Showing papers in "Fractional Calculus and Applied Analysis in 2017"
TL;DR: In this article, several definitions of the Riesz fractional Laplace operator in R^d have been studied, including singular integrals, semigroups of operators, Bochner's subordination, and harmonic extensions.
Abstract: This article reviews several definitions of the fractional Laplace operator (-Delta)^{alpha/2} (0 < alpha < 2) in R^d, also known as the Riesz fractional derivative operator, as an operator on Lebesgue spaces L^p, on the space C_0 of continuous functions vanishing at infinity and on the space C_{bu} of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner's subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain.
372 citations
TL;DR: In this article, a survey analyzes and measures the evolution that occurred during the last five decades in the light of books, journals and conferences dedicated to the theory and applications of fractional calculus, dealing with operations of integration and differentiation of arbitrary (fractional) order and their generalizations.
Abstract: Abstract Since the 60s of last century Fractional Calculus exhibited a remarkable progress and presently it is recognized to be an important topic in the scientific arena. This survey analyzes and measures the evolution that occurred during the last five decades in the light of books, journals and conferences dedicated to the theory and applications of this mathematical tool, dealing with operations of integration and differentiation of arbitrary (fractional) order and their generalizations.
107 citations
TL;DR: In this article, a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂ α t u + Au = f b (u) in Q where 1 1.
Abstract: Let Ω be a C 2-bounded domain of R d , d = 2, 3, and fix Q = (0, T) × Ω with T ∈ (0, +∞]. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂ α t u + Au = f b (u) in Q where 1 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂ α t u + Au = f (t, x) in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data.
92 citations
TL;DR: In this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated, and several interesting properties of the proposed principles are revealed profoundly.
Abstract: Abstract In this paper, from the classical short memory principle under Grünwald-Letnikov definition, several novel short memory principles are presented and investigated. On one hand, the classical principle is extended to Riemann-Liouville and Caputo cases. On the other hand, a special kind of principles are formulated by introducing a discrete argument instead of the continuous time, resulting in principles with fixed memory length or fixed memory step. Apart from these, several interesting properties of the proposed principles are revealed profoundly.
73 citations
TL;DR: In this article, a class of evolution equations with Hilfer fractional derivative was considered and the fixed point theorem and the noncompact measure method were employed to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.
Abstract: Abstract In this paper, we consider a class of evolution equations with Hilfer fractional derivative. By employing the fixed point theorem and the noncompact measure method, we establish a number of new criteria to guarantee the existence and uniqueness of mild solutions when the associated semigroup is compact or not.
73 citations
TL;DR: In this article, the existence and uniqueness of solution for higher-order fractional differential equations with conjugate type integral conditions was proved and the interesting point lies in that the Lipschitz constant is closely associated with the first eigenvalues corresponding to the relevant linear operator.
Abstract: Abstract In this article, we prove the existence and uniqueness of solution for some higher-order fractional differential equations with conjugate type integral conditions. The interesting point lies in that the Lipschitz constant is closely associated with the first eigenvalues corresponding to the relevant linear operator. The discussion is based on the Banach contraction map principle and the theory of u0-positive linear operator.
63 citations
TL;DR: In this paper, an unstructured mesh Galerkin finite element method was used to discretize in space and the Crank-Nicolson scheme was used for discretizing the Caputo time fractional derivative.
Abstract: Most existing research on applying the finite element method to discretize space fractional operators is studied on regular domains using either uniform structured triangular meshes, or quadrilateral meshes. Since many practical problems involve irregular convex domains, such as the human brain or heart, which are difficult to partition well with a structured mesh, the existing finite element method using the structured mesh is less efficient. Research on the finite element method using a completely unstructured mesh on an irregular domain is of great significance. In this paper, a novel unstructured mesh finite element method is developed for solving the time-space fractional wave equation on a two-dimensional irregular convex domain. The novel unstructured mesh Galerkin finite element method is used to discretize in space and the Crank-Nicolson scheme is used to discretize the Caputo time fractional derivative. The implementation of the unstructured mesh Crank-Nicolson Galerkin method (CNGM) is detailed and the stability and convergence of the numerical scheme are analysed. Numerical examples are presented to verify the theoretical analysis. To highlight the ability of the proposed unstructured mesh Galerkin finite element method, a comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme is conducted. The proposed numerical method using an unstructured mesh is shown to be more effective and feasible for practical applications involving irregular convex domains.
61 citations
TL;DR: In this article, the authors derived sufficient conditions on consensus by using the technique of inequality, which showed that the consensus can be achieved for any bounded input time delay, under the assumption that the interconnection topology of all agents has a spanning tree.
