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Fractional-order derivatives defined by continuous kernels are too restrictive
Accepted Manuscript Fractional-order derivatives defined by continuous kernels are too restrictive Martin Stynes
PII: DOI: Reference:
S0893-9659(18)30155-1 https://doi.org/10.1016/j.aml.2018.05.013 AML 5520
To appear in:
Applied Mathematics Letters
Received date : 15 May 2018 Accepted date : 15 May 2018 Please cite this article as: M. Stynes, Fractional-order derivatives defined by continuous kernels are too restrictive, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.05.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
Fractional-order derivatives defined by continuous kernels are too restrictive Martin Stynesa,1 a
Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Haidian District, Beijing 100193, China
Abstract Various new definitions of fractional-order derivatives seek to replace the singular kernel of the Caputo definition by some continuous function. It is shown here that the use of any continuous kernel often places a severe and unnatural restriction on the data that can be used in problems formulated using these new definitions. Keywords: fractional derivative, continuous kernel, Caputo-Fabrizio derivative, Atangana-Baleanu derivative 2010 MSC: 35R11, 35B65 1. Introduction and summary Fractional-order derivatives appear ever more frequently in the modelling of physical processes—see for example [1]. There is much debate about which variant of fractional derivative to incorporate into each model. The traditional choices of Caputo or Rieman-Liouville derivatives [2] have integral operators whose kernels are singular, i.e., they blow up at one limit of integration in the definition of the integral. These singularities can cause difficulties in analysing (and solving numerically) fractional-derivative problems. Consequently various authors have proposed alternative definitions of fractional derivatives with integral operators whose kernels are continuous over the domain of integration, in order to mitigate the analysis of fractional-derivative ∗
Corresponding author. The research for this paper was supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401. Email address: [email protected] (Martin Stynes)
Preprint submitted to Applied Mathematics Letters
May 15, 2018
problems. Two recent examples that have attracted much attention are the Caputo-Fabrizio derivative [3] and the Atangana-Baleanu derivative [4]. Various advantages and disadvantages of these alternative definitions have been discussed by several authors [4, 3, 5, 6]. This discussion will not be presented here; it is our aim, in this short paper, only to point out one significant shortcoming of all continuous-kernel definitions that has not previously been reported in the research literature. Thus, we shall prove that in timefractional initial-boundary value problems—a class of fractional derivative problem that is used widely in modelling—the use of a continuous-kernel fractional derivative often imposes implicitly a severe and unnatural constraint on the data of the problem. 2. Definitions of derivatives, the initial-boundary value problem ¯ and ∂Ω Let Ω be a bounded domain in Rn for some n ≥ 1. Let Ω denote its closure and boundary. Let α ∈ (0, 1). Given a suitable function g(x, t) defined on Ω × [0, T ] for some T > 0, the Caputo fractional temporal derivative Dtα of order α is defined [2] by Z t ∂g(x, s) 1 α (t − s)−α ds for x ∈ Ω, 0 < t ≤ T. Dt g(x, t) := Γ(1 − α) s=0 ∂s
Observe that in this definition the kernel (t − s)−α will blow up as s approaches the upper limit of integration t, and this singularity can be troublesome when analysing this derivative. Thus various alternative definitions have been proposed (e.g., [4, 3]) that take the form Z t ∂g(x, s) α ˜ Dt g(x, t) := C(α) K(t, s) ds for x ∈ Ω, 0 < t ≤ T, (1) ∂s s=0
where C(α) is a positive normalisation constant and the kernel K(s, t) is ¯ × [0, T ]. continuous on Ω In this paper, we shall consider the initial-boundary value problem ˆ αu − D t
n X
n
X ∂ 2u ∂u pij (x, t) + qi (x, t) + r(x, t)u = f (x, t) ∂xi ∂xj ∂xi i,j=1 i=1
(2a)
for (x, t) ∈ Q := Ω × (0, T ], with u(x, t) = ψ(x, t) for (x, t) ∈ ∂Ω × (0, T ], ¯ u(x, 0) = φ(x) for x ∈ Ω, 2
(2b) (2c)
ˆ α is some fractional time derivative of order α, the operator where D t w 7→
n X
pij (x, t)∂ 2 w/∂xi ∂xj +
i,j=1
n X
qi (x, t)∂w/∂xi + r(x, t)w
i=1
is uniformly elliptic on Q, and the functions pij , qi , r, ψ and φ are continuous on the closures of their domains. We assume the minimal amount of compatibility between the initial and boundary data so that any solution u of (2) is ¯ := Ω ¯ × [0, T ]: continuous on Q φ(x) = ψ(x, 0) for all x ∈ ∂Ω.
