Variational formulation of problems involving fractional order differential operators

Bangti Jin; Raytcho Lazarov; Joseph Pasciak

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Variational formulation of problems involving fractional order differential operators

Bangti Jin, Raytcho Lazarov, Joseph Pasciak

arXiv:1307.4795v1Jul 17, 2013math.NAmath.AP

TL;DR

The paper addresses variational formulations and numerical approximation for boundary-value problems with Caputo and Riemann-Liouville derivatives of order α∈(1,2). It establishes stability, regularity pickup, and finite-element error estimates, with numerical results broadly supporting the H^(α/2)-norm estimates while exposing remaining L2 limitations.

Problem
Analysis of these nonlocal, nonsymmetric fractional boundary-value problems is limited, particularly for Caputo formulations and the regularity needed for numerical error analysis.
Method
The paper develops variational formulations, shift theorems, finite-element discretizations, and error estimates for both Riemann-Liouville and Caputo derivatives.
Results
The formulations are stable and their regularity supports finite-element convergence rates; numerical experiments support H^(α/2)(0,1)-norm estimates, while L2(0,1) estimates are one-half order below empirical rates.
Takeaways & Limitations
Regularity pickup enables convergence analysis, but the Riemann-Liouville singularity and limited adjoint regularity constrain some approximation rates.
Takeaways & Limitations
In the Riemann-Liouville case, the singular term x^(α−1) generally limits solution regularity, while L2 error estimates remain one-half order below empirical convergence rates.

Abstract

from arXiv · show

In this work, we consider boundary value problems involving Caputo and Riemann-Liouville fractional derivatives of order $α\in(1,2)$ on the unit interval $(0,1)$. These fractional derivatives lead to non-symmetric boundary value problems, which are investigated from a variational point of view. The variational problem for the Riemann-Liouville case is coercive on the space $H_0^{α/2}(0,1)$ but the solutions are less regular, whereas that for the Caputo case involves different test and trial spaces. The numerical analysis of these problems requires the so-called shift theorems which show that the solutions of the variational problem are more regular. The regularity pickup enables one to establish convergence rates of the finite element approximations. Finally, numerical results are presented to illustrate the error estimates.

1. Introduction

The paper studies fractional differential boundary-value problems motivated by anomalous diffusion, focusing on variational formulations for Riemann-Liouville and Caputo derivatives and their finite-element analysis. It addresses limited theory by establishing stability, regularity pickup, error estimates, and numerical confirmation.

Anomalous diffusion motivates the model because particle motion can involve dependent, heavy-tailed jumps and depart from Gaussian behavior.
Theoretical analysis remains scarce because fractional differential operators are nonlocal and introduce difficulties absent from classical second-order elliptic equations.
The work revisits variational formulations for both Riemann-Liouville and Caputo derivatives and establishes variational stability, shift theorems, finite-element analysis, and error estimates.
The Riemann-Liouville formulation is nonsymmetric but coercive on H^(α/2), while the Caputo formulation uses a different test space with a nonlocal integral constraint.
Regularity pickup is essential for proving optimal finite-element convergence rates, and numerical results are presented to confirm the error estimates.

2. Fractional differential operators on fractional Sobolev spaces

This section develops the fractional Sobolev-space framework needed to interpret Caputo and Riemann-Liouville operators variationally. It extends operators from smooth functions, identifies boundary and integrability constraints, and highlights limitations of these extensions.

Fractional derivatives and integrals are studied as operators on fractional Sobolev spaces to support regularity analysis and finite-element convergence estimates.
The space eH^β(0,1) consists of functions in H^β(0,1) whose zero extension to R belongs to H^β(R).
Left- and right-sided fractional operators are extended continuously from smooth functions to bounded maps involving fractional Sobolev spaces and L2(0,1).
For α ∈(3/2,2), solutions are generally not in eH^α(0,1) because that space imposes additional endpoint derivative conditions.
The continuous extension of the Caputo derivative need not coincide with its formal definition, even for functions where the formal definition is meaningful.