Abstract: Abstract Many phenomena in inter-disciplinary fields can be explained naturally by coordinated behavior of agents with fractional-order dynamics. Under the assumption that the interconnection topology of all agents has a spanning tree, the consensuses of linear and nonlinear fractional-order multi-agent systems with input time delay are studied, respectively. Based on the properties of Mittag-Leffler function, matrix theory, stability theory of fractional-order differential equations, some sufficient conditions on consensus are derived by using the technique of inequality, which shows that the consensus can be achieved for any bounded input time delay. Finally, two numerical examples are given to illustrate the effectiveness of the theoretical results.
61 citations
TL;DR: In this paper, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before, and a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc.
Abstract: Abstract The importance of the concept of stability in fractional order system and control has been recognized for some time now. Recently, it has become evident that many conclusions were drawn, but little consensus was reached. Consequently, there is an urgent need for a much deeper understanding of such a concept. With the definition of fractional order positive definite matrix, a set of equivalent and elegant stability criteria are developed via revisiting a stability criterion we proposed before. All the results are formed in terms of linear matrix inequalities. Afterwards, a series of interesting properties of these criteria are revealed profoundly, including completeness, singularity, conservatism, etc. Eventually, a simulation study is provided to validate the effectiveness of the obtained results.
53 citations
TL;DR: In this paper, the maximum principle for a time-fractional diffusion equation with the homogeneous Dirichlet boundary condition has been derived and the non-negativity of the weak solution has been shown.
Abstract: In this paper, we discuss the maximum principle for a time-fractional diffusion equation $$ \partial_t^\alpha u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_j u(x,t)) + c(x)u(x,t) + F(x,t),\ t>0,\ x \in \Omega \subset {\mathbb R}^n$$ with the Caputo time-derivative of the order $\alpha \in (0,1)$ in the case of the homogeneous Dirichlet boundary condition. Compared to the already published results, our findings have two important special features. First, we derive a maximum principle for a suitably defined weak solution in the fractional Sobolev spaces, not for the strong solution. Second, for the non-negative source functions $F = F(x,t)$ we prove the non-negativity of the weak solution to the problem under consideration without any restrictions on the sign of the coefficient $c=c(x)$ by the derivative of order zero in the spatial differential operator. Moreover, we prove the monotonicity of the solution with respect to the coefficient $c=c(x)$.
52 citations
TL;DR: In this paper, a fast space-time finite difference method for space time fractional diffusion equations by fully utilizing the mathematical structure of the scheme is proposed. But the method is not lossy, as no compression of the underlying numerical scheme has been employed.
Abstract: Abstract We develop a fast space-time finite difference method for space-time fractional diffusion equations by fully utilizing the mathematical structure of the scheme. A circulant block preconditioner is proposed to further reduce the computational costs. The method has optimal-order memory requirement and approximately linear computational complexity. The method is not lossy, as no compression of the underlying numerical scheme has been employed. Consequently, the method retains the stability, accuracy, and, in particular, the conservation property of the underlying numerical scheme. Numerical experiments are presented to show the efficiency and capacity of long time modelling of the new method.
TL;DR: In this article, a survey on inequalities in fractional calculus that have proven to be very useful in analyzing differential equations is presented, in particular the Leibniz inequality for fractional derivatives of Riesz, Riemann-Liouville or Caputo type.
Abstract: Abstract We present a survey on inequalities in fractional calculus that have proven to be very useful in analyzing differential equations. We mention in particular, a “Leibniz inequality” for fractional derivatives of Riesz, Riemann-Liouville or Caputo type and its generalization to the d-dimensional case that become a key tool in differential equations; they have been used to obtain upper bounds on solutions leading to global solvability, to obtain Lyapunov stability results, and to obtain blowing-up solutions via diverging in a finite time lower bounds. We will also mention the weakly singular Gronwall inequality of Henry and its variants, principally by Medved, that plays an important role in differential equations of any kind. We will also recall some “traditional” inequalities involving fractional derivatives or fractional powers of the Laplacian.
TL;DR: In this article, the authors present a survey of non-instantaneous impulses in Caputo fractional differential equations and compare two approaches in interpretation of the solutions, and also some existence results are derived.
Abstract: Abstract Recent modeling of real world phenomena give rise to Caputo type fractional order differential equations with non-instantaneous impulses. The main goal of the survey is to highlight some basic points in introducing non-instantaneous impulses in Caputo fractional differential equations. In the literature there are two approaches in interpretation of the solutions. Both approaches are compared and their advantages and disadvantages are illustrated with examples. Also some existence results are derived.
TL;DR: In this paper, a class of fractional semilinear integro-differential equations of order β ∈ (1, 2) with nonlocal conditions was studied.
