Recently there has been a surge of interest in PDEs involving fractional derivatives in different fields of engineering. In this extended abstract we present some of the results developed in \cite{FV}. We compute the fundamental solution for the three-parameter fractional Laplace operator $\Delta^{(\alpha,\beta,\gamma)}$ with $(\alpha, \beta, \gamma) \in \,]0,1]^3$ by transforming the eigenfunction equation into an integral equation and applying the method of separation of variables. The obtained solutions are expressed in terms of Mittag-Leffer functions. For more details we refer the interested reader to \cite{FV} where it is also presented an operational approach based on the two Laplace transform

Ferreira, Milton dos Santos

Vieira, Nelson Felipe Loureiro

Recently there has been a surge of interest in PDEs involving fractional derivatives in different fields of engineering. In this extended abstract we present some of the results developedin [3]. We  compute the fundamental solution for the three-parameter fractional Laplace operator Δ by transforming the eigenfunction equation into an integral equation and applying the method of separation of variables. The obtained solutions are expressed in terms of Mittag-Leffer functions. For more details we refer the interested reader to [3] where it is also presented an operational approach based on the two Laplace transform

Vieira, Nelson

Online-Publikationssystem der Bauhaus-Universität Weimar

20th International Conference on the Application of ComputerScience and Mathematics in Architecture and Civil EngineeringK. Gu¨rlebeck and T. Lahmer (eds.)Weimar, Germany, 20-22 July 2015EIGENFUNCTIONS AND FUNDAMENTAL SOLUTIONS FOR THEFRACTIONAL LAPLACIAN IN 3 DIMENSIONSM. Ferreira†,‡, N. Vieira‡† School of Technology and ManagementPolytechnic Institute of LeiriaP-2411-901, Leiria, Portugal.Email: milton.ferreira@ipleiria.pt‡ CIDMA - Center for Research and Development in Mathematics and ApplicationsDepartment of Mathematics, University of AveiroCampus Universita´rio de Santiago, 3810-193 Aveiro, Portugal.Email: mferreira@ua.pt, nloureirovieira@gmail.comKeywords: Fractional Laplace operator; Riemann-Liouville fractional derivatives; Eigenfunc-tions; Fundamental solution; Mittag-Leffler function.Abstract. Recently there has been a surge of interest in PDEs involving fractional derivativesin different fields of engineering. In this extended abstract we present some of the results devel-oped in [3]. We compute the fundamental solution for the three-parameter fractional Laplaceoperator ∆(α,β,γ) with (α, β, γ) ∈ ]0, 1]3 by transforming the eigenfunction equation into an in-tegral equation and applying the method of separation of variables. The obtained solutions areexpressed in terms of Mittag-Leffer functions. For more details we refer the interested reader to[3] where it is also presented an operational approach based on the two Laplace transform.301 INTRODUCTIONThe problems with the fractional Laplacian attracted in the last years a lot of attention, dueespecially to their large range of applications. The fractional Laplacian appears in probabilisticframework as well as in mathematical finance as infinitesimal generators of the stable Le´vyprocesses [1]. One can find problems involving the