• Journal of applied mathematics & informatics
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• v.24 no.1_2
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• pp.31-48
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• 2007

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on bypassing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable function $f(x+h)=E_{\alpha}(h^{\alpha}{D_x^{\alpha})f(x)$.

• Kyungpook Mathematical Journal
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• v.20 no.1
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• pp.77-82
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• 1980

There are many definitions of the fractional derivative. It is purpose of this paper to show some results which were got for fractional partial derivative of functions of two variables and to give an application of the fractional partial derivative.

• Journal of Mechanical Science and Technology
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• v.20 no.5
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• pp.658-667
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• 2006

Fractional derivative models, which are used to describe the viscoelastic behavior of material, have received considerable attention. Thus it is necessary to put forward the analysis solutions of dynamic systems containing a fractional derivative. Although previously reported such kind of fractional calculus-based constitutive models, it only handles the particularity of rational number in part, has great limitation by reason of only handling with particular rational number field. Simultaneously, the former study has great unreliability by reason of using the complementary error function which can't ensure uniform real number. In this paper, a new approach is proposed for an analytical scheme for dynamic system of a spring-mass-damper system of single-degree of freedom under general forcing conditions, whose damping is described by a fractional derivative of the order $0<{\alpha}<1$ which can be both irrational number and rational number. The new approach combines the fractional Green's function and Laplace transform of fractional derivative. Analytical examples of dynamic system under general forcing conditions obtained by means of this approach verify the feasibility very well with much higher reliability and universality.

• East Asian mathematical journal
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• v.34 no.3
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• pp.249-263
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• 2018

We aim to establish certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional differential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1101-1121
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• 2008

One proposes an approach to fractional Fourier's transform, or Fourier's transform of fractional order, which applies to functions which are fractional differentiable but are not necessarily differentiable, in such a manner that they cannot be analyzed by using the so-called Caputo-Djrbashian fractional derivative. Firstly, as a preliminary, one defines fractional sine and cosine functions, therefore one obtains Fourier's series of fractional order. Then one defines the fractional Fourier's transform. The main properties of this fractal transformation are exhibited, the Parseval equation is obtained as well as the fractional Fourier inversion theorem. The prospect of application for this new tool is the spectral density analysis of signals, in signal processing, and the analysis of some partial differential equations of fractional order.

• Journal of the Chungcheong Mathematical Society
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• v.30 no.1
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• pp.159-164
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• 2017

This paper deals with linear impulsive differential equations involving the Caputo fractional derivative. We provide exact solutions of nonhomogeneous linear impulsive fractional differential equations with constant coefficients by means of the Mittag-Leffler functions.

• Structural Engineering and Mechanics
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• v.5 no.2
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• pp.177-191
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• 1997

In the paper, a fractional derivative Kelvin-Voigt model describing the dynamic behavior of a special class of fluid viscous dampers, is presented. First of all, in order to verify their mechanical properties, two devices were tested the former behaving as a pure damper (PD device), whereas the latter as an elastic-damping device (ED device). For both, quasi-static and dynamic tests were carried out under imposed displacement control. Secondarily, in order to describe their cyclical behavior, a model composed by an elastic and a damping element connected in parallel was defined. The elastic force was assumed as a linear function of the displacement whereas the damping one was expressed by a fractional derivative of the displacement. By setting an appropriate numerical algorithm, the model parameters (fractional derivative order, damping coefficient and elastic stiffness) were identified by experimental results. The estimated values allowed to outline the main parameter properties on which depend both the elastic as well as the damping behavior of the considered devices.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.183-194
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• 1998

Making use of a certain operator of fractional derivative, a new subclass $L_p({\alpha},{\beta},{\gamma},{\lambda})$) of analytic and p-valent functions is introduced in the present paper. Apart from various coefficient bounds, many interesting and useful properties of this class of functions are given, some of these properties involve, for example, linear combinations and modified Hadamard product of several functions belonging to the class introduced here.

• Bulletin of the Korean Mathematical Society
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• v.47 no.6
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• pp.1225-1234
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• 2010

Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order $O({\Delta}x+{\Delta}t)$ is obtained in the discrete $L_2$ norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

• Journal of applied mathematics & informatics
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• v.18 no.1_2
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• pp.339-350
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• 2005

A space-time fractional advection-dispersion equation (ADE) is a generalization of the classical ADE in which the first-order time derivative is replaced with Caputo derivative of order $\alpha{\in}(0,1]$, and the second-order space derivative is replaced with a Riesz-Feller derivative of order $\beta{\in}0,2]$. We derive the solution of its Cauchy problem in terms of the Green functions and the representations of the Green function by applying its Fourier-Laplace transforms. The Green function also can be interpreted as a spatial probability density function (pdf) evolving in time. We do the same on another kind of space-time fractional advection-dispersion equation whose space and time derivatives both replacing with Caputo derivatives.