Lattice Boltzmann method for the fractional sub‐diffusion equation

A lattice Boltzmann model for the fractional sub‐diffusion equation is presented. By using the Chapman–Enskog expansion and the multiscale time expansion, several higher‐order moments of equilibrium distribution functions and a series of partial differential equations in different time scales are obtained. Furthermore, the modified partial differential equation of the fractional sub‐diffusion equation with the second‐order truncation error is obtained. In the numerical simulations, comparisons between numerical results of the lattice Boltzmann models and exact solutions are given. The numerical results agree well with the classical ones. Copyright © 2015 John Wiley & Sons, Ltd.     

Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions

We give an analytical treatment of a time fractional diffusion equation with Caputo time-fractional derivative in a bounded domain with different boundary conditions. We use the Fourier method of separation of variables and Laplace transform method. The solution is obtained in terms of the Mittag-Leffler-type functions and complete set of eigenfunctions of the Sturm–Liouville problem. Such problems can be used in the context of anomalous diffusion in complex media, as well as for modeling voltammetric experiment in limiting diffusion space.     

Discrete weighted fractional calculus and applications

Nonlinear Dynamics - We introduce fractional (nabla) sums and differences with a weight function and prove some of their properties. We give particular emphasis to the tempered fractional case. In...     

Discrete fractional logistic map and its chaos

A discrete fractional logistic map is proposed in the left Caputo discrete delta’s sense. The new model holds discrete memory. The bifurcation diagrams are given and the chaotic behaviors are numerically illustrated.     

On the physical interpretation of fractional diffusion

Even if the diffusion equation has been widely used in physics and engineering, and its physical content is well understood, some variants of it escape fully physical understanding. In particular, anormal diffusion appears in the so-called fractional diffusion equation, whose main particularity is its non-local behavior, whose physical interpretation represents the main part of the present work.     

Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives

Meccanica - We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the...     

On the fundamental linear fractional order differential equation

This paper deals with the rational function approximation of the irrational transfer function G(s) = \fracX(s)E(s) = \frac1[(t₀s)^2m + 2z(t₀s)^m + 1]G(s) = \frac{X(s)}{E(s)} = \frac{1}{[(\tau _{0}s)^{2m} + 2\zeta (\tau _{0}s)^{m} + 1]} of the fundamental linear fractional order differential equation (t₀)^2m\fracd^2mx(t)dt^2m + 2z(t₀)^m\fracd^mx(t)dt^m + x(t) = e(t)(\tau_{0})^{2m}\frac{d^{2m}x(t)}{dt^{2m}} + 2\zeta(\tau_{0})^{m}\frac{d^{m}x(t)}{dt^{m}} + x(t) = e(t), for 0<m<1 and 0<ζ<1. An approximation method by a rational function, in a given frequency band, is presented and the impulse and the step responses of this fractional order system are derived. Illustrative examples are also presented to show the exactitude and the usefulness of the approximation method.     

The Cattaneo type space-time fractional heat conduction equation

The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient ${{\partial_x}}$ and fractional time derivative instead of the first order partial time derivative ${{\partial_t}}$ . The existence of the unique solution to the initial-boundary value problem corresponding to the generalized model is established in the space of distributions. We also obtain explicit form of the solution and compare it numerically with some limiting cases.     

Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions

An implicit finite difference method is developed for a one-dimensional fractional percolation equation (FPE) with the Dirichlet and fractional boundary conditions. The stability and convergence are discussed for two special cases, i.e., a continued seepage flow with a monotone percolation coefficient and a seepage flow with the fractional Neumann boundary condition. The accuracy and efficiency of the method are checked with two numerical examples.     

Efficient solution of a vibration equation involving fractional derivatives

Fractional order (or, shortly, fractional) derivatives are used in viscoelasticity since the late 1980s, and they grow more and more popular nowadays. However, their efficient numerical calculation is non-trivial, because, unlike integer-order derivatives, they require evaluation of history integrals in every time step. Several authors tried to overcome this difficulty, either by simplifying these integrals or by avoiding them. In this paper, the Adomian decomposition method is applied on a fractionally damped mechanical oscillator for a sine excitation, and the analytical solution of the problem is found. Also, a series expansion is derived which proves very efficient for calculations of transients of fractional vibration systems. Numerical examples are included.     

Darboux problem for a differential equation with fractional derivative

We find conditions for the unique solvability of the problem u _xy (x, y) = f(x, y, u(x, y), (D ₀ ^r u)(x, y)), u(x, 0) = u(0, y) = 0, x ∈ [0, a], y ∈ [0, b], where (D ₀ ^r u)(x, y) is the mixed Riemann-Liouville derivative of order r = (r ₁, r ₂), 0 < r ₁, r ₂ < 1, in the class of functions that have the continuous derivatives u _xy (x, y) and (D ₀ ^r u)(x, y). We propose a numerical method for solving this problem and prove the convergence of the method. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 456–467, October–December, 2005.     

A periodic solution of the diffusion equation

Under some circumstances, autooscillations can arise in an electrochemical system with decreasing characteristics [1–3]. A method for finding the polarization curve P=P() (here is the electrode potential, P=i I c (0, t), where i is the current density, and c(0, t) is the mass concentration at the electrode surface), if the distribut ion in time of the current density is given, is proposed in [1]. In the numerical solution of this problem, which is considered below, considerable computational difficulties were encountered.     

A higher order numerical scheme for generalized fractional diffusion equations

In this article, we develop a higher order approximation for the generalized fractional derivative that includes a scale function z(t) and a weight function w(t). This is then used to solve a generalized fractional diffusion problem numerically. The stability and convergence analysis of the numerical scheme are conducted by the energy method. It is proven that the temporal convergence order is 3 and this is the best result to date. Finally, we present four examples to confirm the theoretical results.     

Optimal control of a class of fractional heat diffusion systems

In this paper, a solution procedure for a class of optimal control problems involving distributed parameter systems described by a generalized, fractional-order heat equation is presented. The first step in the proposed procedure is to represent the original fractional distributed parameter model as an equivalent system of fractional-order ordinary differential equations. In the second step, the necessity for solving fractional Euler–Lagrange equations is avoided completely by suitable transformation of the obtained model to a classical, although infinite-dimensional, state-space form. It is shown, however, that relatively small number of state variables are sufficient for accurate computations. The main feature of the proposed approach is that results of the classical optimal control theory can be used directly. In particular, the well-known “linear-quadratic” (LQR) and “Bang-Bang” regulators can be designed. The proposed procedure is illustrated by a numerical example.     

Constitutive equation with fractional derivatives for the generalized UCM model

A fundamental problem on the constitutive equation with fractional derivatives for the generalized upper convected Maxwell model (UCM) is studied. The existing investigations on the constitutive equation are reviewed and their limitations or deficiencies are highlighted. By utilizing the convected coordinates approach, a mathematically rigorous constitutive equation with fractional derivatives for the generalized UCM model is proposed, which has an explicit expression for the stress tensor. This model can be reduced to the linear generalized Maxwell model with fractional derivatives, the UCM model and some other existing models. In addition, the rheological properties of this proposed model in the start-up of simple shear and elongation flows are investigated. It is shown that this generalized UCM model can describe the various stress evolution processes and the strain hardening effect of the viscoelastic fluids.