3. Strong solutions of fractional order equations

The section constructs strong solutions using fractional integral operators and their smoothing properties. It treats Riemann-Liouville and Caputo cases separately and identifies an unresolved low-regularity regime for the Caputo representation.

Fractional integral operators provide a smoothing property used to construct strong solutions from source terms in fractional Sobolev spaces.
The construction extends source terms with vanishing moments, enabling bounded fractional integral operators to produce functions with the required Sobolev regularity.
For the Riemann-Liouville case with q = 0, setting g = 0I^α f yields a solution because the fractional derivative of g recovers f and the boundary conditions are satisfied.
For the Caputo case with q = 0, a related integral construction gives a solution after choosing β so that α + β ∈(3/2,2).
The Caputo solution representation remains unclear for low-regularity sources f ∈eH^β(0,1) when α + β ≤3/2.

4. Variational formulations of the fractional derivative problems

The paper develops and analyzes variational formulations for Riemann–Liouville and Caputo fractional derivative problems, establishing stability, regularity pickup, and well-posedness results. The Caputo formulation requires distinct trial and test spaces, while the Riemann–Liouville formulation uses a coercive bilinear form on a fractional Sobolev space.

4. Variational formulations of the fractional derivative problems: The section derives variational formulations for both Riemann–Liouville and Caputo fractional derivative problems and establishes regularity pickup needed for later error estimates.The Caputo derivation includes an explicit solution representation and the Riemann–Liouville derivation uses fractional integration identities.
4.1. Derivation of the variational formulations.: The Riemann–Liouville bilinear form is bounded and coercive on eHα/2(0, 1), yielding a variational formulation with the solution and test space identified.The formulation is developed first for q = 0 and then extended to q ≠ 0 by adding the term (qu, v).
4.1. Derivation of the variational formulations.: The Caputo formulation avoids the troublesome boundary term by restricting test functions through the nonlocal condition (x1−α, v) = 0.This produces different trial and test spaces and a well-posed problem with a unique solution for every f ∈ L2(0, 1).
4.2. Variational stability in Riemann-Liouville case.: Under the stated uniqueness assumption, the Riemann–Liouville variational problem has a unique solution for every F ∈ U∗.The result follows from the bijectivity of the operator associated with the bilinear form.
4.2. Variational stability in Riemann-Liouville case.: For f ∈ L2(0, 1), the Riemann–Liouville variational solution is also a strong solution and belongs to eHα−1+β(0, 1) for β ∈ [0, 1/2).The same regularity bound applies to the adjoint solution under the corresponding variational problem.
4.2. Variational stability in Riemann-Liouville case.: The Riemann–Liouville solution generally cannot exceed eHα−1+β(0, 1) because of the singular term xα−1.Improved regularity is possible only under an additional cancellation condition.
4.3. Variational stability in Caputo case.: Under Assumption 4.2, the Caputo variational problem has a unique solution u ∈ U satisfying a(u, v) = ⟨F, v⟩ for all v ∈ V.The corresponding operator maps the trial space into the dual of the distinct test space.
4.3. Variational stability in Caputo case.: For sufficiently regular f and q, the Caputo variational solution is a solution of the original problem and belongs to eHα/2(0, 1) ∩ Hα+β(0, 1).With additional smoothness of q, the solution can achieve full regularity, unlike the generally singular Riemann–Liouville solution.

5. Stability of Finite Element Approximation

The paper develops stable finite element approximations for the Riemann–Liouville and Caputo variational formulations, with error estimates based on finite-element approximation and regularity pickup. The Caputo discretization requires distinct trial and test spaces, while adjoint regularity limits the L2 error rate.