fractional Laplacian in mechanics and inelastostatics, for example, a Signorini obstacle problem originating from linear elasticity [2].The aim of this paper is to present an explicit expression for the family of eigenfunctionsand fundamental solutions of the three-parameter fractional Laplace. For the sake of simplicitywe restrict ourselves to the three dimensional case, however the results can be generalized foran arbitrary dimension. The two dimensional case was already studied in [10]. Connectionsbetween fractional calculus and Clifford analysis were considered recently in [5, 9].2 PRELIMINARIES2.1 Fractional calculus and special functionsLet(Dαa+f)(x) denote the fractional Riemann-Liouville derivative of order α > 0 (see [6])(Dαa+f) (x) =(ddx)n1Γ(n− α)∫ xaf(t)(x− t)α−n+1 dt, n = [α] + 1, a, x > 0. (1)where [α] means the integer part of α.When 0 < α < 1 then (1) takes the form(Dαa+f) (x) =ddx1Γ(1− α)∫ xaf(t)(x− t)α dt. (2)The Riemann-Liouville fractional integral of order α > 0 is given by (see [6])(Iαa+f) (x) =1Γ(α)∫ xaf(t)(x− t)1−α dt, a, x > 0. (3)We recall also the following definition (see [8]):Definition 2.1 A function f ∈ L1(a, b) has a summable fractional derivatives(Dαa+f)(x) if(In−αa+)(x) ∈ ACn([a, b]), where n = 0, 1, . . . , n − 1 and ACn([a, b]) denote the class offunctions f(x), which are continuously differentiable on the segment [a, b] up to order n−1 andf (n−1)(x) is absolutely continuous on [a, b].If a function f admits a summable fractional derivative, then the composition of (1) and (3) canbe written in the form (see [8, Thm. 2.4])(Iαa+ Dαa+f) (x) = f(x)−n−1∑k=0(x− a)α−k−1Γ(α− k)(In−αa+)(n−k−1)(a), n = [α] + 1. (4)Nevertheless we note that Dαa+ Iαa+f = f, for any summable function. This is a particular caseof a more general property (cf. [7, (2.114)])Dαa+(Iγa+f)= Dα−γa+ f, 0 ≤ γ ≤ α. (5)31One important function used in this paper is the two-parameter Mittag-Leffler function Eµ,ν(z)[4], which is defined in terms of the power series byEµ,ν(z) =∞∑n=0znΓ(µn+ ν), µ > 0, ν ∈ R, z ∈ C. (6)In particular, the function Eµ,ν(z) is entire of order ρ = 1µ and type σ = 1. Two important frac-tional integral and differential formulae involving the two-parametric Mittag-Leffler functionare the following (see [4, p.61,p.87])Iαa+((x− a)ν−1Eµ,ν(k(x− a)µ))= (x− a)ν+α−1Eµ,ν+α(k(x− a)µ) (7)Dαa+((x− a)ν−1Eµ,ν(k(x− a)µ))= (x− a)ν−α−1Eµ,ν−α(k(x− a)µ) (8)for all α > 0, µ > 0, ν ∈ R, k ∈ C, a > 0, x > a.The approach developed in Section 3 leads to the solution of linear Abel integral equations ofthe second kind.Theorem 2.2 [4, Thm. 4.2] Let f ∈ L1[a, b], α > 0 and λ ∈ C. Then the integral equationu(x) = f(x) +λΓ(α)∫ xa(x− t)α−1u(t) dt, x ∈ [a, b]has a unique solutionu(x) = f(x) + λ∫ xa(x− t)α−1Eα,α(λ(x− t)α)f(t) dt. (9)3 EIGENFUNCTIONS AND FUNDAMENTAL SOLUTION OF THE FRACTIONALLAPLACE OPERATORConsider the eigenfunction equation for the fractional Laplace operator ∆(α,β,γ)∆(α,β,γ)u(x, y, z) = λu(x, y, z)⇔(D1+αx+0u)(x, y, z) +(D1+βy+0u)(x, y, z) +(D1+γz+0u)(x, y, z) = λ u(x, y, z). (10)where λ ∈ C, (α, β, γ) ∈]0, 1]3, (x, y, z) ∈ Ω = [x0, X0] × [y0, Y0] × [z0, Z0], x0, y0, z0 ≥ 0,X0, Y0, Z0 < ∞, and u(x, y, z) admits summable fractional derivatives D1+αx+0 , D1+βy+0, D1+γz+0.Applying the fractional integral operators I1+αx+0, I1+βy+0and