Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

Almeida, Alexandre

doi:10.1007/s00009-009-0006-7

Cited by 30 publications

(35 citation statements)

References 43 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hypersingular integrals with variable α(x) are studied by Almeida and Samko in their recent paper [AS09]. Hypersingular integrals with variable α(x) are studied by Almeida and Samko in their recent paper [AS09].…”

Section: Hypersingular Integralsmentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

View full text Add to dashboard Cite

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 ...

show abstract

Section: Hypersingular Integralsmentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

View full text Add to dashboard Cite

show abstract

“…In the non-weighted case, the theorem above was proved by Almeida and Samko in [1] in the context of (quasi)-metric measure spaces. The authors give boundedness results with measures either satisfying the doubling condition or the lower Ahlfors condition.…”

Section: For Any Setmentioning

confidence: 93%

“…The operator above is equivalent to the one defined in [1] in the context of a (quasi)-metric space equipped with a lower Ahlfors measure whenever the quasi-distance is a fixed multiple of the euclidean distance. Let s be a real number such that 1 < s < ∞.…”

Section: For Any Setmentioning

confidence: 97%

Boundedness of fractional operators in weighted variable exponent spaces with non doubling measures

2010

View full text Add to dashboard Cite

show abstract

“…They introduced the study of fractional integration and differentiation when the order is not a constant but a function. Afterwards, several works were dedicated to variable order fractional operators, their applications and interpretations (Almeida and Samko 2009;Coimbra 2003;Lorenzo and Hartley 2002). In particular, Samko's variable order fractional calculus turns out to be very useful in mechanics and in the theory of viscous flows (Coimbra 2003;Diaz and Coimbra 2009;Lorenzo and Hartley 2002;Pedro et al 2008;Ramirez andCoimbra 2010, 2011).…”

Section: Variable Order Fractional Operatorsmentioning

confidence: 99%

Fractional Calculus

Malinowska¹,

Odzijewicz²,

Torres³

2015

Advanced Methods in the Fractional Calculus of Variations

View full text Add to dashboard Cite

A brief exposition of fractional order operators and their properties is given. After that, we introduce the notion of generalized fractional operators.Keywords Fractional derivatives and integrals · Generalized fractional derivatives and integrals · Fractional derivatives and integrals of variable order · RiemannLiouville, Hadamard and Caputo operators · Fractional integration by parts · Multidimensional generalized fractional calculus Fractional calculus was introduced on September 30, 1695. On that day, Leibniz wrote a letter to L'Hôpital, raising the possibility of generalizing the meaning of derivatives from integer order to noninteger order derivatives. L'Hôpital wanted to know the result for the derivative of order n = 1/2. Leibniz replied that "one day, useful consequences will be drawn" and, in fact, his vision became a reality. However, the study of noninteger order derivatives did not appear in the literature until 1819, when Lacroix presented a definition of fractional derivative based on the usual expression for the nth derivative of the power function (Lacroix 1819). Within years the fractional calculus became a very attractive subject to mathematicians, and many different forms of fractional (i.e., noninteger) differential operators were introduced: the Grunwald-Letnikow, Riemann-Liouville, Hadamard, Caputo, Riesz (Hilfer 2000;Kilbas et al. 2006;Podlubny 1999;) and the more recent notions of Cresson (2007), Katugampola (2011), Klimek (2005, Kilbas and Saigo (2004) or variable order fractional operators introduced by Samko and Ross (1993).In 2010, an interesting perspective to the subject, unifying all mentioned notions of fractional derivatives and integrals, was introduced in Agrawal (2010) and later studied in Bourdin et al. (2014), Klimek and Lupa (2013), Odzijewicz et al. (2012a, b, 2013a. Precisely, authors considered general operators, which by choosing special kernels, reduce to the standard fractional operators. However, other nonstandard kernels can also be considered as particular cases.This chapter presents preliminary definitions and facts of classical, variable order, and generalized fractional operators.

show abstract

Fractional and Hypersingular Operators in Variable Exponent Spaces on Metric Measure Spaces

Cited by 30 publications

References 43 publications

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Boundedness of fractional operators in weighted variable exponent spaces with non doubling measures

Fractional Calculus

Contact Info

Product

Resources

About