Finite element analysis targets stability and error estimates for discrete variational formulations of the boundary value problem.The analysis covers both Riemann–Liouville and Caputo cases.
The method uses an equally spaced mesh with size h = 1/m and continuous piecewise-linear spaces Uh and Vh.Uh functions vanish at 0 and 1, while Vh functions vanish at 1 and satisfy the integral constraint (x1−α, vh) = 0.
For u ∈ Hγ(0, 1) ∩ eHα/2(0, 1), both spaces achieve Hα/2 approximation error O(hγ−α/2) for α/2 ≤ γ ≤ 2.The estimates are obtained from interpolation properties and a bounded projection onto the constrained space V.
Riemann–Liouville case: The Riemann–Liouville discrete problem has a unique solution for sufficiently small h under the stated assumptions.For q ≠ 0, the proof uses a discrete inf-sup condition established through Schatz’s method, coercivity, an adjoint problem, and a kick-back argument.
Riemann–Liouville case: The Riemann–Liouville finite element approximation satisfies an error estimate for every β ∈ [0, 1/2), with existence and uniqueness established for h ≤ h0.The L2 estimate uses Nitsche’s trick and regularity of the adjoint problem.
Riemann–Liouville case: The Hα/2 error estimate is optimal, whereas the L2 estimate is suboptimal because the adjoint solution has limited regularity.Numerical experiments are reported to confirm this distinction.
Caputo case: For the Caputo case, discrete stability requires a separate inf-sup analysis on Uh × Vh and yields a unique solution for sufficiently small h.The construction uses constrained test functions, interpolation, and a correction enforcing the integral condition.

6. Numerical experiments and discussions

Numerical experiments assess finite element error estimates for Caputo and Riemann-Liouville problems under smooth, nonsmooth, and singular source terms. The results confirm the H^α/2 estimates, while L2 convergence is empirically one-half order better than predicted and depends on solution regularity.

Experimental setup: The experiments use three source terms with differing regularity, uniform meshes, and α values 7/4, 3/2, and 4/3.The reported errors use L2(0,1)- and H^α/2(0,1)-norms.
Example (a): For smooth data, Riemann-Liouville solutions retain an x^α−1 singularity, producing slower H^α/2 convergence than in the Caputo case.The predicted rates are O(h^α/2−1/2) and O(h^2α−1) for Riemann-Liouville, versus O(h^2−α/2) and O(h^3/2) for Caputo in the H^α/2 and L2 norms.
Example (a): The H^α/2 error estimates are fully confirmed, whereas observed L2 convergence is one-half order higher than the theoretical prediction for both derivatives.The higher L2 rate is reported for the smooth-source experiments and agrees with the conjecture associated with the adjoint representation.
Example (b): With f(x)=1, Riemann-Liouville convergence remains limited by solution regularity, while Caputo matches the smooth-source rates for α≥3/2 but slows for α=4/3.For both derivatives, empirical L2 convergence again exceeds the theoretical prediction by one-half order.
Example (c): For the singular source f(x)=x−1/4, Riemann-Liouville rates remain similar to earlier cases, while Caputo convergence slows as α approaches 1.The Caputo observations remain consistent with the theoretical prediction despite the lower source regularity.

7. Conclusions

The work develops and analyzes variational formulations and finite element discretizations for boundary value problems with Riemann–Liouville and Caputo fractional derivatives.

Variational formulations are developed for Riemann–Liouville and Caputo fractional derivatives of order α ∈ (1, 2).
The formulations’ stability and the Sobolev regularity of their variational solutions are established.
Finite element discretizations are developed with convergence rates established in eHα/2(0, 1)- and L2(0, 1)-norms.
The eHα/2(0, 1)-error estimates are fully supported numerically, while the L2(0, 1)-estimates remain one-half order below empirical rates.The authors identify the L2 estimates as requiring further investigation.
Uniform-mesh computations show pronounced spurious oscillations near the origin in the Riemann–Liouville case, especially when α is close to unity.The solution’s inherent xα−1 singularity motivates adaptive mesh refinement or locally enriched solution spaces.