Generalized Differential Transform Method Research Articles

Arrhenius energy effect on the rotating flow of Casson nanofluid with convective conditions and velocity slip effects: Semi‐numerical calculations

AbstractThe present paper focuses on the effects of Arrhenius activation energy on the hydromagnetic rotating flow of Casson nanofluid. Entropy production, viscous dissipation Brownian dispersion, porous medium, velocity slip, magnetic field, convective heat, and mass conditions are taken into consideration. Transformation schemes are used to extract a nonlinear system of the differential equation. The generalized differential transform method (GDTM) is used to compute/obtain the solutions of the present system. Results for the distributions of velocities, nano concentration, and temperature are scrutinized graphically. The accuracy of GDTM is proved via a comparison of numerical and graphical results with the nearest published results by Hayat et al. and found to be excellent. Graphical results are extracted for different values of pertinent parameters versus velocities, nano concentration, and temperature profiles. The results show that more heat is produced due to the process of thermal radiation for which entropy is knowingly improved.

Concentration-dependent viscosity effect on magnetonano peristaltic flow of Powell-Eyring fluid in a divergent-convergent channel

The viscosity of fluids has an essential role in blood circulation through veins and arteries. So, this paper focuses on the influences of dependent-viscosity on the peristaltic flow of Powell-Eyring fluid in a divergent-convergent channel. Viscosity is supposed to vary with the temperature and concentration fluid. A theoretical relation to the concentration-dependent viscosity is introduced mathematically for the Powell-Eyring nanofluid for the first time in our literature. The formulation of the model contains a highly nonlinear system of PDEs that converted to ODEs by physical assumptions to low Reynolds number and long wavelength. A semi-analytical solution was obtained in different theoretical cases of the divergent-convergent channel using the generalized differential transform method (GDTM). Results for velocity profile, temperature distribution, and concentration distribution as well as extra stress tensor profile versus various values flow parameters have been constructed. Results show that the variation parameters of viscosity α, and β can control with flow velocity at each part of channel wall making them improve/get better the blood circulation in veins (Circulatory system). Furthermore, the concentration-dependent viscosity parameter increases the temperature distribution of fluid while it declines the nanoparticle diameter of fluid.

Application of Generalized Differential Transform Method to Special Cases of Riccati Differential Equation of Fractional Order

In this paper, we acquire the inexact solutions of Special cases of Riccati Differential equation of Fractional order using Generalized Differential Transform Method (GDTM). The fractional derivatives are described in the Caputo sense. Accuracy and competence of the proposed method is verified through numerical solution of some special cases of Riccati Differential equation of fractional order. The obtained results reveal that the performance of the proposed method is specific and predictable.

Solution of fractional telegraph equation with Riesz space-fractional derivative

In this study, the so-called generalized differential transform method (GDTM) is developed to derive a semi- analytical solution for fractional partial differential equations which involves Riesz space fractional derivative. We focus primarily on implementing the novel algorithm to fractional telegraph equation with Riesz space-fractional derivative. Some theorems are presented to obtain new algorithm, as well as the error bound is found. This method is dealing with separating the main equation into sub-equations and applying transformation for sub-equations to attain compatible recurrence relations. This process will allow to obtain semi-analytical solution using inverse transformation. To illustrate the reliability and capability of the method, some examples are provided. The results reveal that the algorithm is very effective and uncomplicated.

Analytic study for fractional coupled Burger’s equations via Sumudu transform method

AbstractIn this work, we aim to apply a reliable analytic algorithm based on homotopy perturbation Sumudu transform method (HPSTM) to examine the nonlinear time-fractional coupled Burger’s equations. The approximate analytical solution and some numerical examples show the accuracy and efficiency of the proposed method, which is simple and accurate in comparison to the Adomain decomposition method (ADM), homotopy perturbation method (HPM) and generalized differential transform method (GDTM).

"Approximate solution for the two-dimensional Volterra-Integro differential equation of fractional order by using Generalized Deferential Transform method "

"In this work, an efficient generalized differential transform method (GDTM) is proposed for solving the twodimensional Volterra-Integro differential equation (2-DVIDE) of fractional order. The results of the proposed method are compared with exact solution, a numerical example is considered for testing the accuracy and validity of this method."

Generalized Differential Transform Method for Solving RLC Electric Circuit of Non-Integer Order

AbstractSystematic construction of fractional ordinary differential equations [FODEs] has gained much attention nowadays research because dimensional homogeneity plays a major role in mathematical modeling. In order to keep up the dimension of the physical quantities, we need some auxiliary parameters. When we utilize auxiliary parameters, the FODE turns out to be more intricate. One of such kind of model is non-homogeneous fractional second order RLC circuit. To solve this kind of complicated FODEs, we need proficient modern analytical method. In this paper, we use two different methods, one is modern and the other is traditional, namely generalized differential transform Method (GDTM) and Laplace transform method (LTM) to obtain the analytical solution of non-homogeneous fractional second order RLC circuit. We present the solution in terms of convergent series. Though GDTM and LTM are capable to produce the exact solution of fractional RLC circuit, great strength of GDTM over LTM is that differential transform of initial conditions occupy the coefficients of first two terms in series solution so that we arrive exact solution with few iterations and also, it does not allow the noise terms while computing the coefficients. Due to this, GDTM takes less time to converge than LTM and it has been demonstrated. Furthermost, we discuss the characteristics of non-homogeneous fractional second order RLC circuit through numerical illustrations.

