Higher order numerical methods for solving fractional differential equations

The final publication is available at Springer via http://dx.doi.org/10.1007/s10543-013-0443-3In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0 0. The order of convergence of the numerical method is O(h^3) for α ≥ 1 and O(h^(1+2α)) for 0 < α ≤ 1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results

A hybrid numerical scheme for fractional-order systems

In this work we present a hybrid numerical scheme for the solution of systems of fractional differential equations arising in several fields of engineering. The numerical scheme can deal with both smooth and non-smooth solutions, and, the idea behind the hybrid method is that of approximating the solution as a linear combination of non-polynomial functions in a region near the singularity, and by polynomials in the remaining domain. The numerical method is then used to study fractional RC electrical circuits.The first, third and fourth authors would like to thankthe funding by FCT-Portuguese Foundation for Science and Technology throughscholarship and projects: SFRH/BPD/100353/2014 and UID/Multi/04621/2013, UID/MAT/00297/2013 (Centro de Matem ́atica e Aplica ̧c ̃oes), respectively

Jacobi-Predictor-Corrector Approach for the Fractional Ordinary Differential Equations

We present a novel numerical method, called {\tt Jacobi-predictor-corrector approach}, for the numerical solution of fractional ordinary differential equations based on the polynomial interpolation and the Gauss-Lobatto quadrature w.r.t. the Jacobi-weight function $\omega(s)=(1-s)^{\alpha-1}(1+s)^0$. This method has the computational cost O(N) and the convergent order $IN$, where $N$ and $IN$ are, respectively, the total computational steps and the number of used interpolating points. The detailed error analysis is performed, and the extensive numerical experiments confirm the theoretical results and show the robustness of this method.Comment: 24 pages, 5 figure

Good (and Not So Good) practices in computational methods for fractional calculus

The solution of fractional-order differential problems requires in the majority of cases the use of some computational approach. In general, the numerical treatment of fractional differential equations is much more difficult than in the integer-order case, and very often non-specialist researchers are unaware of the specific difficulties. As a consequence, numerical methods are often applied in an incorrect way or unreliable methods are devised and proposed in the literature. In this paper we try to identify some common pitfalls in the use of numerical methods in fractional calculus, to explain their nature and to list some good practices that should be followed in order to obtain correct results

Fractional compartmental models and multi-term Mittag–Leffler response functions

Systems of fractional differential equations (SFDE) have been increasingly used to represent physical and control system, and have been recently proposed for use in pharmacokinetics (PK) by (J Pharmacokinet Pharmacodyn 36:165–178, 2009) and (J Phamacokinet Pharmacodyn, 2010). We contribute to the development of a theory for the use of SFDE in PK by, first, further clarifying the nature of systems of FDE, and in particular point out the distinction and properties of commensurate versus non-commensurate ones. The second purpose is to show that for both types of systems, relatively simple response functions can be derived which satisfy the requirements to represent single-input/single-output PK experiments. The response functions are composed of sums of single- (for commensurate) or two-parameters (for non-commensurate) Mittag–Leffler functions, and establish a direct correspondence with the familiar sums of exponentials used in PK

Why fractional derivatives with nonsingular kernels should not be used

In recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new

Coupled systems of fractional equations related to sound propagation: analysis and discussion

In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive and also memoryless processes like classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations

Numerical simulation of the fractional Bloch equations

In physics and chemistry, specifically in NMR (nuclear magnetic resonance) or MRI (magnetic resonance imaging), or ESR (electron spin resonance) the Bloch equations are a set of macroscopic equations that are used to calculate the nuclear magnetization M=(,,) as a function of time when relaxation times and are present. Recently, some fractional models have been proposed for the Bloch equations, however, effective numerical methods and supporting error analyses for the fractional Bloch equation (FBE) are still limited. In this paper, the time-fractional Bloch equations (TFBE) and the anomalous fractional Bloch equations (AFBE) are considered. Firstly, we derive an analytical solution for the TFBE with an initial condition. Secondly, we propose an effective predictor-corrector method (PCM) for the TFBE, and the error analysis for PCM is investigated. Furthermore, we derive an effective implicit numerical method (INM) for the anomalous fractional Bloch equations (AFBE), and the stability and convergence of the INM are investigated. We prove that the implicit numerical method for the AFBE is unconditionally stable and convergent. Finally, we present some numerical results that support our theoretical analysis

Detailed error analysis for a fractional adams method with graded meshes

The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5We consider a fractional Adams method for solving the nonlinear fractional differential equation \, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil -1$. Here $\alpha$ may be an arbitrary positive number and $\lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption \, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T], Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{-2})$ if $\alpha > 1$ and $O(N^{-1-\alpha})$ if $\alpha \leq 1$. They also showed that if \, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T], the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if \, ^{C}_{0}D^{\alpha}_{t} y behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results