I1+γy+0from both sides of (10), takinginto account (4) and using Fubini’s Theorem, we get(I1+βy+0I1+γz+0u)(x, y, z) +(I1+αx+0I1+γz+0u)(x, y, z)+(I1+αx+0I1+βy+0u)(x, y, z)− λ(I1+αx+0I1+βy+0I1+γz+0u)(x, y, z)=(x− x0)α−1Γ(α)(I1+βy+0I1+γz+0f0)(y, z) +(x− x0)αΓ(1 + α)(I1+βy+0I1+γz+0f1)(y, z)+(y − y0)β−1Γ(β)(I1−αx+0I1+γz+0h0)(x, z) +(y − y0)βΓ(1 + β)(I1+αx+0I1+γz+0h1)(x, z)+(z − z0)γ−1Γ(γ)(I1−αx+0I1+βy+0g0)(x, y) +(z − z0)γΓ(1 + γ)(I1+αx+0I1+βy+0g1)(x, y), (11)32where we denote the Cauchy’s fractional integral conditions byf0(y, z) =(I1−αx+0u)(x0, y, z), f1(y, z) =(Dαx+0u)(x0, y, z), (12)h0(x, z) =(I1−βy+0u)(x, y0, z), h1(x, z) =(Dβy+0u)(x, y0, z), (13)g0(x, y) =(I1−γz+0u)(x, y, z0), g1(x, y) =(Dγz+0u)(x, y, z0). (14)We now assume that u(x, y, z) = u1(x)u2(y)u3(z). Substituting in (11) and taking into accountthe initial conditions (12), (13), and (14) we obtainu1(x)(I1+βy+0u2(y) I1+γz+0u3(z))+ u2(y)(I1+αx+0u1(x) I1+γz+0u3(z))+ u3(z)(I1+αx+0u1(x) I1+βy+0u2(y))(x, y, z)− λ(I1+αx+0u1)(x)(I1+βy+0u2)(y)(I1+γz+0u3)(z)= a1(x− x0)α−1Γ(α)(I1+βy+0u2(y) I1+γz+0u3(z))+ a2(x− x0)αΓ(1 + α)(I1+βy+0u2(y) I1+γz+0u3(z))+ b1(y − y0)β−1Γ(β)(I1+αx+0u1(x) I1+γz+0u3(z))+ b2(y − y0)βΓ(1 + β)(I1+αx+0u1(x) I1+γz+0u3(z))+ c1(z − z0)γ−1Γ(γ)(I1+αx+0u1(x) I1+βy+0u2(y))+ c2(z − z0)γΓ(1 + γ)(I1+αx+0u1(x) I1+βy+0u2(y)), (15)where ai, bi, ci ∈ C, i = 1, 2, are constants defined by the initial conditions (12), (13), and (14).Supposing that(I1+αx+0u1)(x)(I1+βy+0u2)(y)(I1+γz+0u3)(z) ̸= 0, for (x, y, z) ∈ Ω, we can divide(15) by this factor. Separating the variables we get the following three Abel’s integral equationsof second kind:u1(x)− µ(I1+αx+0u1)(x) = a1(x− x0)α−1Γ(α)+ a2(x− x0)αΓ(1 + α), (16)u2(y) + ν(I1+βy+0u2)(y) = b1(y − y0)β−1Γ(β)+ b2(y − y0)βΓ(1 + β), (17)u3(z) + (µ− λ− ν)(I1+γz+0u3)(z) = c1(z − z0)γ−1Γ(γ)+ c2(z − z0)γΓ(1 + γ), (18)where λ, µ, ν ∈ C are constants. We observe that the equality(I1+αx+0u1)(x)(I1+βy+0u2)(y)(I1+γz+0u3)(z) = 0,agrees with (15), (16), (17), and (18) for at least one point (ξ, η, θ). Solving the latter equa-tions using (9) in Theorem 1.2 and after straightforward computations we obtain a family ofeigenfunctions.33Theorem 3.1 A family of eigenfunctions of the fractional Laplace operator∆(α,β,γ) is given byuλ,µ,ν(x, y, z) = u1(x) u2(y) u3(z) withu1(x) = a1 (x− x0)α−1 E1+α,α(µ(x− x0)1+α)+ a2 (x− x0)α E1+α,1+α(µ(x− x0)1+α)(19)u2(y) = b1 (y − y0)β−1 E1+β,β(−ν(y − y0)1+β)+ b2 (y − y0)β E1+β,1+β(−ν(y − y0)1+β) (20)u3(z) = c1 (z − z0)γ−1 E1+γ,γ((µ− λ− ν)(z − z0)1+γ)+ c2 (z − z0)γ E1+γ,1+γ((µ− λ− ν)(z − z0)1+γ), (21)where λ, µ, ν ∈ C are constants.Corollary 3.2 For λ = 0, u0,µ,ν(x, y, z) = u1(x) u2(y) u3(z) is a family of fundamental solu-tions for the fractional Laplace operator ∆(α,β,γ).Remark 3.3 In the special case of α = β = γ = 1 the functions u1, u2 and u3 take the form:u1(x) = a1 cosh (√µ (x− x0)) + a2√µsinh (√µ (x− x0)) ,u2(y) = b1 cos(√ν (y − y0))+b2√νsin(√ν (y − y0)),u3(z) = c1 cosh(√µ− λ− ν (z − z0))+c2√µ− λ− ν sinh(√µ− λ− ν (z − z0)).which are the components of the fundamental solution of the Laplace operator in R3 obtainedby the method of separation of variables.It is also possible to apply an operational approach based on the two dimensional Laplacetransform to obtain a complete family of eigenfunctions and fundamental solutions