Fawang Liu

Mark M. Meerschaert

Shaher Momani

Nikolai N. Leonenko

Wen Chen

Om P. Agrawal

Fractional Differential Equations

We will consider He&#39;s variational iteration method for solving fractional
Riccati differential equation. This method is based on the use of Lagrange multipliers for
identification of optimal value of a parameter in a functional. This technique provides a
sequence of functions which converges to the exact solution of the problem. The present
method performs extremely well in terms of efficiency and simplicity.

H. Jafari

H. Tajadodi

He&#39;s Variational Iteration Method for Solving Fractional Riccati Differential Equation

In the present review we survey the properties of a transcendental function of the Wright type, nowadays known as <svg style="vertical-align:-0.0pt;width:17.237499px;" id="M2" height="10.325" version="1.1" viewBox="0 0 17.237499 10.325" width="17.237499"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
		<g transform="translate(72,-63.74)">
			<text transform="matrix(1,0,0,-1,-71.95,63.79)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
			</text>

		</g>
	</g>
</svg>-Wright function, entering as a probability density in a
relevant class of self-similar stochastic processes that we generally refer to as time-fractional
diffusion processes. Indeed, the master equations governing these processes
generalize the standard diffusion equation by means of time-integral operators interpreted
as derivatives of fractional order. When these generalized diffusion processes are properly
characterized with stationary increments, the <svg style="vertical-align:-0.0pt;width:17.237499px;" id="M3" height="10.325" version="1.1" viewBox="0 0 17.237499 10.325" width="17.237499"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.325)">
		<g transform="translate(72,-63.74)">
			<text transform="matrix(1,0,0,-1,-71.95,63.79)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝑀</tspan>
			</text>

		</g>
	</g>
</svg>-Wright function is shown to play the
same key role as the Gaussian density in the standard and fractional Brownian motions. 
Furthermore, these processes provide stochastic models suitable for describing phenomena
of anomalous diffusion of both slow and fast types.

Francesco Mainardi

Antonio Mura

Gianni Pagnini

The <svg style="vertical-align:-0.0pt" height="11.175" version="1.1" viewBox="0 0 17.475 11.175" width="17.475" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path id="x1D440" d="M998 650l-8 -28q-71 -4 -86 -16t-22 -69l-50 -397q-3 -28 -4.5 -44t2 -29t6.5 -18.5t17 -10.5t24.5 -6.5t37.5 -3.5l-8 -28h-271l7 28q63 6 78 22t25 90l60 415h-2l-353 -552h-23l-130 536h-2l-70 -254q-44 -158 -47 -188q-5 -38 9 -51t71 -18l-6 -28h-241l8 28&#xA;q45 4 67 18.5t35 45.5q16 38 74 233l52 173q24 79 11.5 98t-89.5 26l6 28h177l136 -508l337 508h172z" /></g></svg>-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

We introduce a formulation for the time-optimal control
problems of systems displaying fractional dynamics in the sense of the
Riemann-Liouville fractional derivatives operator. To propose a solution
to the general time-optimal problem, a rational approximation based on
the Hankel data matrix of the impulse response is considered to emulate
the behavior of the fractional differentiation operator. The original problem is then reformulated according to the new model which can be solved
by traditional optimal control problem solvers. The time-optimal problem is extensively investigated for a double fractional integrator and its
solution is obtained using either numerical optimization time-domain
analysis.

Christophe Tricaud

YangQuan Chen

Time-Optimal Control of Systems with Fractional Dynamics

Riesz potentials (also called Riesz fractional derivatives) and their Hilbert
transforms are computed for the Korteweg-de Vries soliton. They are expressed
in terms of the full-range Hurwitz Zeta functions <svg style="vertical-align:-3.49493pt;width:48.25px;" id="M1" height="15.05" version="1.1" viewBox="0 0 48.25 15.05" width="48.25"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,15.05)">
		<g transform="translate(72,-59.96)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜁</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-65.26,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">+</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-58.77,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑠</tspan>
				<tspan style="font-size: 12.50px; " x="9.6648188" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="14.866067" y="0">𝑎</tspan>
				<tspan style="font-size: 12.50px; " x="21.142572" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg> and <svg style="vertical-align:-3.13504pt;width:48.25px;" id="M2" height="14.6" version="1.1" viewBox="0 0 48.25 14.6" width="48.25"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,14.6)">
		<g transform="translate(72,-60.32)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜁</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-65.26,60.37)">
				<tspan style="font-size: 8.75px; " x="0" y="0">−</tspan>
			</text>

			<text transform="matrix(1,0,0,-1,-58.77,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">(</tspan>
				<tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝑠</tspan>
				<tspan style="font-size: 12.50px; " x="9.6648188" y="0">,</tspan>
				<tspan style="font-size: 12.50px; " x="14.866067" y="0">𝑎</tspan>
				<tspan style="font-size: 12.50px; " x="21.142572" y="0">)</tspan>
			</text>

		</g>
	</g>
</svg>. 
It is proved that these Riesz potentials and their Hilbert transforms are linearly
independent solutions of a Sturm-Liouville problem. Various new
properties are established for this family of functions. The fact that the
Wronskian of the system is positive leads to a new inequality for the Hurwitz
Zeta functions.

