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A generalized definition of Caputo derivatives and its application to fractional ODEs

Lei Li, Jian-Guo Liu

arXiv:1612.05103v5math.CAmath.AP

TL;DR

Common Riemann–Liouville and Caputo derivatives do not have the semigroup property. The paper builds a convolution-group fractional calculus and uses it to generalize Caputo derivatives; it proves broad fractional ODE results and shows that a fractional Hamiltonian system’s energy can decay to zero.

  • Problem

    Common Riemann–Liouville and Caputo fractional derivatives do not have the semigroup property, limiting a group-based calculus.

    Source: §1 · p. 2

  • Method

    The paper constructs a modified Riemann–Liouville calculus from a convolution group and uses it to define generalized Caputo derivatives.

    Source: §1 · p. 2

  • Results

    The framework establishes fractional ODE existence and uniqueness and comparison principles under very weak conditions, and the fractional Hamiltonian system’s energy can decay to zero.

    Source: §1 · pp. 4–5

  • Takeaways & Limitations

    The comparison principle provides essential tools for a priori energy estimates of fractional PDEs.

    Source: §1 · p. 4

    Because time-fractional ODEs are non-Markovian, the standard ODE continuation technique cannot be applied.

    Source: §4 · p. 27

Abstract

from arXiv · show

We extend in this paper the definition of Caputo derivatives of order in $(0,1)$ to a certain class of locally integrable functions using a convolution group. Our strategy is to define a fractional calculus for a certain class of distributions using the convolution group. When acting on causal functions, this fractional calculus agrees with the traditional Riemann-Liouville definition for $t>0$ but includes some singularities at $t=0$ so that the group property holds. Then, making use of this fractional calculus, we introduce the generalized definition of Caputo derivatives. The new definition is consistent with various definitions in literature while reveals the underlying group structure. Since the new definition is valid for a class of locally integrable functions that can blow up in finite time, it provides a framework for solutions to fractional ODEs and fractional PDEs with very weak conditions. The underlying group property makes many properties of Caputo derivatives natural. In particular, it allows us to de-convolve the fractional differential equations to integral equations with completely monotone kernels, which then enables us to prove the general Gr\"onwall inequality (or comparison principle) with the most general conditions. This then provides the essential tools for {\it a priori} energy estimates of fractional PDEs. Some other fundamental results for fractional ODEs are also established within this frame under very weak conditions.

1. Introduction.

The paper uses a convolution group to extend Caputo derivatives to functions with weak regularity, revealing a group structure and supporting fractional ODE and PDE analysis. It also establishes general ODE results and illustrates the framework with a fractional Hamiltonian system whose energy can decay to zero.

Introduction.

  • The modified Riemann–Liouville calculus agrees with the traditional Riemann–Liouville calculus for t > 0 and adds singularities at t = 0 to obtain a group.The authors build it using a convolution group for a class of distributions.Source: §1 · p. 2
  • Traditional Caputo definitions require a first derivative, while an alternative based on integration by parts can have a divergent integral when the function lacks Hölder continuity of order γ.These limitations motivate extensions for broader classes of functions.Source: §1 · pp. 2–3
  • The generalized Caputo definition removes an initial-value singular term and applies to functions that can blow up in finite time.It covers the traditional definition and the alternative in, while its group structure makes derivative properties natural.Source: §1 · pp. 3–4
  • Under very weak conditions, the framework establishes fractional ODE existence and uniqueness and comparison principles, supporting a priori energy estimates for fractional PDEs.The fundamental theorem transforms differential equations of order in (0, 1) into integral equations with completely monotone kernels when the function is right continuous at t = 0.Source: §1 · p. 4
  • The fractional Hamiltonian system does not preserve energy, which can decay to zero.The paper presents this as an example of a fractional ODE.Source: §1 · p. 5

2. Time-continuous groups and fractional calculus.

The section develops fractional calculus from a time-continuous convolution group, defining distributional and modified Riemann–Liouville operators that agree with traditional Riemann–Liouville calculus for causal functions when t > 0, while allowing singularities at t = 0. It also establishes distributional continuity and Sobolev regularity properties of the modified operators, including order-dependent regularity changes under specified exponent conditions.

