Semi-Numerical, Semi-Analytical Approximations of the Rayleigh Equation for Gas-Filled Hyper-Spherical Bubble

2018 ◽  
Vol 16 (01) ◽  
pp. 1850094 ◽  
Author(s):  
Yupeng Qin ◽  
Zhen WANG ◽  
Li Zou ◽  
Mingfeng He
Keyword(s):  
Fourth Order ◽  
Differential Transform Method ◽  
Rayleigh Equation ◽  
Kutta Method ◽  
Transform Method ◽  
Spherical Bubble ◽  
Analytical Approximations ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Runge Kutta

A new semi-numerical, semi-analytical approach based on the differential transform method is proposed to solve the problems of a gas-filled hyper-spherical bubble governed by the Rayleigh equation. Semi-numerical, semi-analytical approximations are constructed for the Rayleigh equation in the form of piecewise functions. The proposed approach is compared with the standard fourth-order Runge–Kutta method and the standard differential transform method, respectively. The results reveal two main benefits of the new approach, one is that it possesses result with higher precision than the standard fourth-order Runge–Kutta method, the other is that it remains valid and accurate for longer time compared to the standard differential transform method. In addition, we also consider the Rayleigh equation in [Formula: see text] dimensions when the surface tension is not zero.

The Multi-Step Differential Transform Method and Its Application to Determine the Solutions of Non-Linear Oscillators

2012 ◽  
Vol 4 (04) ◽  
pp. 422-438 ◽  
Author(s):  
Vedat Suat Ertürk ◽  
Zaid M. Odibat ◽  
Shaher Momani
Keyword(s):  
Fourth Order ◽  
Differential Transform Method ◽  
Kutta Method ◽  
Transform Method ◽  
Promising Tool ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Non Linear ◽  
Runge Kutta

AbstractIn 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.

scholarly journals Analytical Study of Fractional-Order Multiple Chaotic FitzHugh-Nagumo Neurons Model Using Multistep Generalized Differential Transform Method

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Shaher Momani ◽  
Asad Freihat ◽  
Mohammed AL-Smadi
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Analytical Study ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

The multistep generalized differential transform method is applied to solve the fractional-order multiple chaotic FitzHugh-Nagumo (FHN) neurons model. The algorithm is illustrated by studying the dynamics of three coupled chaotic FHN neurons equations with different gap junctions under external electrical stimulation. The fractional derivatives are described in the Caputo sense. Furthermore, we present figurative comparisons between the proposed scheme and the classical fourth-order Runge-Kutta method to demonstrate the accuracy and applicability of this method. The graphical results reveal that only few terms are required to deduce the approximate solutions which are found to be accurate and efficient.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmed Alawneh
Keyword(s):  
Enzyme Kinetics ◽  
Fractional Derivatives ◽  
Differential Transform Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Kutta Method ◽  
Transform Method ◽  
Enzyme Substrate ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential

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.

scholarly journals Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Brahim Benhammouda ◽  
Hector Vazquez-Leal ◽  
Luis Hernandez-Martinez
Keyword(s):  
Approximate Solutions ◽  
Perturbation Parameter ◽  
Convergent Series ◽  
Differential Transform Method ◽  
Transform Method ◽  
Analytical Approximations ◽  
Runge Kutta Method ◽  
Power Series Solutions ◽  
Differential Transform ◽  
Fifth Order

This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. The main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter.

SOLUTION OF HIGHER-ORDER ABEL EQUATIONS BY DIFFERENTIAL TRANSFORM METHOD

2012 ◽  
Vol 23 (09) ◽  
pp. 1250056 ◽  
Author(s):  
SUPRIYA MUKHERJEE ◽  
DEBKALPA GOSWAMI ◽  
BANAMALI ROY
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Approximate Solutions ◽  
Higher Order ◽  
Differential Transform Method ◽  
Transform Method ◽  
Differential Transform ◽  
Abel Equations ◽  
Runge Kutta ◽  
Abel Differential Equations

In this paper, the second-, third- and fourth-order Abel equations are solved. The differential transform method (DTM) is used to compute approximate solutions of the nonlinear ordinary Abel differential equations. The results are compared with the results obtained by the classical Runge–Kutta (RK4) method. Figures are presented to show the reliability and simplicity of the method.

Numerical Approximations to Solution of Biochemical Reaction Model

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Ahmet Yildirim ◽  
Ahmet Gökdogan ◽  
Mehmet Merdan
Keyword(s):  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Transformation Method ◽  
Reaction Model ◽  
Biochemical Reaction ◽  
Graphical Form ◽  
Kutta Method ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform

In this paper, approximate analytical solution of biochemical reaction model is used by the multi-step differential transform method (MsDTM) based on classical differential transformation method (DTM). Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. Results are given explicit and graphical form.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

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