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We discuss a method which provides accurate numerical solutions to fractional-in-time partial differential equations posed on [ 0 , T ] × Ω with Ω ⊂ R d without the excessive memory requirements associated with the nonlocal fractional derivative operator. Our approach combines recent advances in the development and utilization of multivariate sparse spectral methods as well as fast methods for the computation of Gauss quadrature nodes with recursive non-classical methods for the Caputo fractional derivative of general fractional order α > 0. An attractive feature of the method is that it has minimal theoretical overhead when using it on any domain Ω ⊂ R d on which an orthogonal polynomial basis is available. We discuss the memory requirements of the method, present several numerical experiments demonstrating the method's performance in solving time-fractional PDEs on intervals, triangles and disks and derive error bounds which suggest sensible convergence strategies. As an important model problem for this approach we consider a type of wave equation with time-fractional dampening related to acoustic waves in viscoelastic media with applications in the physics of medical ultrasound and outline future research steps required to use such methods for the reverse problem of image reconstruction from sensor data. [ABSTRACT FROM AUTHOR]

Abstract In this report we establish Green's formula for an integro-differential operator, and apply it to describe a class of self-adjoint fractional order differential operators. A found symmetric fractional order Caputo–Riemann–Liouville type operator can be considered as a fractional analogue of the classical Sturm–Liouville operator in some sense. [ABSTRACT FROM AUTHOR]

In this paper, we consider the extended stochastic Navier–Stokes equations with Caputo derivative driven by fractional Brownian motion. We firstly derive the pathwise spatial and temporal regularity of the generalized Ornstein–Uhlenbeck process. Then we discuss the existence, uniqueness, and Hölder regularity of mild solutions to the given problem under certain sufficient conditions, which depend on the fractional order α and Hurst parameter H . The results obtained in this study improve some results in existing literature. [ABSTRACT FROM AUTHOR]

A two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ ( 1 , 2 ) and where a convection term is also present is considered. Its boundary conditions are of Robin type and include Dirichlet boundary conditions as a special case. An explicit formula for the associated Green's function is obtained in terms of two-parameter Mittag-Leffler functions. Some new properties of these Mittag-Leffler functions are derived; from these, one can deduce necessary and sufficient conditions on the boundary conditions that ensure non-negativity of the Green's function and hence a maximum principle for the boundary value problem. In particular, unlike the classical elliptic case α = 2 , Dirichlet boundary conditions will not in general yield a maximum principle. [ABSTRACT FROM AUTHOR]

In this paper, a finite difference scheme is proposed to solve time–space fractional diffusion equation which has second-order accuracy in both time and space direction. The time and space fractional derivatives are considered in the senses of Caputo and Riesz, respectively. First, the centered difference approach is used to approximate the Riesz fractional derivative in space. Then, the obtained fractional ordinary differential equations are transformed into equivalent Volterra integral equations. And then, the trapezoidal rule is utilized to approximate the Volterra integral equations. The stability and convergence of our scheme are proved via mathematical induction method. Finally, numerical experiments are performed to confirm the high accuracy and efficiency of our scheme. [ABSTRACT FROM AUTHOR]

The present work studies the effect of an impulsive perturbation to classical and fractional classes of initial value problems. Scattering properties of the time sequences corresponding to the impulses are studied in the both integer and non-integer order cases. It is shown that, the sign-changing property of the perturbation function determines the mentioned scattering property of the kick times. [ABSTRACT FROM AUTHOR]

In this paper we are interested in the well-posedness of fully nonlinear Cauchy problems in which the time derivative is of Caputo type. We address this question in the framework of viscosity solutions, obtaining the existence via Perron's method, and comparison for bounded sub and supersolutions by a suitable regularization through inf and sup convolution in time. As an application, we prove the steady-state large time behavior in the case of proper nonlinearities and provide a rate of convergence by using the Mittag–Leffler operator. [ABSTRACT FROM AUTHOR]

We propose a simple and accurate numerical scheme for solving the time fractional telegraph (TFT) equation within Caputo type fractional derivative. A fictitious coordinate ϑ is imposed onto the problem in order to transform the dependent variable u ( x , t ) into a new variable with an extra dimension. In the new space with the added fictitious dimension, a combination of method of line and group preserving scheme (GPS) is proposed to find the approximate solutions. This method preserves the geometric structure of the problem. Power and accuracy of this method has been illustrated through some examples of TFT equation. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical method for solving the distributed fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann–Liouville fractional integral operator for hybrid functions is introduced. This operator is then utilized to reduce the solution of the distributed fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional moving-boundary problem for the time-fractional diffusion equation, where the time-fractional derivative of order α ∈ ( 0 , 1 ) is taken in the Caputo sense. A generalization of the Hopf lemma is proved and then used to prove a monotonicity property for the free-boundary when a fractional free-boundary Stefan problem is investigated. [ABSTRACT FROM AUTHOR]

In this paper, an efficient and accurate spectral numerical method is presented for solving second-, fourth-order fractional diffusion-wave equations and fractional wave equations with damping. The proposed method is based on Jacobi tau spectral procedure together with the Jacobi operational matrix for fractional integrals, described in the Riemann–Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought-for spectral approximations. The validity and effectiveness of the method are demonstrated by solving five numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. [ABSTRACT FROM AUTHOR]

Shooting methods are a well established tool for the numerical solution of terminal value problems of fractional order. However, they can be computationally quite expensive because of their iterative nature in which (a) each single iteration may be costly, and (b) the number of iterations can be large. In this paper we propose algorithmic strategies for improving the efficiency of such methods. Our strategies are aimed at simultaneously reducing the cost of each iteration and reducing the number of required iterations. [ABSTRACT FROM AUTHOR]

