scholarly journals An Intriguing Approach to The Fractional Mellin Transform Method

2017 ◽  
Vol 13 (5) ◽  
pp. 7398-7402
Author(s):  
OZLEM OZTURK MIZRAK
Keyword(s):  
Fractional Derivative ◽  
Mellin Transform ◽  
Cosmic Time ◽  
Variable Coefficients ◽  
Erential Equation ◽  
Transform Method ◽  
Mellin Transformation

We propose an adapted Mellin transform method that gives the solution of a fractional di ¤erential equation with variable coefficients in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Mellin transformation, via an example for 0 < alpha < 1; where alpha is the order of fractional derivative. Then, we give an application of the results.

scholarly journals Semi- analytic numerical method for solution of time-space fractional heat and wave type equations with variable coefficients

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 74-86 ◽  
Author(s):  
Rishi Kumar Pandey ◽  
Hradyesh Kumar Mishra
Keyword(s):  
Fractional Derivative ◽  
Series Solution ◽  
Variable Coefficients ◽  
Variable Order ◽  
Wave Type ◽  
Transform Method ◽  
Time Space ◽  
Homotopy Analysis ◽  
Sumudu Transform ◽  
Variable Order Derivative

AbstractThe time and space fractional wave and heat type equations with variable coefficients are considered, and the variable order derivative in He‘s fractional derivative sense are taken. The utility of the homotopy analysis fractional sumudu transform method is shown in the form of a series solution for these generalized fractional order equations. Some discussion with examples are presented to explain the accuracy and ease of the method.

scholarly journals Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

2020 ◽  
Vol 26 (1) ◽  
pp. 35-55
Author(s):  
Abdelkader Kehaili ◽  
Ali Hakem ◽  
Abdelkader Benali
Keyword(s):  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Fractional Differential Equations ◽  
Variable Coefficients ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

scholarly journals Solving the Oscillation Equation With Fractional Order Damping Term Using a New Fourier Transform Method

2017 ◽  
Vol 13 (5) ◽  
pp. 7393-7397
Author(s):  
OZLEM OZTURK MIZRAK
Keyword(s):  
Fourier Transform ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Fourier Transformation ◽  
Cosmic Time ◽  
Damping Term ◽  
Fractional Damping ◽  
Transform Method ◽  
Fourier Transform Method ◽  
Oscillation Equation

We propose an adapted Fourier transform method that gives the solution of an oscillation equation with a fractional damping term in ordinary domain. After we mention a transformation of cosmic time to individual time (CTIT), we explain how it can reduce the problem from fractional form to ordinary form when it is used with Fourier transformation, via an example for 1 < alpha < 2; where alpha is the order of fractional derivative. Then, we give an application of the results.

scholarly journals Series method to solve conformable fractional ric-cati differential equations

2017 ◽  
Vol 6 (1) ◽  
pp. 30
Author(s):  
Mohammed Al Masalmeh
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Boundary Control ◽  
Series Solution ◽  
Stochastic Games ◽  
Variable Coefficients ◽  
Conformable Fractional Derivative ◽  
Series Method ◽  
Previous Definition ◽  
Hyperbolic Boundary

This paper investigates and states some properties of conformable fractional derivative, Further Study and applies the series solution for a case of conformable fractional Riccati deferential equation with variable coefficients “which is arising in stochastic games” or “hyperbolic boundary control." Recently, Prof. Roshdi Khalil introduced a new and interesting definition for the C F D, which is simpler than the previous definition in Caputo and Riemann-Liouville. It leads to many extensions of the classical theorems in calculus.

