scholarly journals Comparative study of two techniques on some nonlinear problems based ussing conformable derivative

2020 ◽  
Vol 9 (1) ◽  
pp. 470-482
Author(s):  
Aniqa Zulfiqar ◽  
Jamshad Ahmad
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Three Dimensional ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Conformable Derivative ◽  
Transform Method ◽  
Fractional Equations ◽  
Differential Transform ◽  
Foam Drainage Equation

AbstractIn this paper, three eminent types of time-fractional nonlinear partial differential equations are considered, which are the fractional foam drainage equation, fractional Gardner equation, and fractional Fornberg–Whitham equation in the sense of conformable fractional derivative. The approximate solutions of these considered problems are constructed and discussed using the conformable fractional variational iteration method and conformable fractional reduced differential transform method. The conformable derivative is one of the admirable choices to handle nonlinear physical problems of different fields of interest. Comparisons of approximate solution obtained by two techniques, to each other and with the exact solutions are also presented and affirm that the considered methods are efficient and reliable techniques to study other nonlinear fractional equations and models in the sense of conformable derivative. To explain the effects of several parameters and variables on the movement, the approximate results are shown in tables and two-and three-dimensional surface graphs.

scholarly journals The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Hassan Kamil Jassim
Keyword(s):  
Fractional Derivative ◽  
Wave Equations ◽  
Three Dimensional ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Derivative Operator ◽  
Transform Method ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Dimensional Diffusion

We used the local fractional variational iteration transform method (LFVITM) coupled by the local fractional Laplace transform and variational iteration method to solve three-dimensional diffusion and wave equations with local fractional derivative operator. This method has Lagrange multiplier equal to minus one, which makes the calculations more easily. The obtained results show that the presented method is efficient and yields a solution in a closed form. Illustrative examples are included to demonstrate the high accuracy and fast convergence of this new method.

scholarly journals Analytic Solution of Steady Three-Dimensional Problem of Condensation Film on Inclined Rotating Disk by Differential Transform Method

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Mohammad Mehdi Rashidi ◽  
Seyyed Amin Mohimanian pour
Keyword(s):  
Three Dimensional ◽  
Rotating Disk ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Temperature Profiles ◽  
Dimensional Problem ◽  
Governing Equations ◽  
Transform Method ◽  
Differential Transform ◽  
Similarity Method

The differential transform method (DTM) is applied to the steady three-dimensional problem of a condensation film on an inclined rotating disk. With similarity method, the governing equations can be reduced to a system of nonlinear ordinary differential equations. The approximate solutions of these equations are obtained in the form of series with easily computable terms. The velocity and temperature profiles are shown and the influence of Prandtl number on the temperature profiles is discussed in detail. The validity of our solutions is verified by the numerical results.

scholarly journals Enhanced Multistage Differential Transform Method: Application to the Population Models

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Taylor Series ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Initial Condition ◽  
Initial Value ◽  
Transform Method ◽  
Large Domain ◽  
Runge Kutta Method ◽  
Differential Transform

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).

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

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Asad Freihat ◽  
Shaher Momani
Keyword(s):  
Fractional Order ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Differential Transform Method ◽  
Time Step ◽  
Transform Method ◽  
Runge Kutta Method ◽  
Differential Transform ◽  
Generalized Differential ◽  
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.

scholarly journals Modified Fractional Reduced Differential Transform Method for the Solution of Multiterm Time-Fractional Diffusion Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Salah Abuasad ◽  
Ishak Hashim ◽  
Samsul Ariffin Abdul Karim
Keyword(s):  
Initial Conditions ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Differential Transform Method ◽  
Diffusion Equations ◽  
Transform Method ◽  
Fractional Diffusion Equations ◽  
Fractional Equations ◽  
Differential Transform ◽  
Reduced Differential Transform Method

In this study, we introduce a new modification of fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for nonhomogeneous linear multiterm time-fractional diffusion equations (MT-TFDEs) of constant coefficients in a bounded domain with suitable initial conditions. Different applications in two and three fractional order terms are given to illustrate our new modification. The approximate solutions are given in the form of series solutions. The results show that the m-FRDTM for MT-TFDEs is a powerful method and can be generalized to other types of multiterm time-fractional equations.

ANALYTICAL APPROXIMATE SOLUTIONS OF (N + 1)-DIMENSIONAL FRACTAL HARRY DYM EQUATIONS

Fractals ◽  
2018 ◽  
Vol 26 (06) ◽  
pp. 1850094
Author(s):  
JIANSHE SUN
Keyword(s):  
Approximate Solutions ◽  
Travelling Wave ◽  
Cantor Sets ◽  
Travelling Wave Solutions ◽  
Fractional Partial Differential Equations ◽  
Transform Method ◽  
Fractional Complex Transform ◽  
Fractal Models ◽  
Differential Transform ◽  
Harry Dym Equation

The new fractal models of the [Formula: see text]-dimensional and [Formula: see text]-dimensional nonlinear local fractional Harry Dym equation (HDE) on Cantor sets are derived and the analytical approximate solutions of the above two new models are obtained by coupling the fractional complex transform via local fractional derivative (LFD) and local fractional reduced differential transform method (LFRDTM). Fractional complex transform for functions of [Formula: see text]-dimensional variables is generalized and the theorems of [Formula: see text]-dimensional LFRDTM are supplementary extended. The travelling wave solutions of the fractal HDE show that the proposed LFRDTM is effective and simple for obtaining approximate solutions of nonlinear local fractional partial differential equations.

scholarly journals A fuzzy solution of nonlinear partial differential equations

2021 ◽  
Vol 5 (1) ◽  
pp. 51-63
Author(s):  
Mawia Osman ◽  
◽  
Zengtai Gong ◽  
Altyeb Mohammed Mustafa ◽  
◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Infinite Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Series Expansions ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Fuzzy Solution

In this paper, the reduced differential transform method (RDTM) is applied to solve fuzzy nonlinear partial differential equations (PDEs). The solutions are considered as infinite series expansions which converge rapidly to the solutions. Some examples are solved to illustrate the proposed method.

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