β−type fractional Sturm‐Liouville Coulomb operator and applied results

In this article, β‐type fractional Sturm‐Liouville Coulomb operator is considered by Hilfer fractional derivative. Fundamental spectral theory is investigated for the aforementioned problem. In this context, it is shown that the operator is self‐adjoint, eigenfunctions correspond to the distinct eigenfunctions are orthogonal, and eigenvalues are real. Furthermore, applications of this problem are given by the Adomian decomposition method and the results are shown with visual graphs.     

Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator

In this paper, using the Riemann‐Liouville fractional integral with respect to another function and the ψ?Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro‐differential equation. In this sense, for this new fractional Volterra integro‐differential equation, we study the Ulam‐Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed‐point theorem. As an application, we present the Ulam‐Hyers stability using the α‐resolvent operator in the Sobolev space .     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation

In this paper, by means of Banach fixed point theorem, we investigate the existence and Ulam–Hyers–Rassias stability of the noninstantaneous impulsive integrodifferential equation by means of ψ‐Hilfer fractional derivative. In this sense, some examples are presented, in order to consolidate the results obtained.     

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L_∞ ‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ? ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L² type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.     

A remark on ψ–Hilfer fractional differential equations with non-instantaneous impulses

We present a reasonable concept for solutions of non-instantaneous impulsive Cauchy problems with a ψ–Hilfer fractional derivative. Also, we provide a new sufficient condition for the existence, uniqueness, and stability of solutions for the given problem.     

Stability Analysis of Distributed-Order Hilfer–Prabhakar Systems Based on Inertia Theory

The notion of a distributed-order Hilfer–Prabhakar derivative is introduced, which reduces in special cases to the existing notions of fractional or distributed-order derivatives. The stability of two classes of distributed-order Hilfer–Prabhakar differential equations, which are generalizations of all distributed or fractional differential equations considered previously, is analyzed. Sufficient conditions for the asymptotic stability of these systems are obtained by using properties of generalized Mittag-Leffler functions, the final-value theorem, and the Laplace transform. Stability conditions for such systems are introduced by using a new definition of the inertia of a matrix with respect to the distributed-order Hilfer–Prabhakar derivative.     

Study on Sobolev type Hilfer fractional integro-differential equations with delay

In this paper, we deal with a class of nonlinear Sobolev type fractional integro-differential equations with delay using Hilfer fractional derivative, which generalized the famous Riemann–Liouville fractional derivative. The definition of mild solutions for studied problem was given based on an operator family generated by the operator pair (A, B) and probability density function. Combining with the techniques of fractional calculus, measure of noncompactness and fixed point theorem, we obtain new existence result of mild solutions with two new characteristic solution operators and the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition. The results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.     

Controllability of Hilfer Fractional Differential Systems with Non-Dense Domain

In this paper, some exact controllability results are investigated for an integral solution of a non-densely defined abstract fractional differential system involving Hilfer fractional derivative. We tend to implement semigroup theory, fractional calculus, and measure of noncompactness to obtain the main results by fixed point technique. Finally, an application is given to illustrate the obtained results.     

Asymptotic expansion for harmonic functions in the half‐space with a pressurized cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as a reduced form of the boundary value problem for the Lamé system, we consider a Neumann problem for harmonic functions in the half‐space with a cavity C. Zero normal derivative is assumed at the boundary of the half‐space; differently, at ?C, the normal derivative of the function is required to be given by an external datum g, corresponding to a pressure term exerted on the medium at ?C. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half‐space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum g, which describes a constant pressure at ?C, we recover a simplified representation based on a polarization tensor. Copyright © 2016 John Wiley & Sons, Ltd.     

A new exploration on existence of Sobolev‐type Hilfer fractional neutral integro‐differential equations with infinite delay

This article is primarily focusing on the existence of Sobolev‐type Hilfer fractional neutral integro‐differential systems via measure of noncompactness. We study our primary outcomes by employing fractional calculus, measure of noncompactness and fixed point technique. First, we discuss the existence of mild solution for the fractional evolution system. Then, we extend our results to discuss the system with nonlocal conditions. Finally, we provide theoretical and practical applications to illustrate the obtained theory.     

Asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations

This article discusses the analyticity and the long‐time asymptotic behavior of solutions to space‐time fractional diffusion‐reaction equations in . By a Laplace transform argument, we prove that the decay rate of the solution as t→∞ is dominated by the order of the time‐fractional derivative. We consider the decay rate also in a bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.     

A new numerical approach for single rational soliton solution of Chen‐Lee‐Liu equation with Riesz fractional derivative in optical fibers

In this paper, time‐splitting spectral approximation technique has been proposed for Chen‐Lee‐Liu (CLL) equation involving Riesz fractional derivative. The proposed numerical technique is efficient, unconditionally stable, and of second‐order accuracy in time and of spectral accuracy in space. Moreover, it conserves the total density in the discretized level. In order to examine the results, with the aid of weighted shifted Grünwald‐Letnikov formula for approximating Riesz fractional derivative, Crank‐Nicolson weighted and shifted Grünwald difference (CN‐WSGD) method has been applied for Riesz fractional CLL equation. The comparison of results reveals that the proposed time‐splitting spectral method is very effective and simple for obtaining single soliton numerical solution of Riesz fractional CLL equation.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

A fractional order epidemic model and simulation for avian influenza dynamics

We present a nonlinear fractional order epidemic model to investigate the spreading dynamical behavior of the avian influenza. The population of the model contains susceptible individuals, asymptomatic but infective latent individuals, and infective individuals. We first establish the existence, uniqueness, nonnegativity, and positive invariance of the solution, then we study the reproduction number of the model and the stability of the disease‐free equilibrium. We observe that the reproduction number varies with the order of the fractional derivative ν. In terms of epidemics, this suggests that varying ν induces a change in the avian's epidemic status. Furthermore, we derive the sufficient conditions for the existence and the stability of the endemic equilibrium. Finally, we carry out some numerical simulations to validate the analytical results. We find from simulations that the solution of the fractional order model tends to a stationary state over a longer period of time with decreasing the value of the fractional derivative, and the size of epidemic decreases with decreasing ν.     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017     

Remarks on some families of fractional-order differential equations

The main purpose of this note is to draw the reader's attention towards some errors and omissions in a recent work involving solutions of some families of fractional-order differential equations, which was published in this Journal (see, for details, [Tomovski ?, Hilfer R, Srivastava HM. Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions. Integral Transforms Spec Funct. 2010;21:797–814]). Several relevant remarks and observations on some other related recent developments on this subject are also presented.     

Well‐posedness of time‐space fractional stochastic evolution equations driven by α‐stable noise

This paper is devoted to the well‐posedness for time‐space fractional Ginzburg‐Landau equation and time‐space fractional Navier‐Stokes equations by α‐stable noise. The spatial regularity and the temporal regularity of the nonlocal stochastic convolution are firstly established, and then the existence and uniqueness of the global mild solution are obtained by the Banach fixed point theorem and Mittag‐Leffler functions, respectively. Numerical simulations for time‐space fractional Ginzburg‐Landau equation are provided to verify the analysis results.