Numerical Technique for a System of Nonlinear Partial Differential Equations of the Time-Fractional Order

In this article, generalized differential transform method (GDTM) has been employed to obtain traveling wave solutions for some Systems of nonlinear wave equations of time-fractional order. This demonstrates the validity and great potential of the generalized differential transform method (GDTM) for solving system of nonlinear fractional partial differential equations NFDEs. Thus NFDEs can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the generalized differential transform method (GDTM). The fractional derivative is described in the Caputo fractional derivative sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms.

Generalized Differential Transform Method for Solving Discrete Complex Cubic Ginzburg–Landau Equation

In this paper, generalized differential transform method (GDTM) is applied to solve discrete complex cubic Ginzburg–Landau (DCCGL) equation which is a famous nonlinear difference-differential equation (NDDE). GDTM approximate solutions for various discrete soliton solutions of DCCGL such as discrete bright soliton, discrete dark soliton, and discrete alternating soliton are obtained. Also this method is successfully employed to obtain approximate solution for dark solitary wave solution of integrable discrete nonlinear Schrödinger (IDNS) equation. Numerical results compared with their corresponding numerical and analytical solutions to show the efficiency and high accuracy of the considered method.

APPLICATION OF GENERALIZED DIFFERENTIAL TRANSFORM METHOD TO FRACTIONAL ORDER RICCATI DIFFERENTIAL EQUATION AND NUMERICAL RESULT

In the present paper, we obtain the approximate solutions of Fractional Order Riccati Differential Equation by Generalized Differential Transform Method (GDTM). The fractional derivatives are described in the Caputo sense. Next we give eight numerical examples for sake of illustration of our method. Finally, we demonstrate our findings by means of two graphs by making use of MATHEMATICA software. AMS Subject Classification: 26A33, 35A22, 65L05, 34A08

Efficient analytic method for solving nonlinear fractional differential equations

In this paper, we propose a new approach of the generalized differential transform method (GDTM) for solving nonlinear fractional differential equations. In GDTM, it is a key to derive a recurrence relation of generalized differential transform (GDT) associated with the solution in the given fractional equation. However, the recurrence relations of complex nonlinear functions such as exponential, logarithmic and trigonometry functions have not been derived before in GDTM. We propose new algorithms to construct the recurrence relations of complex nonlinear functions and apply the GDTM with the proposed algorithms to solve nonlinear fractional differential equations. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed technique is robust and accurate for solving fractional differential equations.

Application of the Multistep Generalized Differential Transform Method to Solve a Time-Fractional Enzyme Kinetics

The multistep differential transform method is first employed to solve a time-fractional enzyme kinetics. This enzyme-substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The fractional derivatives are described in the Caputo sense. A comparative study between the new algorithm and the classical Runge-Kutta method is presented in the case of integer-order derivatives. The results demonstrate reliability and efficiency of the algorithm developed.

Comparing Numerical Methods for Solving Time-Fractional Reaction-Diffusion Equations

Multivariate Padé approximation (MPA) is applied to numerically approximate the solutions of time-fractional reaction-diffusion equations, and the numerical results are compared with solutions obtained by the generalized differential transform method (GDTM). The fractional derivatives are described in the Caputo sense. Two illustrative examples are given to demonstrate the effectiveness of the multivariate Padé approximation (MPA). The results reveal that the multivariate Padé approximation (MPA) is very effective and convenient for solving time-fractional reaction-diffusion equations.

Numerical solutions of time‐fractional Burgers equations

PurposeThe purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.Design/methodology/approachIn these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.FindingsThe paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.Research limitations/implicationsBurgers and coupled Burgers equations with time‐fractional derivative used.Practical implicationsThe implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.Originality/valueThe numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

Adaptation of Differential Transform Method for the Numeric‐Analytic Solution of Fractional‐Order Rössler Chaotic and Hyperchaotic Systems

A new reliable algorithm based on an adaptation of the standard generalized differential transform method (GDTM) is presented. The GDTM is treated as an algorithm in a sequence of intervals (i.e., time step) for finding accurate approximate solutions of fractional‐order Rössler chaotic and hyperchaotic systems. A comparative study between the new algorithm and the classical Runge‐Kutta method is presented in the case of integer‐order derivatives. The algorithm described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

Generalized Differential Transform Method to Space‐Time Fractional Telegraph Equation

We use generalized differential transform method (GDTM) to derive the solution of space‐time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag‐Leffler functions.

On the solutions of time-fractional reaction–diffusion equations

In this paper, a new application of generalized differential transform method (GDTM) has been used for solving time-fractional reaction–diffusion equations. To illustrate the reliability of the method, some examples are provided.

Analytic study on time-fractional Schrödinger equations: exact solutions by GDTM

In this work, we present a framework to obtain exact solutions to linear and nonlinear time-fractional Schrödinger equations. The generalized differential transform method (GDTM) is employed to derive analytical solutions for these equations. Some examples are tested and the results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order.