Two-Dimensional Legendre Wavelets for Solving Variable-Order Fractional Nonlinear Advection-Diffusion Equation with Variable Coefficients

2018 ◽  
Vol 19 (7-8) ◽  
pp. 793-802 ◽  
Author(s):  
M. Hosseininia ◽  
M. H. Heydari ◽  
Z. Avazzadeh ◽  
F. M. Maalek Ghaini
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Main Idea ◽  
Variable Coefficients ◽  
Variable Order ◽  
Two Dimensional ◽  
Legendre Wavelets ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Nonlinear Advection

AbstractThis article studies a numerical scheme for solving two-dimensional variable-order time fractional nonlinear advection-diffusion equation with variable coefficients, where the variable-order fractional derivative is in the Caputo type. The main idea is expanding the solution in terms of the 2D Legendre wavelets (2D LWs) where the variable-order time fractional derivative is discretized. We describe the method using the matrix operators and then implement it for solving various types of fractional advection-diffusion equations. The experimental results show the computational efficiency of the new approach.

scholarly journals Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation

2019 ◽  
Vol 3 (2) ◽  
pp. 14 ◽  
Author(s):  
Ndolane Sene ◽  
Aliou Niang Fall
Keyword(s):  
Diffusion Equation ◽  
Fractional Derivative ◽  
Approximate Solutions ◽  
Fractional Diffusion ◽  
Diffusion Reaction ◽  
Reaction Equation ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Homotopy Perturbation ◽  
Generalized Fractional Derivative

In this paper, the approximate solutions of the fractional diffusion equations described by the fractional derivative operator were investigated. The homotopy perturbation Laplace transform method of getting the approximate solution was proposed. The Caputo generalized fractional derivative was used. The effects of the orders α and ρ in the diffusion processes was addressed. The graphical representations of the approximate solutions of the fractional diffusion equation and the fractional diffusion-reaction equation both described by the Caputo generalized fractional derivative were provided.

scholarly journals Exact Solutions for the Conformable Space-Time Fractional Zeldovich Equation with Time-Dependent Coefficients

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Sami Injrou
Keyword(s):  
Fractional Derivative ◽  
Exact Solutions ◽  
Combustion Process ◽  
Variable Coefficients ◽  
Space Time ◽  
Time Dependent ◽  
Conformable Fractional Derivative ◽  
Fractional Partial Differential Equations ◽  
Rational Solutions ◽  
Different Types

The aim of this paper is to improve a sub-equation method to solve the space-time fractional Zeldovich equation with time-dependent coefficients involving the conformable fractional derivative. As result, we obtain three families of solutions including the hyperbolic, trigonometric, and rational solutions. These solutions may be helpful to explain several phenomena in chemistry, including the combustion process. The study shows that the used method is effective and reliable and can be utilized as a substitution to construct new solutions of different types of nonlinear conformable fractional partial differential equations (NFPDEs) with variable coefficients.

Fractional Derivative Consideration on Nonlinear Viscoelastic Dynamical Behavior Under Statical Pre-Displacement

2005 ◽  
Author(s):  
Masataka Fukunaga ◽  
Nobuyuki Shimizu ◽  
Hiroshi Nasuno
Keyword(s):  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Dynamical Behavior ◽  
Variable Coefficients ◽  
Rate Of Change ◽  
Order Derivative ◽  
Nonlinear Viscoelastic ◽  
Higher Order Derivative ◽  
The Difference ◽  
Different Order

Nonlinear fractional calculus model for the viscoelastic material is examined for oscillation around the off-equilibrium point. The model equation consists of two terms of different order fractional derivatives. The lower order derivative characterizes the slow process, and the higher order derivative characterizes the process of rapid oscillation. The measured difference in the order of the fractional derivative of the material, that the order is higher when the material is rapidly oscillated than when it is slowly compressed, is partly attributed to the difference in the frequency dependence between the two fractional derivatives. However, it is found that there could be possibility for the variable coefficients of the two terms with the rate of change of displacement.

scholarly journals Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Dianchen Lu ◽  
Jie Liu
Keyword(s):  
Homotopy Analysis Method ◽  
Variable Coefficient ◽  
Burgers Equation ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Hyperbolic Function ◽  
Analysis Method ◽  
Function Form ◽  
Transform Method ◽  
Homotopy Analysis

The homotopy analysis method is applied to solve the variable coefficient KdV-Burgers equation. With the aid of generalized elliptic method and Fourier’s transform method, the approximate solutions of double periodic form are obtained. These solutions may be degenerated into the approximate solutions of hyperbolic function form and the approximate solutions of trigonometric function form in the limit cases. The results indicate that this method is efficient for the nonlinear models with the dissipative terms and variable coefficients.

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