DOCSLIB.ORG

Introduction to Fractional Calculus (Based on Lectures by R

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Fractional Calculus (Based on Lectures by R Introduction to fractional calculus (Based on lectures by R. Goren‡o, F. Mainardi and I. Podlubny) R. Vilela Mendes July 2008 () July2008 1/44 Contents - Historical origins of fractional calculus - Fractional integral according to Riemann-Liouville - Caputo fractional derivative - Riesz-Feller fractional derivative - Grünwal-Letnikov - Integral equations - Relaxation and oscillation equations - Fractional di¤usion equation - A nonlinear fractional di¤erential equation. Stochastic solution - Geometrical interpretation of fractional integration () July2008 2/44 Fractional Calculuswas born in1695 n fd dt n What if the order will be n = ½? It will lead to a paradox, from which one day useful consequences will be Start drawn. JJ II G.F.A. de L’Hôpital G.W. Leibniz J I 10 / 90 (1661–1704) (1646–1716) Back Full screen Close End G. W. Leibniz (1695{1697) In the letters to J. Wallis and J. Bernulli (in 1697) Leibniz mentioned the possible approach to fractional-order differ- entiation in that sense, that for non-integer values of n the definition could be the following: dnemx = mnemx; n dx Start JJ II J I 11 / 90 Back Full screen Close End L. Euler (1730) n m d x m n = m(m 1) ::: (m n + 1)x − dxn − − Γ(m + 1) = m(m 1) ::: (m n + 1) Γ(m n + 1) n m − − − d x Γ(m + 1) m n = x − : dxn Γ(m n + 1) − Euler suggested to use this relationship also for negative or 1 non-integer (rational) values of n. Taking m = 1 and n = 2, Euler obtained: Start d1=2x 4x 2 JJ II = r = x1=2 dx1=2 π pπ J I 12 / 90 Back Full screen Close End S. F. Lacroix adopted Euler's derivation for his success- ful textbook ( Trait´edu Calcul Diff´erentiel et du Calcul Int´egral, Courcier, Paris, t. 3, 1819; pp. 409{410). Start JJ II J I 13 / 90 Back Full screen Close End J. B. J. Fourier (1820{1822) The first step to generalization of the notion of differentia- tion for arbitrary functions was done by J. B. J. Fourier (Th´eorieAnalytique de la Chaleur, Didot, Paris, 1822; pp. 499{508). After introducing his famous formula 1 1 1 f(x) = Z f(z)dz Z cos (px pz)dp; 2π − −∞ −∞ Fourier made a remark that Start dnf(x) 1 1 1 π = Z f(z)dzZ cos (px pz + n )dp; JJ II dxn 2π − 2 J I 14 / 90 −∞ −∞ Back and this relationship could serve as a definition of the n-th Full screen order derivative for non-integer n. Close End RiemannಥLiouville definition t d n fτ d τ Dα f t = 1 § · () a t () ¨ ¸ α−n + Γ()n − α dt ³ t − τ 1 © ¹ a () (n− 1 ≤ α < n ) Start JJ II G.F.B. Riemann J. Liouville J I (1826–1866) (1809–1882) 21 / 90 Back Full screen Close End Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . () July2008 3/44 Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) () July2008 3/44 Fractional integral according to Riemann-Liouville According to Riemann-Liouville the notion of fractional integral of order α (α > 0) for a function f (t), is a natural consequence of the well known formula (Cauchy-Dirichlet ?), that reduces the calculation of the n fold primitive of a function f (t) to a single integral of convolution type t n 1 n 1 Ja+f (t) := (t τ) f (τ) dτ , n N (1) (n 1)! a 2 Z vanishes at t = a with its derivatives of order 1, 2, ... , n 1 . Require n f (t) and Ja+f (t) to be causal functions, that is, vanishing for t < 0 . Extend to any positive real value by using the Gamma function, (n 1)! = Γ(n) Fractional Integral of order α>0 (right-sided) t α 1 α 1 Ja+ f (t) := (t τ) f (τ) dτ , α R (2) Γ (α) a Z 2 0 0 De…ne Ja+ := I , Ja+ f (t) = f (t) () July2008 3/44 Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville () July2008 4/44 Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). () July2008 4/44 Fractional integral according to Riemann-Liouville Alternatively (left-sided integral) b α 1 α 1 Jb f (t) := (τ t) f (τ) dτ , α R Γ (α) t Z 2 (a = 0, b = +∞) Riemann (a = ∞, b = +∞) Liouville Let α α J := J0+ Semigroup properties JαJ β = Jα+β , α , β 0 Commutative property J βJα = JαJ β E¤ect on power functions Jαtγ = Γ(γ+1) tγ+α , α > 0 , γ > 1 , t > 0 Γ(γ+1+α) (Natural generalization of the positive integer properties). Introduce the following causal function (vanishing for t < 0 ) α 1 t Φ (t) := + , α > 0 α Γ(α) () July2008 4/44 Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 De…ning the Laplace transform pairs by f (et) f (s) f (s) e Jα f (t) , α > 0 sα e Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α () July2008 5/44 De…ning the Laplace transform pairs by f (t) f (s) f (s) e Jα f (t) , α > 0 sα e Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 e () July2008 5/44 Fractional integral according to Riemann-Liouville Φ (t) Φ (t) = Φ + (t) , α , β > 0 α β α β Jα f (t) = Φ (t) f (t) , α > 0 α Laplace transform ∞ st f (t) := e f (t) dt = f (s) , s C L f g Z0 2 De…ning the Laplace transform pairs by f (et) f (s) f (s) e Jα f (t) , α > 0 sα e () July2008 5/44 Then, de…ne Dα as a left-inverse to Jα. With a positive integer m , m 1 < α m , de…ne: α m m α Fractional Derivative of order α : D f (t) := D J f (t) d m 1 t f (τ) α dtm α+1 m dτ , m 1 < α < m D f (t) := Γ(m α) 0 (t τ) d m ( h dtm fR(t) , i α = m Fractional derivative according to Riemann-Liouville Denote by Dn with n N , the derivative of order n .
Load more

Recommended publications
  • Arxiv:1507.07356V2 [Math.AP]
    TEN EQUIVALENT DEFINITIONS OF THE FRACTIONAL LAPLACE OPERATOR MATEUSZ KWAŚNICKI Abstract. This article discusses several definitions of the fractional Laplace operator ( ∆)α/2 (α (0, 2)) in Rd (d 1), also known as the Riesz fractional derivative − ∈ ≥ operator, as an operator on Lebesgue spaces L p (p [1, )), on the space C0 of ∈ ∞ continuous functions vanishing at infinity and on the space Cbu of bounded uniformly continuous functions. Among these definitions are ones involving singular integrals, semigroups of operators, Bochner’s subordination and harmonic extensions. We collect and extend known results in order to prove that all these definitions agree: on each of the function spaces considered, the corresponding operators have common domain and they coincide on that common domain. 1. Introduction We consider the fractional Laplace operator L = ( ∆)α/2 in Rd, with α (0, 2) and d 1, 2, ... Numerous definitions of L can be found− in− literature: as a Fourier∈ multiplier with∈{ symbol} ξ α, as a fractional power in the sense of Bochner or Balakrishnan, as the inverse of the−| Riesz| potential operator, as a singular integral operator, as an operator associated to an appropriate Dirichlet form, as an infinitesimal generator of an appropriate semigroup of contractions, or as the Dirichlet-to-Neumann operator for an appropriate harmonic extension problem. Equivalence of these definitions for sufficiently smooth functions is well-known and easy. The purpose of this article is to prove that, whenever meaningful, all these definitions are equivalent in the Lebesgue space L p for p [1, ), ∈ ∞ in the space C0 of continuous functions vanishing at infinity, and in the space Cbu of bounded uniformly continuous functions.
