DSpace Collection: 2011
http://hdl.handle.net/10525/1678
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Matrix-Variate Statistical Distributions and Fractional Calculus
http://hdl.handle.net/10525/1687
Title: Matrix-Variate Statistical Distributions and Fractional Calculus<br/><br/>Authors: Mathai, A.; Haubold, H.<br/><br/>Abstract: A connection between fractional calculus and statistical distributiontheory has been established by the authors recently. Some extensions ofthe results to matrix-variate functions were also considered. In the presentarticle, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional differential equations, Mittag-Leffler functions and Fox H-function appear naturally. Some results connected with generalized Mittag-Leffler density and their asymptotic behavior will be considered. Reference is made to applications in physics, particularly super statistics and nonextensive statistical mechanics.<br/><br/>Description: MSC 2010: 15A15, 15A52, 33C60, 33E12, 44A20, 62E15 Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayFractional Derivatives in Spaces of Generalized Functions
http://hdl.handle.net/10525/1686
Title: Fractional Derivatives in Spaces of Generalized Functions<br/><br/>Authors: Stojanović, Mirjana<br/><br/>Abstract: We generalize the two forms of the fractional derivatives (in Riemann-Liouville and Caputo sense) to spaces of generalized functions using appropriate techniques such as the multiplication of absolutely continuous function by the Heaviside function, and the analytical continuation. As an application, we give the two forms of the fractional derivatives of discontinuous functions in spaces of distributions.<br/><br/>Description: MSC 2010: 26A33, 46Fxx, 58C05 Dedicated to 80-th birthday of Prof. Rudolf GorenfloMaximum Principle and Its Application for the Time-Fractional Diffusion Equations
http://hdl.handle.net/10525/1685
Title: Maximum Principle and Its Application for the Time-Fractional Diffusion Equations<br/><br/>Authors: Luchko, Yury<br/><br/>Abstract: In the paper, maximum principle for the generalized time-fractional diffusion equations including the multi-term diffusion equation and the diffusion equation of distributed order is formulated and discussed. In these equations, the time-fractional derivative is defined in the Caputo sense. In contrast to the Riemann-Liouville fractional derivative, the Caputo fractional derivative is shown to possess a suitable generalization of the extremum principle well-known for ordinary derivative. As an application, the maximum principle is used to get some a priori estimates for solutions of initial-boundary-value problems for the generalized time-fractional diffusion equations and then to prove uniqueness of their solutions.<br/><br/>Description: MSC 2010: 26A33, 33E12, 35B45, 35B50, 35K99, 45K05 Dedicated to Professor Rudolf Gorenfloon the occasion of his 80th anniversaryHamilton’s Principle with Variable Order Fractional Derivatives
http://hdl.handle.net/10525/1684
Title: Hamilton’s Principle with Variable Order Fractional Derivatives<br/><br/>Authors: Atanackovic, Teodor; Pilipovic, Stevan<br/><br/>Abstract: We propose a generalization of Hamilton’s principle in which the minimization is performed with respect to the admissible functions and theorder of the derivation. The Euler–Lagrange equations for such minimization are derived. They generalize the classical Euler-Lagrange equation. Also, a new variational problem is formulated in the case when the order of the derivative is defined through a constitutive equation. Necessary conditions for the existence of the minimizer are obtained. They imply various known results in a special cases.<br/><br/>Description: MSC 2010: 26A33, 70H25, 46F12, 34K37 Dedicated to 80-th birthday of Prof. Rudolf GorenfloNonlinear Time-Fractional Differential Equations in Combustion Science
http://hdl.handle.net/10525/1683
Title: Nonlinear Time-Fractional Differential Equations in Combustion Science<br/><br/>Authors: Pagnini, Gianni<br/><br/>Abstract: The application of Fractional Calculus in combustion science to modelthe evolution in time of the radius of an isolated premixed flame ball ishighlighted. Literature equations for premixed flame ball radius are rederived by a new method that strongly simplifies previous ones. These equations are nonlinear time-fractional differential equations of order 1/2with a Gaussian underlying diffusion process. Extending the analysis toself-similar anomalous diffusion processes with similarity parameter ν/2 > 0, the evolution equations emerge to be nonlinear time-fractional differentialequations of order 1−ν/2 with a non-Gaussian underlying diffusion process.<br/><br/>Description: MSC 2010: 34A08 (main), 34G20, 80A25Fractional Fokker-Planck-Kolmogorov type Equations and their Associated Stochastic Differential Equations
http://hdl.handle.net/10525/1682
Title: Fractional Fokker-Planck-Kolmogorov type Equations and their Associated Stochastic Differential Equations<br/><br/>Authors: Hahn, Marjorie; Umarov, Sabir<br/><br/>Abstract: There is a well-known relationship between the Itô stochastic differential equations (SDEs) and the associated partial differential equations called Fokker-Planck equations, also called Kolmogorov equations. The Brownian motion plays the role of the basic driving process for SDEs. This paper provides fractional generalizations of the triple relationship between the driving process, corresponding SDEs and deterministic fractional order Fokker-Planck-Kolmogorov type equations.<br/><br/>Description: MSC 2010: 26A33, 35R11, 35R60, 35Q84, 60H10 Dedicated to 80-th anniversary of Professor Rudolf GorenfloInverse Problem for Fractional Diffusion Equation
http://hdl.handle.net/10525/1681
Title: Inverse Problem for Fractional Diffusion Equation<br/><br/>Authors: Tuan, Vu Kim<br/><br/>Abstract: We prove that by taking suitable initial distributions only finitely many measurements on the boundary are required to recover uniquely the diffusion coefficient of a one dimensional fractional diffusion equation. If a lower bound on the diffusion coefficient is known a priori then even only two measurements are sufficient. The technique is based on possibility of extracting the full boundary spectral data from special lateral measurements.<br/><br/>Description: MSC 2010: 26A33, 33E12, 34K29, 34L15, 35K57, 35R30On a 3D-Hypersingular Equation of a Problem for a Crack
http://hdl.handle.net/10525/1680
Title: On a 3D-Hypersingular Equation of a Problem for a Crack<br/><br/>Authors: Samko, Stefan<br/><br/>Abstract: We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation.<br/><br/>Description: MSC 2010: 45DB05, 45E05, 78A45Professor Rudolf Gorenflo and his Contribution to Fractional Calculus
http://hdl.handle.net/10525/1679
Title: Professor Rudolf Gorenflo and his Contribution to Fractional Calculus<br/><br/>Authors: Luchko, Yury; Mainardi, Francesco; Rogosin, Sergei<br/><br/>Abstract: This paper presents a brief overview of the life story and professionalcareer of Prof. R. Gorenflo - a well-known mathematician, an expert inthe field of Differential and Integral Equations, Numerical Mathematics,Fractional Calculus and Applied Analysis, an interesting conversationalpartner, an experienced colleague, and a real friend. Especially his role inthe modern Fractional Calculus and its applications is highlighted.<br/><br/>Description: MSC 2010: 26A33 Dedicated to Professor Rudolf Gorenflo on the occasion of his 80th anniversary