2.1. A time-continuous convolution group.

  • The distributions gα form a convolution group, with g0 = δ and negative orders defined using distributional derivatives.For n < γ < n + 1, g−γ is defined as D^nδ ∗ g_(n−γ).Source: §2.1 · p. 6

2.2. Time-continuous fractional calculus.

  • The fractional derivatives for distributions supported in [0, ∞) are generalized to distributions in E.Source: §2.2.1 · p. 7

2.2.1. Fractional calculus for distributions.

  • The operators Iα form a group on E, and on finite intervals the calculus is defined by extending distributions into E and pulling the result back.The space E_T is not closed under convolution, motivating this extension-and-restriction construction.Source: §2.2.1 · p. 7

2.2.2. Modified Riemann–Liouville calculus.

  • The modified Riemann–Liouville operators form a group on causal distributions and agree with traditional Riemann–Liouville calculus for t > 0, while adding singularities at t = 0.For example, J−1 includes an atom ϕ(0)δ(t) at the origin.Source: §2.2.2 · p. 10

2.3. Another group for right derivatives.

  • A second convolution group generates right Riemann–Liouville derivatives that depend on future values and are therefore noncausal.Its action is defined for distributions supported on (−∞, M] or functions decaying faster than rational functions at infinity.Source: §2.3 · p. 11

2.4. Regularities of the modified Riemann–Liouville operators.

  • To study regularity while avoiding singularities at t = 0 without requiring much at t = T, the authors introduce the space ˜Hs(0, T).Source: §2.4 · p. 11
  • If vn → f in Hs(0, T) for s ≥ 0, then Jαvn → Jαf in D′(0, T) for every α ∈ R.Source: §2.4 · p. 12
  • When min{s, s + α} ≥ 0, Jα is bounded from H̃s(0, T) to H̃s+α(0, T).The bound has a constant depending on T, s, and α.Source: §2.4 · p. 12
  • For v ∈ H̃0(0, T) = L2(0, T), Jγv(0) is well defined and zero when γ > 1/2; H̃γ(0, T) embeds into C0[0, T] in this range [17].Requiring v ∈ H̃s(0, T) removes singularities at t = 0, yields the stated regularity estimates, and makes Caputo and Riemann–Liouville derivatives coincide.Source: §2.4 · p. 14

3. An extension of Caputo derivatives.

This section develops a generalized Caputo derivative for locally integrable functions, modifying the Riemann–Liouville operator to address singularities at the origin. It relates the definition to a published one under additional Hölder regularity and establishes a fractional-calculus fundamental theorem, a Laplace-transform representation, and further results under weak conditions.

3. An extension of Caputo derivatives.

  • The generalized derivative removes the jump-induced singularity and, with ϕ0=ϕ(0+), agrees with the traditional Caputo derivative under the stated local regularity condition.It is defined using modified Riemann–Liouville operators; the definition also allows derivatives based at any time t=a.Source: §3 · p. 16
  • The convolution-group fundamental theorem deconvolves fractional differential equations into integral equations with completely monotone, nonnegative kernels, enabling a comparison principle for a priori PDE estimates.The section presents this theorem as a tool for fractional ODEs and PDEs.Source: §3 · pp. 16, 18
  • If ϕ≥0, ϕ0=0, and its generalized Caputo derivative is locally integrable and nonpositive, then ϕ(t)=0.The local-integrability condition can be dropped when the inequality is understood distributionally.Source: §3 · p. 18
  • For 0<γ<1, Dγ_cϕ=f has a solution ϕ∈X exactly when f∈Yγ; then its solutions are Jγf+C and belong to Xγ.The passages also characterize Xγ through this solution form.Source: §3 · p. 19
  • If f∈L1[0,T)∩H̃s(0,T) and s≥0 for γ>1/2, or s>1/2−γ for γ≤1/2, there exists ϕ∈C0[0,T) solving Dγ_cϕ=f.For T=∞, the passage additionally allows requiring f∈L1_loc[0,∞).Source: §3 · p. 19
  • For T=∞ and past-supported mollifiers, the derivative is compatible with mollification, and the section establishes an inequality for C1 convex functions.The passage also gives a further inequality when ϕ is continuous at zero and C1 for positive times.Source: §3 · p. 20
  • The definition recovers existing Caputo behavior without changing derivatives on (0, T1], and obtaining the stated mollification limit requires mollifiers supported in negative times.This choice reflects that Caputo derivatives model memory from t = 0+ and remove Riemann–Liouville singularities at t = 0.Source: §3 · pp. 20–21
  • Under additional Hölder regularity, the generalized Caputo derivative recovers definition (4) used by Allen, Caffarelli and Vasseur.The stated condition is ϕ ∈ C^{γ+δ}[0,T) for some δ > 0.Source: §3 · p. 21
  • For ϕ ∈ E(L), Proposition 3.13 gives a Laplace-transform representation of its generalized Caputo derivative for Re(s) > L.The passage defines the transform for the distributional derivative and states that it agrees with the usual transform when the latter exists.Source: §3 · pp. 21–22
  • Caputo derivatives lack the group property, but remove the singularities at t = 0 found in Riemann–Liouville derivatives and resemble ordinary derivatives, making them suitable for initial value problems.Source: §3 · p. 22
  • The following section establishes fractional ODE results whose assumptions are described as sufficiently weak and conclusions as general compared with.Source: §3 · p. 22