(3)
ˆ tα a Caputo or a Riemann-Liouville Problems of the form (2), with D derivative, have been considered in a huge number of papers in the research ˆ α is defined literature. Our interest here is in the variant of (2) when D t using a continuous kernel as in (1), since this formulation of (2a) has begun recently to attract the attention of researchers. Remark 1. For brevity we consider only the case 0 < α < 1, so the initialboundary value problem (2) resembles a classical parabolic problem such as the heat equation, but our results can easily be extended to the case 1 < α < 2, when the problem is akin to a classical second-order hyperbolic problem (e.g., the wave equation); cf. [7]. Furthermore, one can prove analogous results for two-point boundary value problems containing a continuous-kernel fractional derivative; cf. [8]. 3. Consequence of using a continuous-kernel fractional derivative Our analysis is based on the following elementary result, which is inspired by [2, Lemma 3.11]. ˜ α g(x, t) that Lemma 1. Consider a continuous-kernel fractional derivative D t is defined as in (1). Let x ∈ Ω be fixed. Assume that the function ∂g(x, t)/∂t lies in the Lebesgue space L1 [0, T ]. Then ˜ α g(x, t) = 0. lim+ D t
t→0
3
Proof. For any t ∈ (0, T ], Z t ∂g(x, s) ˜α K(t, s) ds Dt g(x, t) = C(α) ∂s s=0 Z t ≤ C(α) max |K(s, t)| ¯ (s,t)∈Ω×[0,T ]
∂g(x, s) ∂s ds. s=0
But ∂g(x, t)/∂t ∈ L1 [0, T ] implies that Z t ∂g(x, s) ds = 0 lim t→0+ s=0 ∂s
by a well-known result for Lebesgue-integrable functions (see, e.g., [9, p.300, Theorem 6]) and the result follows.
It is clear from its proof that Lemma 1 remains valid if the hypothesis on the kernel K is weakened to sup(s,t)∈Ω×[0,T ¯ ] |K(s, t)| < ∞. 2 For functions w ∈ C (Ω), define the differential operator L0 by L0 w(x) := −
n X
i,j=1
pij (x, 0)
n ∂ 2 w(x) X ∂w(x) + qi (x, 0) +r(x, 0)w(x) for x ∈ Ω. ∂xi ∂xj i=1 ∂xi
Our main result can now be derived. Theorem 1. Let u(x, t) be a solution of the initial-boundary value problem ˜ tα u is used in (2a). (2), where a continuous-kernel fractional derivative D Suppose that for each x ∈ Ω, the function ∂u(x, t)/∂t lies in the Lebesgue space L1 [0, T ]. Then the initial data φ(x) = u(x, 0) must satisfy the equation L0 φ = f on Ω. Proof. Consider the limit of equation (2a) as t → 0+ . Lemma 1 shows that limt→0+ Dtα u(x, t) = 0 for each x ∈ Ω. Hence, taking the limit, we get L0 u(x, 0) = f (x, 0) for each x ∈ Ω. The mild assumption that the solution u(x, ·) lies in L1 [0, T ] for each x is satisfied in every application of continuous-kernel fractional derivatives that we have seen. Theorem 1 then shows that the use of a continuouskernel fractional derivative imposes an extra implicit condition on the initial data φ. The next example demonstrates dramatically the effect of this extra condition. 4
Example 1. Consider the fractional heat equation ˜ α v − ∂ 2 v/∂x2 = 0 for (x, t) ∈ (0, 1) × (0, T ], D t ˜ α is a continuous-kernel fractional derivative, the boundary data are where D t v(0, t) = v(π, t) = 0, with the initial data v(x, 0) = φ(x), where φ(x) ∈ C 2 [0, 1] is unspecified except that it satisfies the compatibility condition (3). Assume that for each x, the solution v(x, ·) of this problem lies in L1 [0, T ]. Then Theorem 1 and (3) show that φ must satisfy the conditions −φ00 (x) = 0 on (0, 1),
φ(0) = φ(1) = 0.
