Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

Keyantuo, Valentin; Miana, Pedro J.; Sánchez–Lajusticia, Luis

doi:10.1007/s00020-013-2076-y

Cited by 4 publications

(8 citation statements)

References 30 publications

(54 reference statements)

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“…For more information about recent results on the Cauchy problem for abstract fractional differential-operator equations, we refer the reader to [Baz01,EK04,KJ11,KMSL13]. For more information about recent results on the Cauchy problem for abstract fractional differential-operator equations, we refer the reader to [Baz01,EK04,KJ11,KMSL13].…”

Section: Additional Notesmentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

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Section: Additional Notesmentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

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“…However, this property is not longer true in the general case of (a, k)regularized resolvent families, where loss of regularity is present. This phenomena has been observed for the case of k-convoluted semigroups [9] (in particular for α-times integrated semigroups in [2,22]) and k-convoluted cosine families [23].…”

Section: Laplace Transform In One and Two Variablesmentioning

confidence: 77%

“…The following result is related to [9,Theorem 4.4] and [23,Theorem 3.3]. However, note that both results are not included in this corollary.…”

Section: Laplace Transform In One and Two Variablesmentioning

confidence: 99%

“…More generally one could consider the case of K-convoluted resolvent families, i.e. (g α , (1 * K))-regularized resolvent families for 0 < α < 1 and compare it to the limit case when α → 1 − , see [9,Theorem 4.4].…”

Section: On One Handmentioning

confidence: 99%

“…In contrast, this extension property is not more true for the class of local integrated semigroups ( [2,Example 4.6]). Furthermore, we have that if A generates a local (1 * k)-convoluted semigroup on [0, τ ), then A generates a local (1 * k * n )-convoluted semigroup on an interval [0, nτ ), see [9,Theorem 4.4] ( [23,Theorem 3.3]). In other words, in these cases there is evolution with jumps of regularity and naturally the need of regularize the family of operators appears (in the sense of convolution) in order to have extension.…”

Section: Introductionmentioning

confidence: 99%

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Sharp extensions and algebraic properties for solution families of vector-valued differential equations

Abadías

Lizama²,

Miana

2016

Banach J. Math. Anal.

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In this paper we show the unexpected property that extension from local to global without loss of regularity holds for the solutions of a wide class of vector-valued differential equations, in particular for the class of fractional abstract Cauchy problems in the subdiffusive case. The main technique is the use of the algebraic structure of these solutions, which are defined by new versions of functional equations defining solution families of bounded operators. The convolution product and the double Laplace transform for functions of two variables are useful tools which we apply also to extend these solutions. Finally we illustrate our results with different concrete examples.

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Initial and boundary value problems for fractional order differential equations

Umarov

2015

Developments in Mathematics

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Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

Cited by 4 publications

References 30 publications

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Sharp extensions and algebraic properties for solution families of vector-valued differential equations

Initial and boundary value problems for fractional order differential equations

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