    [Show full text]
  • Riesz Potentials and Liouville Operators on Fractals
    RIESZ POTENTIALS AND LIOUVILLE OPERATORS ON FRACTALS by M. Zähle Mathematical Institute, University of Jena, D-07740 Jena, Germany E-mail: [email protected] Abstract An analogue to the theory of Riesz potentials and Liouville operators in Rn for arbitrary fractal d-sets is developed. Corresponding function spaces agree with traces of euclidean Besov spaces on fractals. By means of associated quadratic forms we construct strongly continuous semigroups with Liouville operators as infinitesimal generator. The case of Dirichlet forms is discussed separately. As an example of related pseudodifferential equations the fractional heat-type equation is solved. Mathematics Subject Classification. Primary 28A80, Secondary 47B07, 35P20 Keywords. fractal d-set, Riesz potential, pseudodifferential operator, fractal Besov space, Dirichlet form 0 Introduction In [17] the Riesz potential of order s of a fractal d-measure µ in Rn with compact support Γ is defined as Z s −(d−s) Iµf(x) := const |x − y| f(y)µ(dy) , f ∈ L2(µ) , where 0 < s < d < n . (Examples for the measure µ are the Haus- dorff measures Hd on arbitrary self-similar sets Γ of dimension d.) Such potentials have a long tradition for the case, where µ is replaced by the Lebesgue measure. For more general µ only a view properties are known. Some references may be found in [17]. We also refer to the fundamental paper [2]. In connection with harmonic analysis and stochastic processes on fractals these potentials appear in a new light. The aim of the present paper is to continue this study in order to get a deeper insight into the interplay between geometry of and analysis on fractal sets and the corresponding properties of the embedding euclidean space.
    [Show full text]
  • Fourier Multipliers and Dirac Operators
    Fourier Multipliers and Dirac Operators Craig A. Nolder and Guanghong Wang Abstract. We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the Π-operator. This opera- tor is the hypercomplex version of the Beurling Ahlfors transform in the plane. The hypercomplex Beuling Ahlfors transform is a direct general- ization of the Beurling Ahlfors transform and reduces to this operator in the plane. We give an integral representation for iterations of the hyper- complex Beurling Ahlfors transform and we present here a bound for the Lp-norm. Such Lp-bounds are essential for applications of the Beurling Ahlfors transformation in the plane. The upper bound presented here is m(p∗ −1) where m is the dimension of the Euclidean space on which the functions are defined, 1 < p < 1 and p∗ = max(p; p=(p − 1)): We use recent estimates on second order Riesz transforms to obtain this result. Using the Fourier multiplier of the Π operator we express this operator as a hypercomplex linear combination of second order Riesz transforms. Keywords. Fourier multipliers, Clifford analysis, Dirac operators, Π op- erator, Beurling Ahlfors transform, Lp-estimates. msc : 47F, 47G, 30G 1. Introduction We determine the Fourier multipliers of the hypercomplex Cauchy transform T and the Vekua Π-operator over Rn+1: Various relations between these op- erators, and the Dirac and Laplacian operator, are easily obtained using these 1 n+1 multipliers. Theorem 3.2 is a convolution identity on C0 (R ) involving the iterated Cauchy kernel and Dirac operator.
    [Show full text]
  • Lecture Notes: Harmonic Analysis
    Lecture notes: harmonic analysis Russell Brown Department of mathematics University of Kentucky Lexington, KY 40506-0027 August 14, 2009 ii Contents Preface vii 1 The Fourier transform on L1 1 1.1 Definition and symmetry properties . 1 1.2 The Fourier inversion theorem . 9 2 Tempered distributions 11 2.1 Test functions . 11 2.2 Tempered distributions . 15 2.3 Operations on tempered distributions . 17 2.4 The Fourier transform . 20 2.5 More distributions . 22 3 The Fourier transform on L2. 25 3.1 Plancherel's theorem . 25 3.2 Multiplier operators . 27 3.3 Sobolev spaces . 28 4 Interpolation of operators 31 4.1 The Riesz-Thorin theorem . 31 4.2 Interpolation for analytic families of operators . 36 4.3 Real methods . 37 5 The Hardy-Littlewood maximal function 41 5.1 The Lp-inequalities . 41 5.2 Differentiation theorems . 45 iii iv CONTENTS 6 Singular integrals 49 6.1 Calder´on-Zygmund kernels . 49 6.2 Some multiplier operators . 55 7 Littlewood-Paley theory 61 7.1 A square function that characterizes Lp ................... 61 7.2 Variations . 63 8 Fractional integration 65 8.1 The Hardy-Littlewood-Sobolev theorem . 66 8.2 A Sobolev inequality . 72 9 Singular multipliers 77 9.1 Estimates for an operator with a singular symbol . 77 9.2 A trace theorem. 87 10 The Dirichlet problem for elliptic equations. 91 10.1 Domains in Rn ................................ 91 10.2 The weak Dirichlet problem . 99 11 Inverse Problems: Boundary identifiability 103 11.1 The Dirichlet to Neumann map . 103 11.2 Identifiability . 107 12 Inverse problem: Global uniqueness 117 12.1 A Schr¨odingerequation .