But these conditions imply that φ ≡ 0. As all the data of this example are now zero, we get v ≡ 0. Thus, using a continuous-kernel fractional derivative forces v ≡ 0; the apparent freedom of choice that one has for φ is illusory. We now generalise Example 1. ¯ the boundary value problem Assumption 1. Assume that for each f ∈ C(Ω), L0 w(x) = f (x, 0) for x ∈ Ω,
w(x) = ψ(x, 0) for x ∈ ∂Ω,
(4)
has at most one solution w(x). For example, Assumption 1 is satisfied if r(x, 0) ≥ 0 for x ∈ Ω, because then L0 satisfies a maximum principle [10, p.72]. More general boundary data than the Dirichlet data of (4) can be handled by invoking [10, p.70, Theorem 9]. Corollary 1. Assume the hypotheses of Theorem 1 and let Assumption 1 be satisfied. Then the initial value function φ of (2c) is determined uniquely by L0 , f and ψ. Proof. By Theorem 1 we have L0 φ(x) = f (x, 0) for all x ∈ Ω, and (3) says that φ(x) = ψ(x, 0) for x ∈ ∂Ω. The result is now immediate from Assumption 1. The conclusion of Corollary 1, with its severe restriction on the choice of φ, is clearly unnatural—the initial data (2c) should not be determined uniquely by the other data of the problem. This unpleasant conclusion has been forced on us by the use of a continuous-kernel fractional derivative in (2a). 5
4. Bibliography [1] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus, Vol. 5 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, models and numerical methods, Second edition. [2] K. Diethelm, The analysis of fractional differential equations, Vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, an application-oriented exposition using differential operators of Caputo type. [3] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2) (2015) 73–85. [4] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, Thermal Science 20 (2) (2016) 763–769. [5] M. D. Ortigueira, J. A. T. Machado, A critical analysis of the caputo-fabrizio operator, Commun. Nonlinear Sci. Numer. Simul.doi:10.1016/j.cnsns.2017.12.001. [6] V. E. Tarasov, No nonlocality. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 62 (2018) 157–163. doi:10.1016/j.cnsns.2018.02.019. [7] M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal. 19 (6) (2016) 1554–1562. doi:10.1515/fca-2016-0080. [8] M. Stynes, A Caputo two-point boundary value problem: existence, uniqueness and regularity of a solution, Model. Anal. Inf. Sist. 23 (3) (2016) 370–376. doi:10.18255/1818-1015-2016-3-370-376. [9] A. N. Kolmogorov, S. V. Fom¯ı n, Introductory real analysis, Dover Publications, Inc., New York, 1975, translated from the second Russian edition and edited by Richard A. Silverman, Corrected reprinting. [10] M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984, corrected reprint of the 1967 original. doi:10.1007/978-1-4612-5282-5.
6
PII: DOI: Reference:
S0893-9659(18)30155-1 https://doi.org/10.1016/j.aml.2018.05.013 AML 5520
To appear in:
Applied Mathematics Letters
Received date : 15 May 2018 Accepted date : 15 May 2018 Please cite this article as: M. Stynes, Fractional-order derivatives defined by continuous kernels are too restrictive, Appl. Math. Lett. (2018), https://doi.org/10.1016/j.aml.2018.05.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Manuscript Click here to view linked References
Fractional-order derivatives defined by continuous kernels are too restrictive Martin Stynesa,1 a
Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Haidian District, Beijing 100193, China
Abstract Various new definitions of fractional-order derivatives seek to replace the singular kernel of the Caputo definition by some continuous function. It is shown here that the use of any continuous kernel often places a severe and unnatural restriction on the data that can be used in problems formulated using these new definitions. Keywords: fractional derivative, continuous kernel, Caputo-Fabrizio derivative, Atangana-Baleanu derivative 2010 MSC: 35R11, 35B65 1. Introduction and summary Fractional-order derivatives appear ever more frequently in the modelling of physical processes—see for example [1]. There is much debate about which variant of fractional derivative to incorporate into each model. The traditional choices of Caputo or Rieman-Liouville derivatives [2] have integral operators whose kernels are singular, i.e., they blow up at one limit of integration in the definition of the integral. These singularities can cause difficulties in analysing (and solving numerically) fractional-derivative problems. Consequently various authors have proposed alternative definitions of fractional derivatives with integral operators whose kernels are continuous over the domain of integration, in order to mitigate the analysis of fractional-derivative ∗
