Class Of Time-fractional Diffusion Equations Research Articles

APPLICATION OF HOSOYA POLYNOMIAL TO SOLVE A CLASS OF TIME-FRACTIONAL DIFFUSION EQUATIONS

In this paper, we study time-fractional diffusion equations such as the time-fractional Kolmogorov equations (TF–KEs) and the time-fractional advection–diffusion equations (TF–ADEs) in the Caputo sense. Here, we have developed the operational matrices (OMs) using the Hosoya polynomial (HP) as basis function for OMs to obtain solution of the TF–KEs and the TF–ADEs. The great benefit of this technique is converting the TF–KEs and the TF–ADEs to algebraic equations, which can be simply solved the problem under study. We provide error bound for the approximation of a bivariate function using the HP. Furthermore, comparison of the numerical results obtained using the proposed technique with the exact solution is done. The results prove that the proposed numerical method is most relevant for solving the TF–KEs and the TF–ADEs and accurate.

A class of time-fractional diffusion equations with generalized fractional derivatives

In this paper, we consider generalized fractional derivatives, which are characterized by a scale function and a weight function. It is proposed to replace the time variable with a new variable, associated with the scale and weight functions, which allows us to reduce the problems for equations with generalized fractional derivatives to problems for equations with the usual fractional derivative. The equations obtained after the above change of variables are quite well studied, so that one can apply well-known effective numerical methods and use the reverse substitution to find solutions to the original problems.

Collocation Method for Time Fractional Diffusion Equation Based on the Chebyshev Polynomials of Second Kind

This paper deals with the numerical solution for a class of time-fractional diffusion equations (TFDE). The fractional time derivative is considered in the Caputo sense of order $$\mu (0<\mu \le 1)$$ . Finite difference approximation is used for time derivative, while Chebyshev polynomials of the second kind are used to approximate the space derivative. The given scheme is simple in use for solving TFDE since the given boundary and initial conditions are taken into account automatically. The advantages of the proposed scheme are its exponential convergence and low computational cost. Also, we discuss error analysis and convergence of the suggested scheme for solving TFDE. The given scheme is examined through some examples and comparison are provided with existed methods which show the efficiency and accuracy of proposed scheme.

Stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations

The aim of this paper is to study the stability and boundedness of solutions of the initial value problem for a class of time-fractional diffusion equations. We first establish a fractional Duhamel principle for the nonhomogeneous time-fractional diffusion equation. Then based on it and the superposition principle, the solution of the above initial value problem is represented. Finally, we obtain the stability and boundedness of the solution and present an illustrative example.

Optimized Overlapping Schwarz Waveform Relaxation for a Class of Time-Fractional Diffusion Problems

For parabolic PDEs with integer-order temporal derivative, if we use the Schwarz waveform relaxation (SWR) algorithm with Robin transmission conditions as the solver, the so-called equioscillation principle is an important concept to get a good Robin parameter, which has a significant effect on the convergence rate of the algorithm. Surprisingly, as we show in this paper such a principle may result in rather disappointing Robin parameter for the SWR algorithm when we use it to solve time-fractional PDEs. For a class of time-fractional diffusion equations, by analyzing a new min–max problem we get much better Robin parameter, which is found very close to the best one that we can make through numerical optimizations and numerical experiments. To use the SWR algorithm in practice, we apply the kernel reduction technique proposed recently by Baffet and Hesthaven to treat the convolutions with kernel function of the form \(\mathscr {K}_{\gamma }(t)=t^{-\gamma }/\varGamma (1-\gamma )\), where \(\gamma \in (0, 1)\). For time-fractional PDEs with this kind of kernel function, the kernel reduction technique results in efficient one-step numerical schemes. Numerical results obtained by using this technique confirm our theoretical conclusions very well.