for the frac-tional Laplace operator. This was done in detail in [3].Acknowledgement: This work was supported by Portuguese funds through the CIDMA -Center for Research and Development in Mathematics and Applications, and the PortugueseFoundation for Science and Technology (“FCT–Fundac¸a˜o para a Cieˆncia e a Tecnologia”),within project UID/MAT/ 0416/2013. N. Vieira was also supported by FCT via the FCT Re-searcher Program 2014 (Ref: IF/00271/2014).REFERENCES[1] D. Applebaum: Le´vy processes: from probability to finance and quantum groups. NoticesAm. Math. Soc., 51-No.11, 1336-1347, 2004.[2] L.A. Caffarelli, S. Salsa and L. Silvestre: Regularity estimates for the solution and the freeboundary of the obstacle problem for the fractional Laplacian. Invent. Math., 171-No.2,425-461, 2008.34[3] M. Ferreira and N. Vieira: Eigenfunctions and fundamental solution of the fractionalLaplace and Dirac operators: the Riemann-Liouville case. Submitted.[4] R. Gorenflo, A.A. Kilbas, F. Mainardi and S. Rogosin: Mittag-Leffler functions. Theoryand applications. Springer Monographs in Mathematics, Springer, Berlin, 2014.[5] U. Ka¨hler and N. Vieira: Fractional Clifford analysis. S. Bernstein, U. Ka¨hler, I. Sabadiniand F. Sommen eds. Hypercomplex Analysis: New perspectives and applications, Trendsin Mathematics, Birkha¨user, Basel, 2014.[6] A. Kilbas, H.M. Srivastava and J.J. Trujillo: Theory and applications of fractional dif-ferential equations. North-Holland Mathematics Studies-Vol.204, Elsevier, Amsterdam,2006.[7] I. Podlubny: Fractional differential equations. An introduction to fractional derivatives,fractional differential equations, to methods of their solution and some of their applica-tions. Mathematics in Science and Engineering-Vol.198, Academic Press, San Diego, CA,1999.[8] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional integrals and derivatives: theoryand apllications. Gordon and Breach, New York, NY, 1993.[9] N. Vieira: Fischer Decomposition and Cauchy-Kovalevskaya extension in fractional Clif-ford analysis: the Riemann-Liouville case. Proc. Edinb. Math. Soc., II. Ser., in press.[10] S. Yakubovich: Eigenfunctions and fundamental solutions of the fractional two-parameterLaplacian. Int. J. Math. Math. Sci., Article ID 541934, 18p, 2010.35

EIGENFUNCTIONS AND FUNDAMENTAL SOLUTIONS FOR THE FRACTIONAL LAPLACIAN IN 3 DIMENSIONS

Repositório Institucional da Universidade de Aveiro

Eigenfunctions and Fundamental Solutions for theFractional Laplacian in 3 dimensionsM. Ferreiray;z N. Vieirazy School of Technology and Management,Polytechnic Institute of LeiriaP-2411-901, Leiria, Portugal.z CIDMA - Center for Research and Development in Mathematics and ApplicationsDepartment of Mathematics, University of AveiroCampus Universitario de Santiago, 3810-193 Aveiro, Portugal.E-mails: milton.ferreira@ipleiria.pt,nloureirovieira@gmail.comAbstractRecently there has been a surge of interest in PDEs involving fractional derivatives in dierent elds ofengineering. In this extended abstract we present some of the results developed in [3]. We compute thefundamental solution for the three-parameter fractional Laplace operator (;;) with (; ; ) 2 ]0; 1]3 bytransforming the eigenfunction equation into an integral equation and applying the method of