Corresponding author. The research for this paper was supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401. Email address: [email protected] (Martin Stynes)
Preprint submitted to Applied Mathematics Letters
May 15, 2018
problems. Two recent examples that have attracted much attention are the Caputo-Fabrizio derivative [3] and the Atangana-Baleanu derivative [4]. Various advantages and disadvantages of these alternative definitions have been discussed by several authors [4, 3, 5, 6]. This discussion will not be presented here; it is our aim, in this short paper, only to point out one significant shortcoming of all continuous-kernel definitions that has not previously been reported in the research literature. Thus, we shall prove that in timefractional initial-boundary value problems—a class of fractional derivative problem that is used widely in modelling—the use of a continuous-kernel fractional derivative often imposes implicitly a severe and unnatural constraint on the data of the problem. 2. Definitions of derivatives, the initial-boundary value problem ¯ and ∂Ω Let Ω be a bounded domain in Rn for some n ≥ 1. Let Ω denote its closure and boundary. Let α ∈ (0, 1). Given a suitable function g(x, t) defined on Ω × [0, T ] for some T > 0, the Caputo fractional temporal derivative Dtα of order α is defined [2] by Z t ∂g(x, s) 1 α (t − s)−α ds for x ∈ Ω, 0 < t ≤ T. Dt g(x, t) := Γ(1 − α) s=0 ∂s
Observe that in this definition the kernel (t − s)−α will blow up as s approaches the upper limit of integration t, and this singularity can be troublesome when analysing this derivative. Thus various alternative definitions have been proposed (e.g., [4, 3]) that take the form Z t ∂g(x, s) α ˜ Dt g(x, t) := C(α) K(t, s) ds for x ∈ Ω, 0 < t ≤ T, (1) ∂s s=0
where C(α) is a positive normalisation constant and the kernel K(s, t) is ¯ × [0, T ]. continuous on Ω In this paper, we shall consider the initial-boundary value problem ˆ αu − D t
n X
n
X ∂ 2u ∂u pij (x, t) + qi (x, t) + r(x, t)u = f (x, t) ∂xi ∂xj ∂xi i,j=1 i=1
(2a)
for (x, t) ∈ Q := Ω × (0, T ], with u(x, t) = ψ(x, t) for (x, t) ∈ ∂Ω × (0, T ], ¯ u(x, 0) = φ(x) for x ∈ Ω, 2
(2b) (2c)
ˆ α is some fractional time derivative of order α, the operator where D t w 7→
n X
pij (x, t)∂ 2 w/∂xi ∂xj +
i,j=1
n X
qi (x, t)∂w/∂xi + r(x, t)w
i=1
is uniformly elliptic on Q, and the functions pij , qi , r, ψ and φ are continuous on the closures of their domains. We assume the minimal amount of compatibility between the initial and boundary data so that any solution u of (2) is ¯ := Ω ¯ × [0, T ]: continuous on Q φ(x) = ψ(x, 0) for all x ∈ ∂Ω.
(3)
ˆ tα a Caputo or a Riemann-Liouville Problems of the form (2), with D derivative, have been considered in a huge number of papers in the research ˆ α is defined literature. Our interest here is in the variant of (2) when D t using a continuous kernel as in (1), since this formulation of (2a) has begun recently to attract the attention of researchers. Remark 1. For brevity we consider only the case 0 < α < 1, so the initialboundary value problem (2) resembles a classical parabolic problem such as the heat equation, but our results can easily be extended to the case 1 < α < 2, when the problem is akin to a classical second-order hyperbolic problem (e.g., the wave equation); cf. [7]. Furthermore, one can prove analogous results for two-point boundary value problems containing a continuous-kernel fractional derivative; cf. [8]. 3. Consequence of using a continuous-kernel fractional derivative Our analysis is based on the following elementary result, which is inspired by [2, Lemma 3.11]. ˜ α g(x, t) that Lemma 1. Consider a continuous-kernel fractional derivative D t is defined as in (1). Let x ∈ Ω be fixed. Assume that the function ∂g(x, t)/∂t lies in the Lebesgue space L1 [0, T ]. Then ˜ α g(x, t) = 0. lim+ D t
t→0
3
Proof. For any t ∈ (0, T ], Z t ∂g(x, s) ˜α K(t, s) ds Dt g(x, t) = C(α) ∂s s=0 Z t ≤ C(α) max |K(s, t)| ¯ (s,t)∈Ω×[0,T ]
∂g(x, s) ∂s ds. s=0
But ∂g(x, t)/∂t ∈ L1 [0, T ] implies that Z t ∂g(x, s) ds = 0 lim t→0+ s=0 ∂s
by a well-known result for Lebesgue-integrable functions (see, e.g., [9, p.300, Theorem 6]) and the result follows.
