# 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.

Abstract: An initial–boundary value problem with a Riemann–Liouville-Caputo space fractional derivative of order α ∈ ( 1 , 2 ) is considered, where the boundary conditions are reflecting. 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. The solution of the general problem is approximated using a finite difference scheme defined on a uniform mesh and the error analysis is given in detail for typical solutions which have a weak singularity near the spatial boundary x = 0 . It is proved that the scheme converges with first order in the maximum norm. Numerical results are given that corroborate our theoretical results for the order of convergence of the difference scheme, the approach of the solution to steady state, and mass conservation.

## Summary (1 min read)

### 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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