separationof variables. The obtained solutions are expressed in terms of Mittag-Leer functions. For more details werefer the interested reader to [3] where it is also presented an operational approach based on the two Laplacetransform.Keywords: Fractional Laplace operator; Riemann-Liouville fractional derivatives; Eigenfunctions; Fun-damental solution; Mittag-Leer function.MSC 2010: Primary 35R11; Secondary 30G35 26A33 35P10 35A22 35A081 IntroductionThe problems with the fractional Laplacian attracted in the last years a lot of attention, due especially totheir large range of applications. The fractional Laplacian appears in probabilistic framework as well as inmathematical nance as innitesimal generators of the stable Levy processes [1]. One can nd problems involvingthe fractional Laplacian in mechanics and in elastostatics, for example, a Signorini obstacle problem originatingfrom linear elasticity [2].The aim of this paper is to present an explicit expression for the family of eigenfunctions and fundamentalsolutions of the three-parameter fractional Laplace. For the sake of simplicity we restrict ourselves to the threedimensional case, however the results can be generalized for an arbitrary dimension. The two dimensionalcase was already studied in [10]. Connections between fractional calculus and Cliord analysis were consideredrecently in [5, 9].Accepted author's manuscript (AAM) published in Digital Proceedings, 20th International Conference on theApplications of Computer Science and Mathematics in Architecture and Civil Engineering, K. Gurlebeck andT. Lahmer (eds.), July 20{22 2015, Bauhaus-University Weimar, 30{35. The nal publication is available athttps://e-pub.uni-weimar.de/opus4/frontdoor/index/index/docId/2451.12 Preliminaries2.1 Fractional calculus and special functionsLet Da+f(x) denote the fractional Riemann-Liouville derivative of order  > 0 (see [6])(Da+f) (x) =ddxn1 (n  )Z xaf(t)(x  t) n+1 dt; n = [] + 1; a; x > 0: (1)where [] means the integer part of : When 0 <  < 1 then (1) takes the form(Da+f) (x) =ddx1 (1  )Z xaf(t)(x  t) dt: (2)The Riemann-Liouville fractional integral of order  > 0 is given by (see [6])(Ia+f) (x) =1 ()Z xaf(t)(x  t)1  dt; a; x > 0: (3)We recall also the following denition (see [8]):Denition 2.1 A function f 2 L1(a; b) has a summable fractional derivatives Da+f(x) if In a+(x) 2ACn([a; b]), where n = 0; 1; : : : ; n 1 and ACn([a; b]) denote the class of functions f(x), which are continuouslydierentiable on the segment [a; b] up to order n  1 and f (n 1)(x) is absolutely continuous on [a; b].If a function f admits a summable fractional derivative, then the composition of (1) and (3) can be written inthe form (see [8, Thm. 2.4])(Ia+ Da+f) (x) = f(x) n 1Xk=0(x  a) k 1 (  k) In a+(n k 1)(a); n = [] + 1: (4)Nevertheless we note that Da+ Ia+f = f; for any summable function. This is a particular case of a more generalproperty (cf. [7, (2.114)])Da+ Ia+f= D a+ f; 0    : (5)One important function used in this paper is the two-parameter Mittag-Leer function E;(z) [4], which isdened in terms of the power series byE;(z) =1Xn=0zn (n+ );  > 0;  2 R; z 2 C: (6)In particular, the function E;(z) is entire of order  =1 and type  = 1. Two important fractional integral anddierential formulae involving the