A semi-discrete finite element method for a class of time-fractional diffusion equations

As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection-diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

Fractional calculus and its applications

Fractional calculus was formulated in 1695, shortly after the development of classical calculus. The earliest systematic studies were attributed to Liouville, Riemann, Leibniz, etc. [1,2]. For a long time, fractional calculus has been regarded as a pure mathematical realm without real applications. But, in recent decades, such a state of affairs has been changed. It has been found that fractional calculus can be useful and even powerful, and an outline of the simple history about fractional calculus, especially with applications, can be found in Machado et al. [3]. Now, fractional calculus and its applications is undergoing rapid developments with more and more convincing applications in the real world [4,5]. This Theme Issue, including one review article and 12 research papers, can be regarded as a continuation of our first special issue of European Physical Journal Special Topics in 2011 [4], and our second special issue of International Journal of Bifurcation and Chaos in 2012 [5]. These selected papers were mostly reported in The Fifth Symposium on Fractional Derivatives and Their Applications (FDTA'11) that was held in Washington DC, USA in 2011. The first paper presents an overview of chaos synchronization of coupled fractional differential systems. A list of coupling schemes are presented, including one-way coupling, Pecora–Carroll coupling, active–passive decomposition coupling, bidirectional coupling and other unidirectional coupling configurations. Meanwhile, several extended concepts of synchronizations are introduced, namely projective synchronization, hybrid projective synchronization, function projective synchroni- zation, generalized synchronization and generalized projective synchronization. Corresponding to different kinds of synchronization schemes, various analysis methods are presented and discussed [6]. The rest of the papers can be roughly grouped into three parts: three papers for fundamental theories of fractional calculus [7–9], five papers for fractional modelling with applications [10–14] and four papers for numerical approaches [15–18]. In the theory part, three papers focus on the existence of the solutions to the considered classes of nonlinear fractional systems, the equivalence system of the multiple-rational-order fractional system, and the reflection symmetry with applications to the Euler–Lagrange equations [7–9]. Baleanu et al. [7] use fixed-point theorems to prove the existence and uniqueness of the solutions to a class of nonlinear fractional differential equations with different boundary-value conditions. Li et al. [8] apply the properties of the fractional derivatives to change the multiple-rational-order system into the fractional system with the same order. Such a reduction makes it convenient for stability analysis and numerical simulations. The reflection symmetry and its applications to the Euler–Lagrange equations in fractional mechanics are investigated in Klimek [9], where an illustrative example is presented. The part on fractional modelling with applications consists of five papers [10–14]. Chen et al. [10] establish a fractional variational optical flow model for motion estimation from video sequences, where the experiments demonstrate the validity of the generalization of derivative order. Another fractional modelling in heat transfer with heterogeneous media is studied in Sierociuk et al. [11]. In the following paper, two-particle dispersion is explored in the context of the anomalous diffusion, where two modelling approaches related to time subordination are considered and unified in the framework of self-similar stochastic processes [12]. The last two papers in this part emphasize the applications of fractional calculus [13,14], where a novel method for the solution of linear constant coefficient fractional differential equations of any commensurate order is introduced in the former paper, and where the CRONE control-system design toolbox for the control engineering community is presented in the latter paper. The last four papers in part three are attributed to numerical approaches [15–18]. Sun et al. [15] construct a semi-discrete finite-element method for a class of temporal-fractional diffusion equations. On the other hand, an implicit numerical algorithm for the spatial- and temporal-fractional Bloch–Torrey equation is established, where stability and convergence are also considered [16]. In Fukunaga & Shimizu [17], a high-speed scheme for the numerical approach of fractional differentiation and fractional integration is proposed. In the last paper, Podlubny et al. [18] further develop Podlubny's matrix approach to discretization of non-integer derivatives and integrals, where non-equidistant grids, variable step lengths and distributed orders are considered. We try our best to organize this Theme Issue in order to offer fresh stimuli for the fractional calculus community to further promote and develop cutting-edge research on fractional calculus and its applications.

The fractional finite Hankel transform and its applications in fractal space

In the present work, a generalized finite Hankel transform is derived which is useful in solving equations in fractal dimension df and involving a fractal diffusion coefficient D0r−θ. The corresponding inversion formula is established and some properties are given. Then, the transform is successfully used to solve a class of time-fractional diffusion equations in fractional spatial dimension with an absorbent term and Schrödinger equation in fractional-dimensional space. Green's functions and exact wave function of the above problems are found.