New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Reduced Differential Transform Method for Solving Linear and Nonlinear Goursat Problem

In this paper a new method for solving Goursat problem is introduced using Reduced Differential Transform Method (RDTM). The approximate analytical solution of the problem is calculated in the form of series with easily computable components. The comparison of the methodology presented in this paper with some other well known techniques demonstrates the effectiveness and power of the newly proposed methodology.

Differential transform method for solving Richards' equation

An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.

Solving shock wave with discontinuity by enhanced differential transform method(EDTM)

An enhanced differential transform method （EDTM）, which introduces the Pad@ technique into the standard differential transform method （DTM）, is proposed. The enhanced method is applied to the analytic treatment of the shock wave. It accelerates the convergence of the series solution and provides an exact Dower series solution.

Reduced Differential Transform Method for Solving Nonlinear Biomathematics Models

In this paper,we study the approximate solutions for some of nonlinear Biomathematics models via the e-epidemic SI1I2R model characterizing the spread of viruses in a computer network and SIR childhood disease model.The reduced differential transforms method(RDTM)is one of the interesting methods for finding the approximate solutions for nonlinear problems.We apply the RDTM to discuss the analytic approximate solutions to the SI1I2R model for the spread of virus HCV-subtype and SIR childhood disease model.We discuss the numerical results at some special values of parameters in the approximate solutions.We use the computer software package such as Mathematical to find more iteration when calculating the approximate solutions.Graphical results and discussed quantitatively are presented to illustrate behavior of the obtained approximate solutions.

Application of the Hybrid Differential Transform Method to the Nonlinear Equations

In this paper, a hybrid method is introduced briefly to predict the behavior of the non-linear partial differential equations. The method is hybrid in the sense that different numerical methods, differential transform and finite differences, are used in different subdomains. Our aim of this approach is to combine the flexibility of differential transform and the efficiency of finite differences. An explicit hybrid method for the transient response of inhomogeneous nonlinear partial differential equations is presented;applying finite difference scheme on the fixed grid size is used to approximate the space discretisation, whereas the differential transform method is used for time operator. Comparison of the efficiency of the different approaches is a very important aspect of this study. In our test cases, the hybrid approach is faster than the corresponding highly optimized finite difference method in two dimensional computations. We compared our hybrid approach’s results with the exact and/or numerical solutions of PDE which obtained from Adomian Decomposition Method. Results show that the hybrid approach may be an important tool to reduce the execution time and memory requirements for large scale computations and get remarkable results in predicting the solutions of nonlinear initial value problems.

Modified Differential Transform Method for Solving Black-Scholes Pricing Model of European Option Valuation Paying Continuous Dividends

.Option pricing is a major problem in quantitative finance.The Black-Scholes model proves to be an effective model for the pricing of options.In this paper a com-putational method known as the modified differential transform method has been em-ployed to obtain the series solution of Black-Scholes equation with boundary condi-tions for European call and put options paying continuous dividends.The proposed method does not need discretization to find out the solution and thus the computa-tional work is reduced considerably.The results are plotted graphically to establish the accuracy and efficacy of the proposed method.

Approximate Solutions of Nonlinear Fractional Kolmogorov-Petrovskii-Piskunov Equations Using an Enhanced Algorithm of the Generalized Two-Dimensional Differential Transform Method

By constructing the iterative formula with a so-called convergence-control parameter, the generalized two-dimensional differential transform method is improved. With the enhanced technique, the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations are dealt analytically and approximate solutions are derived. The results show that the employed approach is a promising tool for solving many nonlinear fractional partial differential equations. The algorithm described in this work is expected to be employed to solve more problems in fractional calculus.

Solution of the time-fractional generalized Burger-Fisher equation using the fractional re duce d differential transform method

“The time-fractional generalized Burger-Fisher equation(TF-GBFE)”is used in various applied sciences and physical applications,including simulation of gas dynamics,financial mathematics,fluid mechan-ics,and ocean engineering.This equation represents a concept for the coordination of reaction systems,as well as advection,and conveys the understanding of dissipation.The Fractional Reduced Differential Transform Method(FRDTM)is used to evaluate“the time-fractional generalized Burger-Fisher equation(TF-GBFE).”Todeterminethemethod’s validity,whenthesolutionsareobtained,theyarecorrelatedto exact solutions ofα=1 order,and even for various values ofα.Three-dimensional graphs are used to depict the solutions.Additionally,the analysis of exact and FRDTM solutions indicates that the proposed approach is very accurate.

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 fourthorder Runge-Kutta method(RK4)reveal that the proposed technique is a promising tool to solve non-linear oscillators.

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

A Study on the Effects of Internal Heat Generation on the Thermal Performance of Solid and Porous Fins using Differential Transformation Method

In this study,the impacts of internal heat generation on heat transfer enhancement of porous fin is theoretical investigated using differential transform method.The parametric studies reveal that porosity enhances the fin heat dissipating capacity but the internal heat generation decreases the heat enhancement capacity of extended surface.Also,it is established that when the internal heat parameter increases to some certain values,some negative effects are recorded where the fin stores heat rather than dissipating it.This scenario defeats the prime purpose of the cooling fin.Additionally,it is established in the present study that the limiting value of porosity parameter for thermal stability for the passive device increases as internal heat parameter increases.This shows that although the internal heat parameter can help assist higher range and value of thermal stability of the fin,it produces negative effect which greatly defeats the ultimate purpose of the fin.The results in the work will help in fin design for industrial applications where internal heat generation is involved.

Mean Square Solutions of Second-Order Random Differential Equations by Using the Differential Transformation Method

The differential transformation method (DTM) is applied to solve the second-order random differential equations. Several examples are represented to demonstrate the effectiveness of the proposed method. The results show that DTM is an efficient and accurate technique for finding exact and approximate solutions.