two-parametric Mittag-Leer function are the following (see [4, p.61,p.87])Ia+ (x  a) 1E;(k(x  a))= (x  a)+ 1E;+(k(x  a)) (7)Da+ (x  a) 1E;(k(x  a))= (x  a)  1E; (k(x  a)) (8)for all  > 0;  > 0;  2 R; k 2 C; a > 0; x > a:The approach developed in Section 3 leads to the solution of linear Abel integral equations of the second kind.Theorem 2.2 [4, Thm. 4.2] Let f 2 L1[a; b];  > 0 and  2 C: Then the integral equationu(x) = f(x) + ()Z xa(x  t) 1u(t) dt; x 2 [a; b]has a unique solutionu(x) = f(x) + Z xa(x  t) 1E;((x  t))f(t) dt: (9)23 Eigenfunctions and fundamental solution of the fractional LaplaceoperatorConsider the eigenfunction equation for the fractional Laplace operator (;;)(;;)u(x; y; z) = u(x; y; z),D1+x+0u(x; y; z) +D1+y+0u(x; y; z) +D1+z+0u(x; y; z) =  u(x; y; z): (10)where  2 C, (; ; ) 2]0; 1]3, (x; y; z) 2  = [x0; X0]  [y0; Y0]  [z0; Z0], x0; y0; z0  0, X0; Y0; Z0 < 1, andu(x; y; z) admits summable fractional derivatives D1+x+0, D1+y+0, D1+z+0. Applying the fractional integral operatorsI1+x+0, I1+y+0and I1+y+0from both sides of (10), taking into account (4) and using Fubini's Theorem, we getI1+y+0I1+z+0u(x; y; z) +I1+x+0I1+z+0u(x; y; z)+I1+x+0I1+y+0u(x; y; z)  I1+x+0I1+y+0I1+z+0u(x; y; z)=(x  x0) 1 ()I1+y+0I1+z+0f0(y; z) +(x  x0) (1 + )I1+y+0I1+z+0f1(y; z)+(y   y0) 1 ()I1 x+0I1+z+0h0(x; z) +(y   y0) (1 + )I1+x+0I1+z+0h1(x; z)+(z   z0) 1 ()I1 x+0I1+y+0g0(x; y) +(z   z0) (1 + )I1+x+0I1+y+0g1(x; y); (11)where we denote the Cauchy's fractional integral conditions byf0(y; z) =I1 x+0u(x0; y; z); f1(y; z) =Dx+0u(x0; y; z); (12)h0(x; z) =I1 y+0u(x; y0; z); h1(x; z) =Dy+0u(x; y0; z); (13)g0(x; y) =I1 z+0u(x; y; z0); g1(x; y) =Dz+0u(x; y; z0): (14)We now assume that u(x; y; z) = u1(x)u2(y)u3(z). Substituting in (11) and taking into account the initialconditions (12), (13), and (14) we obtainu1(x)I1+y+0u2(y) I1+z+0u3(z)+ u2(y)I1+x+0u1(x) I1+z+0u3(z)+ u3(z)I1+x+0u1(x) I1+y+0u2(y)(x; y; z)  I1+x+0u1(x)I1+y+0u2(y)I1+z+0u3(z)= a1(x  x0) 1 ()I1+y+0u2(y) I1+z+0u3(z)+ a2(x  x0) (1 + )I1+y+0u2(y) I1+z+0u3(z)+ b1(y   y0) 1 ()I1+x+0u1(x) I1+z+0u3(z)+ b2(y   y0) (1 + )I1+x+0u1(x) I1+z+0u3(z)+ c1(z   z0) 1 ()I1+x+0u1(x) I1+y+0u2(y)+ c2(z   z0) (1 + )I1+x+0u1(x) I1+y+0u2(y); (15)where ai; bi; ci 2 C, i = 1; 2, are constants dened by the initial conditions (12), (13), and (14). Supposing thatI1+x+0u1(x)I1+y+0u2(y)I1+z+0u3(z) 6= 0, for (x; y; z) 2 , we can divide (15) by this factor. Separating thevariables we get the following three Abel's integral equations of second kind:u1(x)  I1+x+0u1(x) = a1(x  x0) 1 ()+ a2(x  x0) (1 + ); (16)u2(y) + I1+y+0u2(y) = b1(y   y0) 1 ()+ b2(y   y0) (1 + ); (17)u3(z) + (    )I1+z+0u3(z) = c1(z   z0) 1 ()+ c2(z   z0) (1 + ); (18)where ; ;  2 C are constants. We observe that the equalityI1+x+0u1(x)I1+y+0u2(y)I1+z+0u3(z) = 0;agrees with (15), (16), (17), and (18) for at least one point (; ; ). Solving the latter equations using (9) inTheorem 1.2 and after straightforward computations we obtain a family of eigenfunctions.3Theorem 3.1 A family of eigenfunctions of the fractional Laplace operator (;;) is given by u;;(x; y; z) =u1(x) u2(y) u3(z) withu1(x) = a1 (x  x0) 1 E1+; (x  x0)1++ a2 (x  x0) E1+;1+ (x  x0)1+(19)u2(y) = b1 (y   y0) 1 E1+;  (y   y0)1++ b2 (y   y0) E1+;1+  (y   y0)1+ (20)u3(z) = c1 (z   z0) 1 E1+; (    )(z   