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.

scholarly journals Analytic Solution for theRLElectric Circuit Model in Fractional Order

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
P. V. Shah ◽  
A. D. Patel ◽  
I. A. Salehbhai ◽  
A. K. Shukla
Keyword(s):  
Differential Equation ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Analytic Solution ◽  
Electrical Circuit ◽  
Special Cases ◽  
The Laplace Transform ◽  
Fractional Differential

This paper provides an analytic solution ofRLelectrical circuit described by a fractional differential equation of the order0<α≤1. We use the Laplace transform of the fractional derivative in the Caputo sense. Some special cases for the different source terms have also been discussed.

scholarly journals The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Katica (Stevanovic) Hedrih
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Constitutive Relation ◽  
Continuous Functions ◽  
Beam Dynamic ◽  
Fractional Differential ◽  
Elastic Form ◽  
Partial Fractional Differential Equation ◽  
Derivative Order

We considered the problem on transversal oscillations of two-layer straight bar, which is under the action of the lengthwise random forces. It is assumed that the layers of the bar were made of nonhomogenous continuously creeping material and the corresponding modulus of elasticity and creeping fractional order derivative of constitutive relation of each layer are continuous functions of the length coordinate and thickness coordinates. Partial fractional differential equation and particular solutions for the case of natural vibrations of the beam of creeping material of a fractional derivative order constitutive relation in the case of the influence of rotation inertia are derived. For the case of natural creeping vibrations, eigenfunction and time function, for different examples of boundary conditions, are determined. By using the derived partial fractional differential equation of the beam vibrations, the almost sure stochastic stability of the beam dynamic shapes, corresponding to thenth shape of the beam elastic form, forced by a bounded axially noise excitation, is investigated. By the use of S. T. Ariaratnam's idea, as well as of the averaging method, the top Lyapunov exponent is evaluated asymptotically when the intensity of excitation process is small.

scholarly journals A Study of Fractional Differential Equation with a Positive Constant Coefficient via Hilfer Fractional Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Hoa Ngo Van ◽  
Vu Ho
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Positive Constant ◽  
Constant Coefficient ◽  
Initial Conditions ◽  
Hilfer Fractional Derivative ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Rassias Stability

The aim of the paper is to consider the existence and uniqueness of solution of the fractional differential equation with a positive constant coefficient under Hilfer fractional derivative by using the fixed-point theorem. We also prove the bounded and continuous dependence on the initial conditions of solution. Besides, Hyers–Ulam stability and Hyers–Ulam–Rassias stability are discussed. Finally, we provide an example to demonstrate our main results.

scholarly journals Existence of Positive Solution for BVP of Nonlinear Fractional Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Jun-Rui Yue ◽  
Jian-Ping Sun ◽  
Shuqin Zhang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Fractional Derivative ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Nonlinear Fractional Differential Equation ◽  
Krasnoselskii Fixed Point Theorem ◽  
Fractional Differential

We consider the following boundary value problem of nonlinear fractional differential equation:(CD0+αu)(t)=f(t,u(t)),  t∈[0,1],  u(0)=0,   u′(0)+u′′(0)=0,  u′(1)+u′′(1)=0, whereα∈(2,3]is a real number, CD0+αdenotes the standard Caputo fractional derivative, andf:[0,1]×[0,+∞)→[0,+∞)is continuous. By using the well-known Guo-Krasnoselskii fixed point theorem, we obtain the existence of at least one positive solution for the above problem.

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 Uniqueness of Solutionfor Nonlinear Implicit Fractional Differential Equation

2016 ◽  
Vol 12 (11) ◽  
pp. 6807-6811
Author(s):  
Haribhau Laxman Tidke
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Initial Condition ◽  
Fractional Differential

We study the uniquenessof solutionfor nonlinear implicit fractional differential equation with initial condition involving Caputo fractional derivative. The technique used in our analysis is based on an application of Bihari and Medved inequalities.

scholarly journals A fractional differential equation with multi-point strip boundary condition involving the Caputo fractional derivative and its Hyers–Ulam stability

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehboob Alam ◽  
Akbar Zada ◽  
Ioan-Lucian Popa ◽  
Alireza Kheiryan ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Fractional Differential Equation ◽  
Caputo Fractional Derivative ◽  
Integral Boundary Conditions ◽  
Ulam Stability ◽  
Integral Boundary ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Fractional Differential

AbstractIn this work, we investigate the existence, uniqueness, and stability of fractional differential equation with multi-point integral boundary conditions involving the Caputo fractional derivative. By utilizing the Laplace transform technique, the existence of solution is accomplished. By applying the Bielecki-norm and the classical fixed point theorem, the Ulam stability results of the studied system are presented. An illustrative example is provided at the last part to validate all our obtained theoretical results.

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 Analytical solutions of incommensurate fractional differential equation systems with fractional order $ 1 < \alpha, \beta < 2 $ via bivariate Mittag-Leffler functions

2021 ◽  
Vol 7 (2) ◽  
pp. 2281-2317
Author(s):  
Yong Xian Ng ◽  
◽  
Chang Phang ◽  
Jian Rong Loh ◽  
Abdulnasir Isah ◽  
...  
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Fractional Differential Equation ◽  
Fractional Order ◽  
Successive Approximations ◽  
Explicit Analytical Solution ◽  
Special Cases ◽  
Fractional Differential ◽  
Alpha Beta ◽  
Published Paper

<abstract><p>In this paper, we derive the explicit analytical solution of incommensurate fractional differential equation systems with fractional order $ 1 &lt; \alpha, \beta &lt; 2 $. The derivation is extended from a recently published paper by Huseynov et al. in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>, which is limited for incommensurate fractional order $ 0 &lt; \alpha, \beta &lt; 1 $. The incommensurate fractional differential equation systems were first converted to Volterra integral equations. Then, the Mittag-Leffler function and Picard's successive approximations were used to obtain the analytical solution of incommensurate fractional order systems with $ 1 &lt; \alpha, \beta &lt; 2 $. The solution will be simplified via some combinatorial concepts and bivariate Mittag-Leffler function. Some special cases will be discussed, while some examples will be given at the end of this paper.</p></abstract>

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