In this work, we establish the energy stability of high-order L2-type schemes for time-fractional phase-field equations. We propose a reformulation of the discrete L2 operator, show the monotonicity of some associated coefficients, and then obtain two positive definiteness properties for the L2 operator. Based on these two properties, we show that the energy is bounded by the initial energy for the L2 scalar auxiliary variable schemes of time-fractional gradient flows. Furthermore, a fractional energy law can be established for the L2 implicit-explicit scheme of the time-fractional Allen–Cahn equation. Several numerical experiments are provided to verify the stability as well as the convergence. • A reformulation of L2 approximation is introduced for monotonicity. • Two positive definiteness properties of L2 operator are proposed. • The energy stability of a second order L2 SAV scheme and a 3 − α order L2 IMEX scheme is established. [ABSTRACT FROM AUTHOR]

In this paper we consider a boundary-value problem for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides dependent on time. We prove the one-valued solvability of this problem, and provide the coercive estimates of the solution. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present a simple and efficient computational algorithm for solving eigenvalue problems of high fractional-order differential equations with variable coefficients. The method of solution is based on utilizing the series solution to convert the governing fractional differential equation into a linear system of algebraic equations. Then, the eigenvalues can be calculated by finding the roots of the corresponding characteristic polynomial. Notice that this class of eigenvalue problems is very promising to the solution of linear fractional partial differential equations (FPDE). The numerical results demonstrate reliability and efficiency of the proposed algorithm. Based on our simulations some theoretical conjectures are reported. [Copyright &y& Elsevier]

We study the large-time behavior in all L p norms of solutions to a heat equation with a Caputo α -time derivative posed in R N (0 < α < 1). These are known as subdiffusion equations. The initial data are assumed to be integrable, and, when required, to be also in L p. We find that the decay rate in all L p norms, 1 ≤ p ≤ ∞ , depends greatly on the space-time scale under consideration. This result explains in particular the so called "critical dimension phenomenon" (cf. [21]). Moreover, we find the final profiles (that strongly depend on the scale). The most striking result states that in compact sets the final profile (in all L p norms) is a multiple of the Newtonian potential of the initial datum. Our results are very different from the ones for classical diffusion equations and show that, in accordance with the physics they have been proposed for, these are good models for particle systems with sticking and trapping phenomena or fluids with memory. [ABSTRACT FROM AUTHOR]

Abstract: Fractional derivatives provide a general approach for modeling transport phenomena occurring in diverse fields. This article describes a Least Squares Spectral Method for solving advection–dispersion equations using Caputo or Riemann–Liouville fractional derivatives. A Gauss–Lobatto–Jacobi quadrature is implemented to approximate the singularities in the integrands arising from the fractional derivative definition. Exponential convergence rate of the operator is verified when increasing the order of the approximation. Solutions are calculated for fractional-time and fractional-space differential equations. Comparisons with finite difference schemes are included. A significant reduction in storage space is achieved by lowering the resolution requirements in the time coordinate. [Copyright &y& Elsevier]

Abstract: Multi-term fractional diffusion-wave equation along with the homogeneous/non-homogeneous boundary conditions has been solved using the method of separation of variables. It is observed that, unlike in the one term case, solution of multi-term fractional diffusion-wave equation is not necessarily non-negative, and hence does not represent anomalous diffusion of any kind. [Copyright &y& Elsevier]

Abstract: Anomalous diffusion is one of the most ubiquitous phenomena in nature, and it is present in a wide variety of physical situations, for instance, transport of fluid in porous media, diffusion of plasma, diffusion at liquid surfaces, etc. The fractional approach proved to be highly effective in a rich variety of scenarios such as continuous time random walk models, generalized Langevin equations, or the generalized master equation. To investigate the subdiffusion of anomalous diffusion, it would be useful to study a time fractional Fokker–Planck equation. In this paper, firstly the time fractional, the sense of Riemann–Liouville derivative, Fokker–Planck equation is transformed into a time fractional ordinary differential equation (FODE) in the sense of Caputo derivative by discretizing the spatial derivatives and using the properties of Riemann–Liouville derivative and Caputo derivative. Then combining the predictor–corrector approach with the method of lines, the algorithm is designed for numerically solving FODE with the numerical error , and the corresponding stability condition is got. The effectiveness of this numerical algorithm is evaluated by comparing its numerical results for with the ones of directly discretizing classical Fokker–Planck equation, some numerical results for time fractional Fokker–Planck equation with several different fractional orders are demonstrated and compared with each other, moreover for the convergent order in space is confirmed and the numerical results with different time step sizes are shown. [Copyright &y& Elsevier]

Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases, fractional-order-dependent as well as fractional-order-independent characterisations of stability and instability properties are obtained, in terms of the coefficients of the multi-term fractional differential equation. The theoretical results are exemplified for the particular cases of the Basset and Bagley-Torvik equations, as well as for a multi-term fractional differential equation of an inextensible pendulum with fractional damping terms, and for a fractional harmonic oscillator. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Strichartz type estimates for space-time fractional Schrödinger equations. By making use of the Besov norm and the Mittag-Leffler function, we show that the regularity of the solutions depends on the ratio of the orders of time and space derivatives. In particular, if the ratio equals to 2, it follows that the original Strichartz estimates can be recovered. We also show that the regularity of the solution is sharp in the sense of the Besov norm. [ABSTRACT FROM AUTHOR]

We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup. [ABSTRACT FROM AUTHOR]