    [Show full text]
  • Bilateral Tempered Fractional Derivatives
    S S symmetry Article Bilateral Tempered Fractional Derivatives Manuel Duarte Ortigueira 1,* and Gabriel Bengochea 2 1 Centre of Technology and Systems-UNINOVA, NOVA School of Science and Technology of NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal 2 Academia de Matemática, Universidad Autónoma de la Ciudad de México, Ciudad de México 04510, Mexico; [email protected] * Correspondence: [email protected] Abstract: The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and shows that it cannot be considered as a derivative. Keywords: tempered fractional derivative; one-sided tempered fractional derivative; bilateral tem- pered fractional derivative; tempered riesz potential MSC: Primary 26A33; Secondary 34A08; 35R11 1. Introduction In a recent paper [1], we presented a unified formulation for the one-sided Tempered Fractional Calculus, that includes the classic, tempered, substantial, and shifted fractional operators [2–9]. Citation: Ortigueira, M.D.; Here, we continue in the same road by presenting a study on the two-sided tempered Bengochea, G. Bilateral Tempered operators that generalize and include the one-sided. The most interesting is the tempered Fractional Derivatives. Symmetry Riesz potential that was proposed in analogy with the one-sided tempered derivatives [10]. 2021, 13, 823. https://doi.org/ However, a two-sided tempering was introduced before, in the study of the called variance 10.3390/sym13050823 gamma processes [11,12], in Statistical Physics for modelling turbulence, under the concept Academic Editors: Francisco Martínez of truncated Lévy flight [8,13–17], and for defining the Regular Lévy Processes of Expo- González and Mohammed KA Kaabar nential type [2,10,18].
    [Show full text]
  • An $ L^ 1$-Type Estimate for Riesz Potentials
    An L1-type estimate for Riesz potentials 1 2 3 Armin Schikorra∗ , Daniel Spector† , and Jean Van Schaftingen‡ 1Department of Mathematics, University of Basel, Basel, Switzerland 2Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 3Institut de Recherche en Mathématique et en Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium Abstract In this paper we establish new L1-type estimates for the classical Riesz potentials of order α (0,N): ∈ Iαu N/(N α) N C Ru 1 N N . k kL − (R ) ≤ k kL (R ;R ) This sharpens the result of Stein and Weiss on the mapping properties of 1 N Riesz potentials on the real Hardy space (R ) and provides a new fam- H ily of L1-Sobolev inequalities for the Riesz fractional gradient. 1 Introduction and Main Results Let N 2 and define the Riesz potential I of order α (0,N) by its action on ≥ α ∈ a measurable function u via the convolution 1 u(y) Iαu(x) (Iα u)(x) := N α dy, ≡ ∗ γ(α) ˆ N x y R | − | − N/2 α α N α whenever it is well-defined. Here, γ(α) = π 2 Γ( 2 )/Γ( −2 ) is a normal- arXiv:1411.2318v4 [math.FA] 1 Sep 2016 ization constant [24, p. 117] that ensures that the Riesz potentials satisfy the semigroup property Iα+βu = IαIβu, for α, β > 0, such that α + β < N, for u in a suitable class of functions. p N The study of the mapping properties of Iα on L (R ) was initiated by Sobolev, who proved the following fundamental theorem about integrals of the potential type in 1938 [21, p.
    [Show full text]
  • Fractional Calculus and Special Functions
    LECTURE NOTES ON MATHEMATICAL PHYSICS Department of Physics, University of Bologna, Italy URL: www.fracalmo.org FRACTIONAL CALCULUS AND SPECIAL FUNCTIONS Francesco MAINARDI Department of Physics, University of Bologna, and INFN Via Irnerio 46, I{40126 Bologna, Italy. [email protected] [email protected] Contents (pp. 1 { 62) Abstract . p. 1 A. Historical Notes and Introduction to Fractional Calculus . p. 2 B. The Liouville-Weyl Fractional Calculus . p. 8 C. The Riesz-Feller Fractional Calculus . p.12 D. The Riemann-Liouville Fractional Calculus . p.18 E. The Gr¨unwald-Letnikov Fractional Calculus . p.22 F. The Mittag-Leffler Functions . p.26 G. The Wright Functions . p.42 References . p.52 The present Lecture Notes are related to a Mini Course on Introduction to Fractional Calculus delivered by F. Mainardi, at BCAM, Bask Cen- tre for Applied Mathematics, in Bilbao, Spain on March 11-15, 2013, see http://www.bcamath.org/en/activities/courses. The treatment reflects the research activity of the Author carried out from the academic year 1993/94, mainly in collaboration with his students and with Rudolf Gorenflo, Professor Emeritus of Mathematics at the Freie Universt¨at,Berlin. ii Francesco MAINARDI c 2013 Francesco Mainardi FRACTIONAL CALCULUS AND SPECIAL FUNCTIONS 1 FRACTIONAL CALCULUS AND SPECIAL FUNCTIONS Francesco MAINARDI Department of Physics, University of Bologna, and INFN Via Irnerio 46, I{40126 Bologna, Italy. [email protected] [email protected] Abstract The aim of these introductory lectures is to provide the reader with the essentials of the fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes.
