# A finite difference method for an initial–boundary value problem with a Riemann–Liouville–Caputo spatial fractional derivative

TL;DR: A fractional Friedrichs’ inequality is derived and is used to prove that the problem approaches a steady-state solution when the source term is zero, and it is proved that the scheme converges with first order in the maximum norm.

About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2021-01-01 and is currently open access. It has received 2 citations till now. The article focuses on the topics: Fractional calculus & Boundary value problem.

## Summary (1 min read)

Jump to: [1. Introduction] – [2. Some properties of the solution] – [3. Finite difference scheme] – [4. Error estimate for scheme] and [5. Numerical experiments]

### 1. Introduction

- The problem considered in this paper is inspired by [1], which gives a lengthy discussion of various types of fractional initial-boundary value problem and the boundary conditions that are appropriate for each type.
- Here the authors assume that the function v is such that the definitions make sense.
- In [1] the quantity Dα−1C,x u is called the Caputo fractional flux.
- Three numerical examples are given in Section 5, to illustrate their theoretical results.
- Note that C can take different values in different places.

### 2. Some properties of the solution

- In this section the authors first derive a Friedrichs’ inequality for Caputo derivatives (Lemma 1).
- Lemma 1 (Friedrichs’ inequality for Caputo derivatives).
- A related result was obtained in [1, but using a very technical argument.
- Corollary 1 (Stability and uniqueness of solution).

### 3. Finite difference scheme

- ,N. To discretise the spatial derivative in (1a) the authors follow [8], where the two-point boundary value problem corresponding to (1a) was considered.
- The time derivative in (1a) is discretised by the backward Euler method.
- Thus, the diagonal entries of A are positive and its off-diagonal entries are nonpositive.
- This completes the inductive step and the proof.

### 4. Error estimate for scheme

- In this section an error bound for their difference scheme is derived.
- To convert these bounds to an error estimate for the computed solution, the authors shall employ a barrier function whose construction is discussed in Section 4.3.
- For the time-dependent problem (1), one expects that the classical time derivative will not affect the behaviour of the spatial derivatives.
- The mesh function {Ψ̃nm}M,Nm=0,n=0 is called a discrete barrier function.

### 5. Numerical experiments

- The authors present numerical results for three examples.
- In the first example, the exact solution is known and satisfies the bounds (17); its numerical results illustrate the error estimates of Theorem 1.
- In the third example, convergence to steady state and mass conservation are discussed.
- The computed orders of converge again agree with Theorem 1.

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##### Citations

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TL;DR: In this paper, the authors derived physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach, and theoretical properties, including well-posedness and steady state solutions, were reviewed.

Abstract: This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.

39 citations

##### References

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TL;DR: In this paper, the authors discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order, and investigate the dependence of the solution on the order of the differential equation and on the initial condition.

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TL;DR: In this paper, a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented, and appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs.

Abstract: In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

707 citations

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TL;DR: The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading.

Abstract: A reaction-diffusion problem with a Caputo time derivative of order $\alpha\in (0,1)$ is considered. The solution of such a problem is shown in general to have a weak singularity near the initial time $t=0$, and sharp pointwise bounds on certain derivatives of this solution are derived. A new analysis of a standard finite difference method for the problem is given, taking into account this initial singularity. This analysis encompasses both uniform meshes and meshes that are graded in time, and includes new stability and consistency bounds. The final convergence result shows clearly how the regularity of the solution and the grading of the mesh affect the order of convergence of the difference scheme, so one can choose an optimal mesh grading. Numerical results are presented that confirm the sharpness of the error analysis.

573 citations

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TL;DR: In this article, a nonlocal model based on fractional derivatives (FDs) is proposed to describe nondiffusive transport in magnetically confined plasmas, and an α-weighted explicit/implicit numerical integration scheme based on the Grunwald-Letnikov representation of the regularized fractional diffusion operator in flux conserving form is presented.

Abstract: A class of nonlocal models based on the use of fractional derivatives (FDs) is proposed to describe nondiffusive transport in magnetically confined plasmas. FDs are integro-differential operators that incorporate in a unified framework asymmetric non-Fickian transport, non-Markovian (“memory”) effects, and nondiffusive scaling. To overcome the limitations of fractional models in unbounded domains, we use regularized FDs that allow the incorporation of finite-size domain effects, boundary conditions, and variable diffusivities. We present an α-weighted explicit/implicit numerical integration scheme based on the Grunwald-Letnikov representation of the regularized fractional diffusion operator in flux conserving form. In sharp contrast with the standard diffusive model, the strong nonlocality of fractional diffusion leads to a linear in time response for a decaying pulse at short times. In addition, an anomalous fractional pinch is observed, accompanied by the development of an uphill transport region where the “effective” diffusivity becomes negative. The fractional flux is in general asymmetric and, for steady states, it has a negative (toward the core) component that enhances confinement and a positive component that increases toward the edge and leads to poor confinement. The model exhibits the characteristic anomalous scaling of the confinement time, τ, with the system’s size, L, τ∼Lα, of low-confinement mode plasma where 1<α<2 is the order of the FD operator. Numerical solutions of the model with an off-axis source show that the fractional inward transport gives rise to profile peaking reminiscent of what is observed in tokamak discharges with auxiliary off-axis heating. Also, cold-pulse perturbations to steady sates in the model exhibit fast, nondiffusive propagation phenomena that resemble perturbative experiments.

113 citations

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TL;DR: In this paper, a spectral type approximation method for the solution of the steady-state fractional diffusion equation is proposed and studied, where the Jacobi polynomials are pseudo eigenfunctions for the diffusion operator.

Abstract: In this article we investigate the solution of the steady-state fractional diffusion equation on a bounded domain in $\real^{1}$. From an analysis of the underlying model problem, we postulate that the fractional diffusion operator in the modeling equations is neither the Riemann-Liouville nor the Caputo fractional differential operators. We then find a closed form expression for the kernel of the fractional diffusion operator which, in most cases, determines the regularity of the solution. Next we establish that the Jacobi polynomials are pseudo eigenfunctions for the fractional diffusion operator. A spectral type approximation method for the solution of the steady-state fractional diffusion equation is then proposed and studied.

98 citations