Theory and Semi-Analytical Study of Micropolar Fluid Dynamics through a Porous Channel

In this work,We are looking at the characteristics of micropolar flow in a porous channel that’s being driven by suction or injection.The working of the fluid is described in the flowmodel.We can reduce the governing nonlinear partial differential equations(PDEs)to a model of coupled systems of nonlinear ordinary differential equations using similarity variables(ODEs).In order to obtain the results of a coupled system of nonlinear ODEs,we discuss a method which is known as the differential transform method(DTM).The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs.To observe beast agreement between analytical method and numerical method,we compare our result with the Rung-Kutta method of order four(RK4).We also provide simulation plots to the obtained result by using Mathematica.Onthese plots,we discuss the effect of different parameters which arise during the calculation of the flow model equations.

Application of differential transformation method (DTM) for heat and mass transfer in a porous channel

In the present paper a differential transformation method(DTM)is used to obtain the solution of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall.The comparison between the results from the differential transfomiation method and numerical method are in well agreement which proofs the capability of this method for solving such problems.After this validity,results are investigated for the velocity and temperature for various values of Reynolds number,Prandtl number and power law index.

Thermal analysis of a constructal T-shaped porous fin with simultaneous heat and mass transfer

The present work establishes an analytical model for computing the temperature distribution, fin efficiency and optimum design parameters of a constructal T-shaped porous fin operating in fully wet condition. For more practical results, this study considers a cubic polynomial relationship between the humidity ratio of saturated air and the corresponding fin surface temperature. The temperature distribution has been determined by solving the highly non-linear governing equations using a semi-analytical transformation technique called Differential Transform Method. A comparison of the results with that of a numerical model shows that this transformation method is a very efficient and convenient tool for solution of non-linear problems. The effects of various geometric, thermo-physical and psychometric parameters on the temperature distribution, fin efficiency and optimum design condition have been investigated. Also, a comparison has been presented between solid and porous fins and the results point out that by selecting an appropriate value of porosity, the heat transfer rate can be increased than the corresponding solid fin.

Dynamical Behaviors of Nonlinear Coronavirus (COVID-19) Model with Numerical Studies

The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious diseases.For the new strain of coronavirus(COVID-19),there is no vaccine to protect people and to prevent its spread so far.Instead,control strategies associated with health care,such as social distancing,quarantine,travel restrictions,can be adopted to control the pandemic of COVID-19.This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods:the homotopy perturbation method(HPM)and the modied reduced differential transform method(MRDTM).We invoke a novel signal ow graph that is used to describe the COVID-19 model.Through our mathematical studies,it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus.The numerical simulation results are in reasonable agreement with the study predictions.The free equilibrium and stability point for the COVID-19 model are investigated.Also,the existence of a uniformly stable solution is proved.

Approximate solutions to MHD Falkner-Skan flow over permeable wall

The magnetohydrodynamic （MHD） Falkner-Skan boundary layer flow over a permeable wall in the presence of a transverse magnetic field is examined. The approximate solutions and skin friction coefficients of the MHD boundary layer flow are obtained by using a method that couples the differential transform method （DTM） with the Pade approximation called DTM-Pade. The approximate solutions are expressed in the form of a power series that can be easily computed with an iterative procedure. The approximate solutions are tabulated, plotted for the values of different parameters and compared with the numerical ones obtained by employing the shooting technique. It is found that the approximate solution agrees very well with the numerical solution, showing the reliability and validity of the present work. Moreover, the effects of various physical parameters on the boundary layer flow are presented graphically and discussed.

Free Convection of a Viscous Electrically Conducting Fluid Past a Stretching Surface

Free convection of a viscous electrically conducting liquid past a vertical stretching surface is investigated in the presence of a transverse magnetic field.Natural convection is driven by both thermal and solutal buoyancy.The original partial differential equations governing the problem are turned into a set of ordinary differential equations through a similar variables transformation.This alternate set of equations is solved through a Differential Transform Method(DTM)and the Pade approximation.The response of the considered physical system to the non-dimensional parameters accounting for the relative importance of different effects is assessed considering different situations.

Bifurcation and Chaos Analysis of Nonlinear Rotor System with Axial-grooved Gas-lubricated Journal Bearing Support

Axial-grooved gas-lubricated journal bearings have been widely applied to precision instrument due to their high accuracy, low friction, low noise and high stability. The rotor system with axial-grooved gas-lubricated journal bearing support is a typical nonlinear dynamic system. The nonlinear analysis measures have to be adopted to analyze the behaviors of the axial-grooved gas-lubricated journal bearing-rotor nonlinear system as the linear analysis measures fail. The bifurcation and chaos of nonlinear rotor system with three axial-grooved gas-lubricated journal bearing support are investigated by nonlinear dynamics theory. A time-dependent mathematical model is established to describe the pressure distribution in the axial-grooved compressible gas-lubricated journal bearing. The time-dependent compressible gas-lubricated Reynolds equation is solved by the differential transformation method. The gyroscopic effect of the rotor supported by gas-lubricated journal bearing with three axial grooves is taken into consideration in the model of the system, and the dynamic equation of motion is calculated by the modified Wilson-0-based method. To analyze the unbalanced responses of the rotor system supported by finite length gas-lubricated journal bearings, such as bifurcation and chaos, the bifurcation diagram, the orbit diagram, the Poincar6 map, the time series and the frequency spectrum are employed. The numerical results reveal that the nonlinear gas film forces have a significant influence on the stability of rotor system and there are the rich nonlinear phenomena, such as the periodic, period-doubling, quasi-periodic, period-4 and chaotic motion, and so on. The proposed models and numerical results can provide a theoretical direction to the design of axial-grooved gas-lubricated journal bearing-rotor system.