    [Show full text]
  • Nonlinear Potentials, Local Solutions to Elliptic Equations and Rearrangements
    Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. X (2011), 335-361 Nonlinear potentials, local solutions to elliptic equations and rearrangements ANDREA CIANCHI Abstract. A sharp rearrangement estimate for the nonlinear Havin-Maz’ya po- tentials is established. In particular, this estimate leads to a characterization of those rearrangement invariant spaces between which the nonlinear potentials are bounded. In combination with results from [24] and [18], it also enables us to derive local bounds for solutions to quasilinear elliptic PDE’s and for their gra- dient in rearrangement form. As a consequence, the local regularity of solutions to elliptic equations and for their gradient in arbitrary rearrangement invariant spaces is reduced to one-dimensional Hardy-type inequalities. Applications to the special cases of Lorentz and Orlicz spaces are presented. Mathematics Subject Classification (2010): 31C15 (primary); 35B45 (sec- ondary). 1. Introduction The Riesz potential of order α ∈ (0, n) of a locally integrable function f in Rn, n ≥ 1, is defined as f (y) ( ) = ∈ n Iα f x −α dy for x R . (1.1) n | − |n R x y The operator Iα plays a role in various branches of analysis, including potential theory, harmonic analysis, Sobolev spaces, partial differential equations. The link to the theory of elliptic PDE’s is classical. Indeed, if n > 2 and ωn denotes the measure of the unit ball in n, then the function 1 I f (x) is the Newtonian R n(n−2)ωn 2 potential in Rn, namely the unique solution decaying to 0 at infinity to the Poisson equation −u = f in Rn.
    [Show full text]
  • A Renormalized Riesz Potential and Applications
    A renormalized Riesz potential and applications Mihai Putinar Abstract. The convolution in Rn with |x|−n is a very singular operator. Endowed with a proper normalization, and regarded as a limit of Riesz potentials, it is equal to Dirac’s distribution δ. However, a different normalization turns the non-linear operator: −2 −n Ef = exp( |x| ∗ f), |Sn−1| into a remarkable transformation. Its long history (in one dimension) and some of its recent applications in higher dimensions make the subject of this exposition. A classical extremal problem studied by A. A. Markov is related to the operation E in one real variable. Later, the theory of the spectral shift of self-adjoint perturbations was also based on E. In the case of two real variables, the transform E has ap- peared in operator theory, as a determinantal-characteristic function of certain close to normal operators. The interpretation of E in com- plex coordinates reveals a rich structure, specific only to the plane setting. By exploiting an inverse spectral function problem for hy- ponormal operators, applications of this exponential transform to im- age reconstruction, potential theory and fluid mechanics have recently been discovered. In any number of dimensions, the transformation E, applied to characteristic functions of domains Ω, can be regarded as the geometric mean of the distance function to the boundary. This interpretation has a series of unexpected geometric and analytic con- sequences. For instance, for a convex algebraic Ω, it turns out that the operation E is instrumental in converting finite external data (such as field measurements or tomographic pictures) into an equation of the boundary.