Abstract: Abstract In this paper, we study a class of fractional semilinear integro-differential equations of order β ∈ (1,2] with nonlocal conditions. By using the solution operator, measure of noncompactness and some fixed point theorems, we obtain the existence of local and global mild solutions for the problem. The results presented in this paper improve and generalize many classical results. An example about fractional partial differential equations is given to show the application of our theory.
TL;DR: This paper considers corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc.
Abstract: Abstract In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich’s fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.
TL;DR: In this paper, Bessel models of linear viscoelasticity have been studied for short times and behave like fractional Maxwell bodies of order 1/2, and the authors discuss general properties of such models.
Abstract: In this paper we discuss some general properties of viscoelastic models defined in terms of constitutive equations involving infinitely many derivatives (of integer and fractional order). In particular, we consider as a working example the recently developed Bessel models of linear viscoelasticity that, for short times, behave like fractional Maxwell bodies of order 1/2.
TL;DR: In this article, the authors provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.
Abstract: We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie symmetries in the case of an arbitrary finite number of independent variables. We also prove the Lie theorem in the case of fractional differential equations, showing that the proper space for the analysis of these symmetries is the infinite dimensional jet space.
TL;DR: In this paper, the authors considered the controllability of fractional differential systems of Sobolev type in Banach spaces and derived approximate controllable results for the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2.
Abstract: Abstract This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on the norm continuity and compactness of some resolvent operators (also called solution operators). And then via the obtained properties on resolvent operators and fixed point technique, we give some approximate controllability results for Sobolev type fractional differential systems in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. Particularly, the existence or compactness of an operator E−1 is not necessarily needed in our results.
TL;DR: The first part of the important Abel's paper (Abel, N. H., Auflosung einer mechanischen ausgabe) was translated to English by as mentioned in this paper.
Abstract: This is the paper "Niels Henrik Abel and the birth of fractional calculus", Podlubny, I., Magin, R. L., Trymorush I., Fractional Calculus and Applied Analysis, vol.20, no.5, pp.1068-1075, 2017 (this https URL) with the supplements to it.
Hereby translation to English of the first part of the important Abel's paper (Abel, N. H., Oplosning af et par opgaver ved hjelp af bestemte integraler, Magazin for Naturvidenskaberne, Aargang I, Bind 2, Christiania, 1823) and another important Abel's paper (Abel, N. H., Auflosung einer mechanischen ausgabe. Journal fur die Reine und Angewandte Mathematik, Bind I, 153-157, Berlin, 1826) are provided.
To preserve the original flavor and the historical development of mathematical notation, the formulas are typeset in the original manner.
TL;DR: In this paper, a singular, fractional critical problem (Pλ) was studied, and the existence of multiple positive solutions for different values of λ was shown using variational methods.
Abstract: Abstract In this work we study the following singular, fractional critical problem (Pλ)(−Δ)su=λuγ+uqinΩ;u<0inΩ,u=0,inRN∖Ω, $$\begin{array}{*{20}{c}} {({{\text{P}}_\lambda })\left\{ {\begin{array}{*{20}{l}} {{{( - {\Delta})}^s}u = \frac{\lambda }{{{u^\gamma }}} + {u^q}\quad{\text{in}}\; {\Omega};} \\ {u \lt 0\quad {\text{in}} \;\Omega,} \\ {u = 0,\quad{\text{in}} \;{\mathbb{R}^N} \setminus {\Omega},} \end{array}} \right.} \end{array}$$ where Ω ⊂ ℝN(N ≥ 3) is a bounded domain with smooth boundary ∂Ω,N < 2s, 0 < 1, λ < 0, 0<γ<1
TL;DR: In this article, the inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered, and the general representation of regular solution in rectangular domain is obtained.
Abstract: The inhomogenous time-fractional telegraph equation with Caputo derevatives with constant coefficients is considered. For considered equation the general representation of regular solution in rectangular domain is obtained, and the fundamental solution is presented. Using this representation and the properties of fundamental solution, the Cauchy problem and the basic problems in half-strip and rectangular domains are studied. For Cauchy problem the theorems of existence and uniqueness of solution are proved, and the explicit form of solution is constructed. The solutions of the investigated problems are constructed in terms of the appropriate Green functions, which are also constructed an explicit form.
TL;DR: In this paper, the accurate relationship between the averaging procedure of a smooth function over 1D- fractal sets and the fractional integral of the RL-type is established, and numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed.