z0)1++ c2 (z   z0) E1+;1+ (    )(z   z0)1+; (21)where ; ;  2 C are constants.Corollary 3.2 For  = 0; u0;;(x; y; z) = u1(x) u2(y) u3(z) is a family of fundamental solutions for thefractional Laplace operator (;;).Remark 3.3 In the special case of  =  =  = 1 the functions u1, u2 and u3 take the form:u1(x) = a1 cosh (p (x  x0)) + a2psinh (p (x  x0)) ;u2(y) = b1 cos p (y   y0)+b2psin p (y   y0);u3(z) = c1 coshp     (z   z0)+c2p     sinhp     (z   z0):which are the components of the fundamental solution of the Laplace operator in R3 obtained by the method ofseparation of variables.It is also possible to apply an operational approach based on the two dimensional Laplace transform toobtain a complete family of eigenfunctions and fundamental solutions for the fractional Laplace operator. Thiswas done in detail in [3].Acknowledgement: This work was supported by Portuguese funds through the CIDMA - Center forResearch and Development in Mathematics and Applications, and the Portuguese Foundation for Science andTechnology (\FCT{Fundac~ao para a Cie^ncia e a Tecnologia"), within project UID/MAT/ 0416/2013. N. Vieirawas also supported by FCT via the FCT Researcher Program 2014 (Ref: IF/00271/2014).References[1] D. Applebaum: Levy processes: from probability to nance and quantum groups. Notices Am. Math. Soc., 51-No.11,1336-1347, 2004.[2] L.A. Caarelli, S. Salsa and L. Silvestre: Regularity estimates for the solution and the free boundary of the obstacleproblem for the fractional Laplacian. Invent. Math., 171-No.2, 425-461, 2008.[3] M. Ferreira and N. Vieira: Eigenfunctions and fundamental solution of the fractional Laplace and Dirac operators:the Riemann-Liouville case. Submitted.[4] R. Goreno, A.A. Kilbas, F. Mainardi and S. Rogosin: Mittag-Leer functions. Theory and applications. SpringerMonographs in Mathematics, Springer, Berlin, 2014.[5] U. Kahler and N. Vieira: Fractional Cliord analysis. S. Bernstein, U. Kahler, I. Sabadini and F. Sommen eds.Hypercomplex Analysis: New perspectives and applications, Trends in Mathematics, Birkhauser, Basel, 2014.[6] A. Kilbas, H.M. Srivastava and J.J. Trujillo: Theory and applications of fractional dierential equations. North-Holland Mathematics Studies-Vol.204, Elsevier, Amsterdam, 2006.[7] I. Podlubny: Fractional dierential equations. An introduction to fractional derivatives, fractional dierential equa-tions, to methods of their solution and some of their applications. Mathematics in Science and Engineering-Vol.198,Academic Press, San Diego, CA, 1999.4[8] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional integrals and derivatives: theory and apllications. Gordonand Breach, New York, NY, 1993.[9] N. Vieira: Fischer Decomposition and Cauchy-Kovalevskaya extension in fractional Cliord analysis: the Riemann-Liouville case. Proc. Edinb. Math. Soc., II. Ser., in press.[10] S. Yakubovich: Eigenfunctions and fundamental solutions of the fractional two-parameter Laplacian. Int. J. Math.Math. Sci., Article ID 541934, 18p, 2010.5

Eigenfunctions and fundamental solutions for the Fractional Laplacian in 3 dimensions

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Eigenfunctions and fundamental solutions for the Fractional Laplacian in 3 dimensions

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