    [Show full text]
  • On the Role of Riesz Potentials in Poisson's Equation and Sobolev
    On the role of Riesz potentials in Poisson’s equation and Sobolev embeddings Rahul Garg Daniel Spector [email protected] [email protected] Department of Mathematics Technion - Israel Institute of Technology Abstract In this paper, we study the mapping properties of the classical Riesz potentials acting on Lp-spaces. In the supercritical exponent, we obtain new “almost” Lipschitz continuity estimates for these and related poten- tials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity esti- mates for distributional solutions to Poisson’s equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980. Contents 1 Introduction and Statement of Results 1 1.1 Historical Background . .2 1.2 Statement of Results . .4 2 Potential Estimates 7 3 Regularity for Solutions to Poisson’s Equation 12 4 Sobolev Embedding Theorem in the Supercritical Exponent 14 arXiv:1404.1563v1 [math.FA] 6 Apr 2014 1 Introduction and Statement of Results For each 0 < α < N, one defines the Riesz potential γ(N, α) I (x) := . α x N α | | − Then the operator Iα can be defined through its action on suitable subspaces of the space of Lebesgue measurable functions by convolution in the sense that f(y) Iαf(x) := (Iα f)(x) = γ(N, α) N α dy, (1.1) ∗ ˆ N x y R | − | − 1 whenever it is well-defined. Here, the precise definition of the constant is N α Γ( −2 ) γ(N, α) := N , 2 α α π 2 Γ( 2 ) from which one obtains two key properties of the potentials: the semigroup prop- erty and its connection with the Laplacian (see, for example, Stein [20][Chapter 5, p.118]).
    [Show full text]
  • Riesz Potential Versus Fractional Laplacian
    Riesz potential versus fractional Laplacian Manuel D Ortigueira, Taous-Meriem Laleg-Kirati and J A Tenreiro Machado Abstract. This paper starts by introducing the Gru¨nwald–Letnikov derivative, the Riesz potential and the problem of generalizing the Laplacian. Based on these ideas, the generalizations of the Laplacian for 1D and 2D cases are studied. It is presented as a fractional version of the Cauchy–Riemann conditions and, finally, it is discussed with the n-dimensional Laplacian. Keywords: nonlinear dynamics 1. Introduction The Laplacian is an important operator in physics and engineering. However, the standard definition does not cope with domains that are neither homogeneous, nor isotropic. Nowadays, researchers in applied sciences are interested in finding a definition of fractional Laplacian suitable for studying both isotropic and anisotropic media. In recent studies the fractional Laplacian was implemented by means of the inverse of the Riesz potential [1,2]. However, it is not straightforward that this is the single option when thinking about several distinct types of applications. In fact, in nD applications the inverse of the Riesz potential cannot be expressed as a sum of partial derivatives and is only defined using the Fourier transform inverse of an operator. An alternative option is to follow the vectorial calculus that allows finding a Laplacian as the sum of partial derivatives [3–5]. In an attempt to gain a deeper insight, in this paper we develop a heuristic analysis of the Riesz potential, in the perspective of what we are expecting in a Laplacian. As a collateral result, a possible generalization of the Cauchy–Riemann conditions is obtained.
    [Show full text]
  • Fractional Versions of the Fundamental Theorem of Calculus
    Applied Mathematics, 2013, 4, 23-33 http://dx.doi.org/10.4236/am.2013.47A006 Published Online July 2013 (http://www.scirp.org/journal/am) Fractional Versions of the Fundamental Theorem of Calculus Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira Department of Applied Mathematics, University of Campinas, Campinas, Brazil Email: [email protected], [email protected] Received April 27, 2013; revised May 27, 2013; accepted June 7, 2013 Copyright © 2013 Eliana Contharteze Grigoletto, Edmundo Capelas de Oliveira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT The concept of fractional integral in the Riemann-Liouville, Liouville, Weyl and Riesz sense is presented. Some prop- erties involving the particular Riemann-Liouville integral are mentioned. By means of this concept we present the frac- tional derivatives, specifically, the Riemann-Liouville, Liouville, Caputo, Weyl and Riesz versions are discussed. The so-called fundamental theorem of fractional calculus is presented and discussed in all these different versions. Keywords: Fractional Integral; Fractional Derivative; Riemann-Liouville Derivative; Liouville Derivative; Caputo Derivative; Weyl Derivative and Riesz Derivative 1. Introduction much research on its applications as there is on the cal- culus of integer order. Several applications in all areas of Fractional calculus, a popular name used to denote the knowledgement are collected, presented and discussed in calculus of non integer order, is as old as the calculus of different books as follow [7-12]. integer order as created independently by Newton and The main objective of this paper is to explain what is Leibniz.
    [Show full text]