Abstract: Abstract In this paper the accurate relationships between the averaging procedure of a smooth function over 1D- fractal sets and the fractional integral of the RL-type are established. The numerical verifications are realized for confirmation of the analytical results and the physical meaning of these obtained formulas is discussed. Besides, the generalizations of the results for a combination of fractal circuits having a discrete set of fractal dimensions were obtained. We suppose that these new results help understand deeper the intimate links between fractals and fractional integrals of different types. These results can be used in different branches of the interdisciplinary physics, where the different equations describing the different physical phenomena and containing the fractional derivatives and integrals are used.
TL;DR: In this paper, the fundamental solution of the multidimensional time-fractional telegraph equation is studied in terms of a multivariate Mittag-Leffler function.
Abstract: Abstract In this paper we study the fundamental solution (FS) of the multidimensional time-fractional telegraph equation where the time-fractional derivatives of orders α ∈]0,1] and β ∈]1,2] are in the Caputo sense. Using the Fourier transform we obtain an integral representation of the FS in the Fourier domain expressed in terms of a multivariate Mittag-Leffler function. The Fourier inversion leads to a double Mellin-Barnes type integral representation and consequently to a H-function of two variables. An explicit series representation of the FS, depending on the parity of the dimension, is also obtained. As an application, we study a telegraph process with Brownian time. Finally, we present some moments of integer order of the FS, and some plots of the FS for some particular values of the dimension and of the fractional parameters α and β.
TL;DR: In this paper, the authors studied linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivative, with respect to a nonnegative density function.
Abstract: Abstract In the present work we study linear systems with distributed delays and distributed order fractional derivatives based on Caputo type single fractional derivatives, with respect to a nonnegative density function. For the initial problem of this kind of systems, existence, uniqueness and a priory estimate of the solution are proved. As an application of the obtained results, we establish sufficient conditions for global asymptotic stability of the zero solution of the investigated types of systems.
TL;DR: In this paper, the authors apply the classical control theory to a variable order fractional differential system in a bounded domain and show that the considered optimal control problem has a unique solution, where the performance index is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable order.
Abstract: Abstract In this paper, we apply the classical control theory to a variable order fractional differential system in a bounded domain. The Fractional Optimal Control Problem (FOCP) for variable order differential system is considered. The fractional time derivative is considered in a Caputo sense. We first study by using the Lax-Milgram Theorem, the existence and the uniqueness of the solution of the variable order fractional differential system in a Hilbert space. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as a function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable order. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Caputo derivative, we obtain an optimality system for the optimal control. Some examples are analyzed in details.
TL;DR: In this paper, the authors studied the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form C, H D a r x ( t ) = e( t ) + ∫ a t ln t s r − 1 k ( t, s ) f ( s, x ( s ) ) d s s, a > 1, r ǫ is as above.
Abstract: Abstract In this paper, the authors study the asymptotic behavior of solutions of higher order fractional differential equations with Caputo-type Hadamard derivatives of the form C , H D a r x ( t ) = e ( t ) + f ( t , x ( t ) ) , a > 1 , $$\\begin{equation*}^{C,H}\\mathcal{D}_{a}^{r}x(t)=e(t)+f(t,x(t)), \\quad a\\gt1, \\end{equation*}$$ where r = n+α–1, α ∊ (0,1), and n ∊ℤ+. They also apply their technique to investigate the oscillatory and asymptotic behavior of solutions of the related integral equation x ( t ) = e ( t ) + ∫ a t ln t s r − 1 k ( t , s ) f ( s , x ( s ) ) d s s , a > 1 , r is as above . $$\\begin{equation*}x(t)=e(t)+\\int\\limits_{a}^{t}\\left( \\ln \\frac{t}{s}\\right) ^{r-1}k(t,s)f(s,x(s))\\frac{ds}{s}, \\quad a\\gt1, \\quad r\\textrm{ is as above}. \\end{equation*}$$
TL;DR: In this article, an optimized PID-based non-precedent tilt control via both direct fractional-order PID design and via fractionalorder-order based loop shaping that reduces effect of lags in the design model is presented.
Abstract: Advances in the use of fractional order calculus in control theory in- creasingly make their way into control applications such as in the process industry, electrical machines, mechatronics/robotics, albeit at a slower rate into control applications in automotive and railway systems We present work on advances in high-speed rail vehicle tilt control design enabled by use of fractional order methods Analytical problems in rail tilt control still exist especially on simplified tilt using non-precedent sensor information (rather than use of the more complex precedence (or preview) schemes) Challenges arise due to suspension dynamic interactions (due to strong coupling between roll and lateral dynamic modes) and the sensor measurement We explore optimized PID-based non-precedent tilt control via both direct fractional-order PID design and via fractional-order based loop shaping that reduces effect of lags in the design model The impact of fractional order design methods on tilt performance (track curve following vs ride quality) trade off is particularly emphasized Simulation results illustrate superior benefit by utilizing fractional order-based tilt control design