It is clear from its proof that Lemma 1 remains valid if the hypothesis on the kernel K is weakened to sup(s,t)∈Ω×[0,T ¯ ] |K(s, t)| < ∞. 2 For functions w ∈ C (Ω), define the differential operator L0 by L0 w(x) := −
n X
i,j=1
pij (x, 0)
n ∂ 2 w(x) X ∂w(x) + qi (x, 0) +r(x, 0)w(x) for x ∈ Ω. ∂xi ∂xj i=1 ∂xi
Our main result can now be derived. Theorem 1. Let u(x, t) be a solution of the initial-boundary value problem ˜ tα u is used in (2a). (2), where a continuous-kernel fractional derivative D Suppose that for each x ∈ Ω, the function ∂u(x, t)/∂t lies in the Lebesgue space L1 [0, T ]. Then the initial data φ(x) = u(x, 0) must satisfy the equation L0 φ = f on Ω. Proof. Consider the limit of equation (2a) as t → 0+ . Lemma 1 shows that limt→0+ Dtα u(x, t) = 0 for each x ∈ Ω. Hence, taking the limit, we get L0 u(x, 0) = f (x, 0) for each x ∈ Ω. The mild assumption that the solution u(x, ·) lies in L1 [0, T ] for each x is satisfied in every application of continuous-kernel fractional derivatives that we have seen. Theorem 1 then shows that the use of a continuouskernel fractional derivative imposes an extra implicit condition on the initial data φ. The next example demonstrates dramatically the effect of this extra condition. 4
Example 1. Consider the fractional heat equation ˜ α v − ∂ 2 v/∂x2 = 0 for (x, t) ∈ (0, 1) × (0, T ], D t ˜ α is a continuous-kernel fractional derivative, the boundary data are where D t v(0, t) = v(π, t) = 0, with the initial data v(x, 0) = φ(x), where φ(x) ∈ C 2 [0, 1] is unspecified except that it satisfies the compatibility condition (3). Assume that for each x, the solution v(x, ·) of this problem lies in L1 [0, T ]. Then Theorem 1 and (3) show that φ must satisfy the conditions −φ00 (x) = 0 on (0, 1),
φ(0) = φ(1) = 0.
But these conditions imply that φ ≡ 0. As all the data of this example are now zero, we get v ≡ 0. Thus, using a continuous-kernel fractional derivative forces v ≡ 0; the apparent freedom of choice that one has for φ is illusory. We now generalise Example 1. ¯ the boundary value problem Assumption 1. Assume that for each f ∈ C(Ω), L0 w(x) = f (x, 0) for x ∈ Ω,
w(x) = ψ(x, 0) for x ∈ ∂Ω,
(4)
has at most one solution w(x). For example, Assumption 1 is satisfied if r(x, 0) ≥ 0 for x ∈ Ω, because then L0 satisfies a maximum principle [10, p.72]. More general boundary data than the Dirichlet data of (4) can be handled by invoking [10, p.70, Theorem 9]. Corollary 1. Assume the hypotheses of Theorem 1 and let Assumption 1 be satisfied. Then the initial value function φ of (2c) is determined uniquely by L0 , f and ψ. Proof. By Theorem 1 we have L0 φ(x) = f (x, 0) for all x ∈ Ω, and (3) says that φ(x) = ψ(x, 0) for x ∈ ∂Ω. The result is now immediate from Assumption 1. The conclusion of Corollary 1, with its severe restriction on the choice of φ, is clearly unnatural—the initial data (2c) should not be determined uniquely by the other data of the problem. This unpleasant conclusion has been forced on us by the use of a continuous-kernel fractional derivative in (2a). 5
4. Bibliography [1] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus, Vol. 5 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017, models and numerical methods, Second edition. [2] K. Diethelm, The analysis of fractional differential equations, Vol. 2004 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010, an application-oriented exposition using differential operators of Caputo type. [3] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl. 1 (2) (2015) 73–85. [4] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel, Thermal Science 20 (2) (2016) 763–769. [5] M. D. Ortigueira, J. A. T. Machado, A critical analysis of the caputo-fabrizio operator, Commun. Nonlinear Sci. Numer. Simul.doi:10.1016/j.cnsns.2017.12.001. [6] V. E. Tarasov, No nonlocality. No fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 62 (2018) 157–163. doi:10.1016/j.cnsns.2018.02.019. [7] M. Stynes, Too much regularity may force too much uniqueness, Fract. Calc. Appl. Anal. 19 (6) (2016) 1554–1562. doi:10.1515/fca-2016-0080. [8] M. Stynes, A Caputo two-point boundary value problem: existence, uniqueness and regularity of a solution, Model. Anal. Inf. Sist. 23 (3) (2016) 370–376. doi:10.18255/1818-1015-2016-3-370-376. [9] A. N. Kolmogorov, S. V. Fom¯ı n, Introductory real analysis, Dover Publications, Inc., New York, 1975, translated from the second Russian edition and edited by Richard A. Silverman, Corrected reprinting. [10] M. H. Protter, H. F. Weinberger, Maximum principles in differential equations, Springer-Verlag, New York, 1984, corrected reprint of the 1967 original. doi:10.1007/978-1-4612-5282-5.
6