4. Time fractional ordinary differential equations.

This section develops existence, uniqueness, comparison, monotonicity, and energy results for time-fractional ODEs in the generalized Caputo framework, characterizing strong solutions by an integral equation and examining local and global behavior under continuity and Lipschitz conditions. It also gives a global-existence criterion and a fractional Hamiltonian example with convergence proved for orders up to one half; convergence for larger orders remains unresolved.

Time fractional ordinary differential equations

  • A strong solution is equivalent to a function in X with the prescribed initial value that solves the associated integral equation.Strong solutions require the fractional derivative and the right-hand side to be locally integrable, so the differential equation holds almost everywhere.Source: §4 · p. 23
  • For a linear fractional ODE with continuous forcing of suitably subexponential growth, there is a unique strong solution in E(L); the formula also yields a solution in X for any continuous forcing.Uniqueness in E(L) follows from uniqueness of the Laplace transform; the solution formula uses an integrable kernel that may be singular at t = 0.Source: §4 · pp. 23–24
  • If f is continuous and locally Lipschitz in its second variable near the initial value, the IVP has a unique local strong solution that is continuous with respect to the initial value.The solution exists on the interval specified by the theorem and is continuous there.Source: §4 · p. 24
  • If f is continuous and locally Lipschitz in its second variable on every bounded time interval and across all real values, the unique continuous solution exists globally.The result follows by applying the local existence theorem on arbitrary finite intervals.Source: §4 · p. 26
  • For locally Lipschitz f on a domain (α, β), there is a unique solution on a maximal interval [0, Tb) that stays within that domain.The standard ODE continuation argument cannot be used directly: fractional ODEs are non-Markovian, so concatenating solutions at an endpoint need not produce a solution.Source: §4 · pp. 26–27
  • A locally integrable function that is a nonpositive distribution is nonpositive almost everywhere, supporting order arguments for fractional ODEs.The section also gives an order result for Jγ applied to ordered distributions when both outputs are locally integrable.Source: §4 · p. 27
  • For continuous f that is locally Lipschitz and nondecreasing in its second variable, distributional subsolutions and solutions preserve order on their common interval of existence.The theorem also states the corresponding reversed-order conclusion for the reversed inequality.Source: §4 · pp. 28–29

4. Time fractional ordinary differential equations.

  • Under the stated nonnegativity and regularity conditions, the continuous solution of D^γ_c v = f(v) is nondecreasing until its blowup time.Proposition 4.11 assumes f is nondecreasing and locally Lipschitz, with f(0) ≥ 0, and v(0) ≥ 0.Source: §4 · p. 29
  • For convex energies with locally Lipschitz gradients, Proposition 4.12 establishes energy estimates for fractional gradient flows and a corresponding system with an anti-Hermitian operator.The passage does not include the displayed estimate itself, so its exact inequality is not stated here.Source: §4 · p. 30
  • If E(v) tends to infinity as |v| tends to infinity, the solutions covered by Proposition 4.12 exist globally, so T_b = ∞.This is Corollary 4.13 under Proposition 4.12's conditions.Source: §4 · p. 31

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