scholarly journals A Note on Integrating Factors of a Conformable Fractional Differential Equation

2020 ◽  
Vol 10 (4) ◽  
pp. 143-152
Author(s):  
F. Martínez ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Simple Formula ◽  
Conformable Fractional Derivative ◽  
Integrating Factor ◽  
Euler’S Theorem ◽  
Special Cases ◽  
Fractional Differential ◽  
Integrating Factors

Recently exact fractional differential equations have been introduced, using the conformable fractional derivative. In this paper, we propose and prove some new results on the integrating factor. We introduce a conformable version of several classical special cases for which the integrating factor can be determined. Specifically, the cases we will consider are where there is an integrating factor that is a function of only x, or a function of only y, or a simple formula of x and y. In addition, using the Conformable Euler's Theorem on homogeneous functions, an integration factor for the conformable homogeneous differential equations is established. Finally, the above results apply in some interesting examples.

scholarly journals Analytical solutions for conformable fractional Bratu-type equations

2018 ◽  
Vol 7 (1) ◽  
pp. 15 ◽  
Author(s):  
Mousa Ilie ◽  
Jafar Biazar ◽  
Zainab Ayati
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Conformable Fractional Derivative ◽  
Fractional Differential

Solving fractional differential equations have a prominent function in different science such as physics and engineering. Therefore, are different definitions of the fractional derivative presented in recent years. The aim of the current paper is to solve the fractional differential equation by a semi-analytical method based on conformable fractional derivative. Fractional Bratu-type equations have been solved by the method and to show its capabilities. The obtained results have been compared with the exact solution.

scholarly journals SEPARABLE LOCAL FRACTIONAL DIFFERENTIAL EQUATIONS

Fractals ◽  
2016 ◽  
Vol 24 (02) ◽  
pp. 1650021 ◽  
Author(s):  
KIRAN M. KOLWANKAR
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Local Scaling ◽  
Fractal Sets ◽  
Method Of Solution ◽  
Fractional Differential ◽  
Simple Equations

The concept of local fractional derivative was introduced in order to be able to study the local scaling behavior of functions. However it has turned out to be much more useful. It was found that simple equations involving these operators naturally incorporate the fractal sets into the equations. Here, the scope of these equations has been extended further by considering different possibilities for the known function. We have also studied a separable local fractional differential equation along with its method of solution.

scholarly journals On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

2020 ◽  
Vol 40 (2) ◽  
pp. 227-239
Author(s):  
John R. Graef ◽  
Said R. Grace ◽  
Ercan Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Caputo Fractional Derivative ◽  
Nonoscillatory Solutions ◽  
Fractional Differential

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

scholarly journals On a Fractional Operator Combining Proportional and Classical Differintegrals

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 360 ◽  
Author(s):  
Dumitru Baleanu ◽  
Arran Fernandez ◽  
Ali Akgül
Keyword(s):  
Integral Operator ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Integral ◽  
Inverse Operator ◽  
Fractional Operator ◽  
Special Cases ◽  
Non Local ◽  
Fractional Differential

The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f ( t ) , by a fractional integral operator applied to the derivative f ′ ( t ) . We define a new fractional operator by substituting for this f ′ ( t ) a more general proportional derivative. This new operator can also be written as a Riemann–Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann–Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.

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

2017 ◽  
Vol 6 (2) ◽  
pp. 49 ◽  
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.  

scholarly journals Kamenev Type Oscillatory Criteria for Linear Conformable Fractional Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-10 ◽  
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.

scholarly journals Frenet frame with respect to conformable derivative

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1541-1550
Author(s):  
Uğur Gözütok ◽  
Hüsnü Çoban ◽  
Yasemin Sağıroğlu
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fundamental Theorem ◽  
Geometric Interpretation ◽  
Frenet Frame ◽  
Conformable Fractional Derivative ◽  
Conformable Derivative ◽  
Fractional Differential ◽  
Curvature And Torsion

Conformable fractional derivative is introduced by the authors Khalil at al in 2014. In this study, we investigate the frenet frame with respect to conformable fractional derivative. Curvature and torsion of a conformable curve are defined and the geometric interpretation of these two functions is studied. Also, fundamental theorem of curves is expressed for the conformable curves and an example of the curve corresponding to a fractional differential equation is given.

scholarly journals Hille and Nehari Type Oscillation Criteria for Conformable Fractional Differential Equations

2021 ◽  
pp. 578-587
Author(s):  
T. Gayathri ◽  
M. Sathish Kumar ◽  
V. Sadhasivam
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Conformable Fractional Derivative ◽  
Oscillation Criteria ◽  
All Solutions ◽  
Fractional Differential ◽  
Type Oscillation ◽  
Comparison Technique

In this paper, we develop the Hille and Nehari Type criteria for the oscillation of all solutions to the Fractional Differential Equations involving Conformable fractional derivative. Some new oscillatory criteria are obtained by using the Riccati transformations and comparison technique. We show the validity and effectiveness of our results by providing various examples.

scholarly journals He’s fractional derivative for heat conduction in a fractal medium arising in silkworm cocoon hierarchy

2015 ◽  
Vol 19 (4) ◽  
pp. 1155-1159 ◽  
Author(s):  
Fu-Juan Liu ◽  
Zheng-Biao Li ◽  
Sheng Zhang ◽  
Hong-Yan Liu
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Natural Phenomenon ◽  
Fractal Medium ◽  
Complex Transformation ◽  
Special Cases ◽  
Fractional Differential ◽  
Silkworm Cocoon

He?s fractional derivative is adopted in this paper to study the heat conduction in fractal medium. The fractional complex transformation is applied to convert the fractional differential equation to ordinary different equation. Boltzmann transform and wave transform are used to further simplify the governing equation for some special cases. Silkworm cocoon are used as an example to elucidate its natural phenomenon.

Integrating Factors of Ordinary Differential Equations of the Second Order: A Geometrical Interpretation

1943 ◽  
Vol 27 (276) ◽  
pp. 159-165
Author(s):  
H. Wallis Chapman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Constant ◽  
Second Order ◽  
Geometrical Interpretation ◽  
Integrating Factor ◽  
Integrating Factors

A Well-Known method of integrating the simple differential equation of the second order y″ + k 2 y = 0 .....1.1 consists in multiplying by an integrating factor 2y′ and integrating directly, when we obtain y′2 + k 2 y 2 = k 2 a 2, .....1.2 where a is an arbitrary constant.

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