Vladimir Varlamov

Riesz Potentials for Korteweg-de Vries Solitons and Sturm-Liouville Problems


					 Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space
fractional Fokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputo time fractional derivative (CTFD) of order <svg style="vertical-align:-0.33858pt;width:28.325001px;" id="M1" height="8.8500004" version="1.1" viewBox="0 0 28.325001 8.8500004" width="28.325001"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,8.85)">
		<g transform="translate(72,-64.92)">
			<text transform="matrix(1,0,0,-1,-71.95,65.31)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝛼</tspan>
				<tspan style="font-size: 12.50px; " x="10.515023" y="0">∈</tspan>
			</text>

		</g>
	</g>
</svg> (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order <svg style="vertical-align:-2.29482pt;width:29.025px;" id="M2" height="11.2875" version="1.1" viewBox="0 0 29.025 11.2875" width="29.025"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,11.2875)">
		<g transform="translate(72,-62.97)">
			<text transform="matrix(1,0,0,-1,-71.95,65.31)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan>
				<tspan style="font-size: 12.50px; " x="11.077658" y="0">∈</tspan>
			</text>

		</g>
	</g>
</svg> (1, 2]. Approximating the CTFD and RSFD using the L1-algorithm and shifted Gr&#xFC;nwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Qianqian Yang

Ian Turner

Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term

The paper gives sufficient conditions on the existence of periodic solution for a class of compound singular fractional differential systems with delay, involving Nishimoto fractional derivative. Furthermore, for the particular functions, the necessary conditions on the existence of periodic solution are also derived. Especially, for two-dimensional compound singular fractional differential equation with delay, the criteria of existence of periodic solution are obtained. Finally, two examples are presented to verify the validity of criteria.

XuTing Wei

XuanZhu Lu

The Periodic Solutions of the Compound Singular Fractional Differential System with Delay

We review the application of differential operators of noninteger order to
the modeling of dynamic systems. We compare all the definitions of Variable Order
(VO) operators recently proposed in literature and select the VO operator that has
the desirable property of continuous transition between integer and non-integer order
derivatives. We use the selected VO operator to connect the meaning of functional order
to the dynamic properties of a viscoelastic oscillator. We conclude that the order of
differentiation of a single VO operator that represents the dynamics of a viscoelastic
oscillator in stationary motion is a normalized phase shift. The normalization constant
is found by taking the difference between the order of the inertial term (2) and the order
of the spring term (0) and dividing this difference by the angular phase shift between
acceleration and position in radians (<svg style="vertical-align:-0.16302pt;width:9.625px;" id="M1" height="7.0124998" version="1.1" viewBox="0 0 9.625 7.0124998" width="9.625"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,7.0125)">
		<g transform="translate(72,-66.39)">
			<text transform="matrix(1,0,0,-1,-71.95,66.6)">
				<tspan style="font-size: 12.50px; " x="0" y="0">𝜋</tspan>
			</text>

		</g>
	</g>
</svg>), so that the normalization constant is simply
<svg style="vertical-align:-0.17555pt;width:21.7875px;" id="M2" height="10.9125" version="1.1" viewBox="0 0 21.7875 10.9125" width="21.7875"  xmlns="http://www.w3.org/2000/svg">
	<g transform="matrix(1.25,0,0,-1.25,0,10.9125)">
		<g transform="translate(72,-63.27)">
			<text transform="matrix(1,0,0,-1,-71.95,63.5)">
				<tspan style="font-size: 12.50px; " x="0" y="0">2</tspan>
				<tspan style="font-size: 12.50px; " x="6.2515001" y="0">/</tspan>
				<tspan style="font-size: 12.50px; " x="9.727334" y="0">𝜋</tspan>
			</text>

		</g>
	</g>
</svg>.

Lynnette E. S. Ramirez

Carlos F. M. Coimbra

On the Selection and Meaning of Variable Order Operators for Dynamic Modeling

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit expression of the solution is given in this paper. At last, we show the asymptotic solution of the differential equations of fractional order and corresponding truncated error in theory.

X. Huang

X. Lu

The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations

We study a Dirichlet boundary value problem for Langevin equation involving two fractional orders. Langevin equation has been widely used to describe the evolution of physical phenomena in fluctuating environments. However, ordinary Langevin equation does not provide the correct description of the dynamics for systems in complex media. In order to overcome this problem and describe dynamical processes in a fractal medium, numerous generalizations of Langevin equation have been proposed. One such generalization replaces the ordinary derivative by a fractional derivative in the Langevin equation. This gives rise to the fractional Langevin equation with a single index. Recently, a new type of Langevin equation with two different fractional orders has been introduced which provides a more flexible model for fractal processes as compared with the usual one characterized by a single index. The contraction mapping principle and Krasnoselskii&#39;s fixed point theorem are applied to prove the existence of solutions of the problem in a Banach space.

Bashir Ahmad

Juan J. Nieto

Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions

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International Journal of Differential Equations

Fractional Differential Equations

Fractional Differential Equations

Articles

Fractional Differential Equations

He's Variational Iteration Method for Solving Fractional Riccati Differential Equation

The -Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey

Time-Optimal Control of Systems with Fractional Dynamics

Riesz Potentials for Korteweg-de Vries Solitons and Sturm-Liouville Problems

Stability and Convergence of an Effective Numerical Method for the Time-Space Fractional Fokker-Planck Equation with a Nonlinear Source Term

The Periodic Solutions of the Compound Singular Fractional Differential System with Delay

On the Selection and Meaning of Variable Order Operators for Dynamic Modeling

The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary Differential Equations

Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with Dirichlet Boundary Conditions