertical asymmetric channel through a porous medium in the presence of heat ...

Peristaltic Blood Flow with Gold Nanoparticles on a Carreau Nanofluid through a Non-Darcian Porous Medium

The article investigates the influences of a variable thermal conductivity and wall slip on a peristaltic motion of Carreau nanofluid. The model is concerned with heat and mass transfer inside asymmetric channel. The blood is considered as the base Carreau non-Newtonian fluid and gold (Au) as nanoparticles stressed upon. The Fronchiener effect of the non-Darcian medium is taken in consideration. The system is stressed upon a strong magnetic field and the Hall currents are completed. The problem is modulated mathematically by a system of non-linear partial differential equations which describe the fluid velocity, temperature and concentration. The system is reformulated under the approximation of long wavelength and low Reynolds number. It is solved on using multi-step differential transform method (Ms-DTM) as a semi-analytical method. A gold nanoparticle has increased the temperature distribution which is of great importance in destroying the cancer cells.

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.

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

Applications of Adaptive Multi Step Differential Transform Method to Singular Perturbation Problems Arising in Science and Engineering

In this paper, piecewise-analytical and numerical solutio ns of singular perturbation initial-value problems are obt ained by an adaptive multi-step differential transform method (MsDTM). The principle of the method is introduced, and then applied to different types of practical problems arising in science an d engineering. Analytical and numerical solutions are obtained using piecewise convergent series with easily computable components over a sequence of variable-length sub-intervals. Numerical results are compared to those obtained by the classical MsDTM and the Runge-Kutta method. The results demonstrate the reliability and efficie ncy of the method in solving the considered problems.

Numerical Solution of MHD Flow of Micropolar Fluid with Heat and Mass Transfer towards a Stagnation Point on a Vertical Plate

The paper investigates the numerical solution of problem of magnetohydrodynamic (MHD) micropolar fluid flow with heat and mass transfer towards a stagnation point on a vertical plate. In this study, we consider both strong concentrations (n = 0) and weak concentrations (n = 1/2). The governing equations have been transformed into nonlinear ordinary differential equations by applying the similarity transformation and have been solved numerically by using the finite difference method (FDM) and analytically by using (DTM). The effects of various governing parameters, namely, material parameter, radiation parameter, magnetic parameter, Prandtl number, Schmidt number, chemical reaction parameter and Soret number on the velocity, microrotation, temperature and concentration have been computed and discussed in detail through some figures and tables. In order to verify the accuracy of the present results, we have compared these results with the analytical solutions by using the differential transform method (DTM) and the multi-step differential transform method (MDTM). It is observed that this approximate numerical solution is in good agreement with the analytical solution.