Series method to solve conformable fractional ric-cati  differential equations

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.

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A Vector Series Solution for a Class of Hyperbolic System of Caputo Time-Fractional Partial Differential Equations With Variable Coefficients

Frontiers in Physics ◽

10.3389/fphy.2021.525250 ◽

2021 ◽

Vol 9 ◽

Author(s):

Ahmad El-Ajou ◽

Zeyad Al-Zhour

Keyword(s):

Partial Differential Equations ◽

Power Series ◽

Differential Equations ◽

Fractional Derivative ◽

Hyperbolic System ◽

Series Solution ◽

Fractional Power ◽

Variable Coefficients ◽

Fractional Partial Differential Equations ◽

Partial Differential

In this paper, we introduce a series solution to a class of hyperbolic system of time-fractional partial differential equations with variable coefficients. The fractional derivative has been considered by the concept of Caputo. Two expansions of matrix functions are proposed and used to create series solutions for the target problem. The first one is a fractional Laurent series, and the second is a fractional power series. A new approach, via the residual power series method and the Laplace transform, is also used to find the coefficients of the series solution. In order to test our proposed method, we discuss four interesting and important applications. Numerical results are given to authenticate the efficiency and accuracy of our method and to test the validity of our obtained results. Moreover, solution surface graphs are plotted to illustrate the effect of fractional derivative arrangement on the behavior of the solution.

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A novel series method for fractional diffusion equation within Caputo fractional derivative

Thermal Science ◽

10.2298/tsci16s3695y ◽

2016 ◽

Vol 20 (suppl. 3) ◽

pp. 695-699 ◽

Cited By ~ 2

Author(s):

Sheng-Ping Yan ◽

Wei-Ping Zhong ◽

Xiao-Jun Yang

Keyword(s):

Diffusion Equation ◽

Fractional Derivative ◽

Caputo Fractional Derivative ◽

Series Solution ◽

Expansion Method ◽

Fractional Diffusion ◽

Fractional Diffusion Equation ◽

Series Method ◽

Time Fractional Diffusion Equation ◽

Series Expansion Method

In this paper, we suggest the series expansion method for finding the series solution for the time-fractional diffusion equation involving Caputo fractional derivative.

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

International Journal of Differential Equations ◽

10.1155/2020/9312830 ◽

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.

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

Open Physics ◽

10.1515/phys-2017-0009 ◽

2017 ◽

Vol 15 (1) ◽

pp. 74-86 ◽

Cited By ~ 4

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.

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Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

Discrete Dynamics in Nature and Society ◽

10.1155/2018/2394735 ◽

2018 ◽

Vol 2018 ◽

pp. 1-9 ◽

Cited By ~ 3

Author(s):

Bochao Chen ◽

Li Qin ◽

Fei Xu ◽

Jian Zu

Keyword(s):

Power Series ◽

Differential Equations ◽

Variable Coefficients ◽

Series Solutions ◽

Series Method ◽

Power Series Method ◽

Residual Power Series Method ◽

Residual Power Series ◽

Analytical Series ◽

Residual Power

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

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General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative

International Journal of Applied Mathematical Research ◽

10.14419/ijamr.v6i2.7014 ◽

2017 ◽

Vol 6 (2) ◽

pp. 49 ◽

Cited By ~ 13

Author(s):

Zainab Ayati ◽

Jafar Biaar ◽

Mousa Ilei

Keyword(s):

General Solution ◽

Differential Equations ◽

Fractional Derivative ◽

Ordinary Differential Equations ◽

Fractional Differential Equations ◽

Riccati Equations ◽

Nonlinear Ordinary Differential Equations ◽

Conformable Fractional Derivative ◽

Numerical Examples ◽

Fractional Differential

This paper is aimed to develop two well-known nonlinear ordinary differential equations, Bernoulli and Riccati equations to fractional form. General solution to fractional differential equations are detected, based on conformable fractional derivative. For each equation, numerical examples are presented to illustrate the proposed approach.

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A simple technique for constructing exact solutions to nonlinear differential equations with conformable fractional derivative

Optical and Quantum Electronics ◽

10.1007/s11082-017-1105-5 ◽

2017 ◽

Vol 49 (8) ◽

Cited By ~ 10

Author(s):

Melike Kaplan ◽

Ahmet Bekir ◽

Mehmet Naci Ozer

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Exact Solutions ◽

Simple Technique ◽

Nonlinear Differential Equations ◽

Conformable Fractional Derivative

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

Global Journal of Pure and Applied Sciences ◽

10.4314/gjpas.v26i1.6 ◽

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.

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Kamenev Type Oscillatory Criteria for Linear Conformable Fractional Differential Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2019/2310185 ◽

2019 ◽

Vol 2019 ◽

pp. 1-10 ◽

Cited By ~ 3

Author(s):

Jing Shao ◽

Zhaowen Zheng

Keyword(s):

Differential Equations ◽

Fractional Derivative ◽

Fractional Differential Equations ◽

Average Method ◽

Oscillation Theory ◽

Conformable Fractional Derivative ◽

Oscillation Criteria ◽

Fractional Differential ◽

Type Oscillation ◽

Integral Average

Using integral average method and properties of conformable fractional derivative, new Kamenev type oscillation criteria are given firstly for conformable fractional differential equations, which improve known results in oscillation theory. Examples are also given to illustrate the effectiveness of the main results.

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