DSpace Collection: 2007
http://hdl.handle.net/10525/1245
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On a Class of Fractional Type Integral Equations in Variable Exponent Spaces
http://hdl.handle.net/10525/1323
Title: On a Class of Fractional Type Integral Equations in Variable Exponent Spaces<br/><br/>Authors: Rafeiro, Humberto; Samko, Stefan<br/><br/>Abstract: We obtain a criterion of Fredholmness and formula for the Fredholm index of a certain class of one-dimensional integral operators M with a weak singularity in the kernel, from the variable exponent Lebesgue space L^p(·) ([a, b], ?) to theSobolev type space L^α,p(·) ([a, b], ?) of fractional smoothness. We also give formulas of closed form solutions ϕ ∈ L^p(·)of the 1st kind integral equation M0ϕ = f, known as the generalized Abel equation, with f ∈ L^α,p(·), in dependence on thevalues of the variable exponent p(x) at the endpoints x = a and x = b.<br/><br/>Description: 2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30Generalized Fractional Evolution Equation
http://hdl.handle.net/10525/1322
Title: Generalized Fractional Evolution Equation<br/><br/>Authors: Da Silva, J. L.; Erraoui, M.; Ouerdiane, H.<br/><br/>Abstract: In this paper we study the generalized Riemann-Liouville (resp. Caputo)time fractional evolution equation in infinite dimensions. We show that theexplicit solution is given as the convolution between the initial conditionand a generalized function related to the Mittag-Leffler function. The fundamental solution corresponding to the Riemann-Liouville time fractionalevolution equation does not admit a probabilistic representation while forthe Caputo time fractional evolution equation it is related to the inversestable subordinators.<br/><br/>Description: 2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20On q–Analogues of Caputo Derivative and Mittag–Leffler Function
http://hdl.handle.net/10525/1321
Title: On q–Analogues of Caputo Derivative and Mittag–Leffler Function<br/><br/>Authors: Rajkovic, Predrag; Marinkovic, Sladjana; Stankovic, Miomir<br/><br/>Abstract: Based on the fractional q–integral with the parametric lower limit ofintegration, we consider the fractional q–derivative of Caputo type. Especially, its applications to q-exponential functions allow us to introduceq–analogues of the Mittag–Leffler function. Vice versa, those functions canbe used for defining generalized operators in fractional q–calculus.<br/><br/>Description: Mathematics Subject Classification: 33D60, 33E12, 26A33Inversion Formulas for the q-Riemann-Liouville and q-Weyl Transforms Using Wavelets
http://hdl.handle.net/10525/1320
Title: Inversion Formulas for the q-Riemann-Liouville and q-Weyl Transforms Using Wavelets<br/><br/>Authors: Fitouhi, Ahmed; Bettaibi, Néji; Binous, Wafa<br/><br/>Abstract: This paper aims to study the q-wavelets and the continuous q-wavelettransforms, associated with the q-Bessel operator for a fixed q ∈]0, 1[. Usingthe q-Riemann-Liouville and the q-Weyl transforms, we give some relationsbetween the continuous q-wavelet transform, studied in [3], and the continuous q-wavelet transform associated with the q-Bessel operator, and wededuce formulas which give the inverse operators of the q-Riemann-Liouvilleand the q-Weyl transforms.<br/><br/>Description: Mathematics Subject Classification: 42A38, 42C40, 33D15, 33D60Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey
http://hdl.handle.net/10525/1319
Title: Time-Fractional Derivatives in Relaxation Processes: A Tutorial Survey<br/><br/>Authors: Mainardi, Francesco; Gorenflo, Rudolf<br/><br/>Abstract: The aim of this tutorial survey is to revisit the basic theory of relaxationprocesses governed by linear differential equations of fractional order. Thefractional derivatives are intended both in the Rieamann-Liouville senseand in the Caputo sense. After giving a necessary outline of the classicatheory of linear viscoelasticity, we contrast these two types of fractionaderivatives in their ability to take into account initial conditions in theconstitutive equations of fractional order. We also provide historical noteson the origins of the Caputo derivative and on the use of fractional calculusin viscoelasticity.<br/><br/>Description: 2000 Mathematics Subject Classification: 26A33, 33E12, 33C60, 44A10,45K05, 74D05,Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative
http://hdl.handle.net/10525/1318
Title: Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative<br/><br/>Authors: Luchko, Yury; Trujillo, Juan<br/><br/>Abstract: The Caputo fractional derivative is one of the most used definitions of afractional derivative along with the Riemann-Liouville and the Grünwald-Letnikov ones. Whereas the Riemann-Liouville definition of a fractionalderivative is usually employed in mathematical texts and not so frequentlyin applications, and the Grünwald-Letnikov definition – for numerical approximation of both Caputo and Riemann-Liouville fractional derivatives, the Caputo approach appears often while modeling applied problems bymeans of fractional derivatives and fractional order differential equations.In the mathematical texts and applications, the so called Erdélyi-Kober(E-K) fractional derivative, as a generalization of the Riemann-Liouvillefractional derivative, is often used, too. In this paper, we investigate someproperties of the Caputo-type modification of the Erdélyi-Kober fractionalderivative. The relation between the Caputo-type modification of the E-Kfractional derivative and the classical E-K fractional derivative is the sameas the relation between the Caputo fractional derivative and the Riemann-Liouville fractional derivative, i.e. the operations of integration and differentiation are interchanged in the corresponding definitions. Here, some new properties of the classical Erdélyi-Kober fractional derivative and the respective ones of its Caputo-type modification are presented together.<br/><br/>Description: 2000 Math. Subject Classification: 26A33; 33E12, 33E30, 44A15, 45J05Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue
http://hdl.handle.net/10525/1317
Title: Caputo Derivatives in Viscoelasticity: A Non-Linear Finite-Deformation Theory for Tissue<br/><br/>Authors: Freed, Alan; Diethelm, Kai<br/><br/>Abstract: The popular elastic law of Fung that describes the non-linear stress-strain behavior of soft biological tissues is extended into a viscoelastic material model that incorporates fractional derivatives in the sense of Caputo. This one-dimensional material model is then transformed into athree-dimensional constitutive model that is suitable for general analysis.The model is derived in a configuration that differs from the current, orspatial, configuration by a rigid-body rotation; it being the polar configuration. Mappings for the fractional-order operators of integration and differentiation between the polar and spatial configurations are presented as a theorem. These mappings are used in the construction of the proposed viscoelastic model.<br/><br/>Description: Mathematics Subject Classification: 26A33, 74B20, 74D10, 74L15On q-Laplace Transforms of the q-Bessel Functions
http://hdl.handle.net/10525/1316
Title: On q-Laplace Transforms of the q-Bessel Functions<br/><br/>Authors: Purohit, S.; Kalla, S.<br/><br/>Abstract: The present paper deals with the evaluation of the q-Laplace transformsof a product of basic analogues of the Bessel functions. As applications,several useful special cases have been deduced.<br/><br/>Description: Mathematics Subject Classification: 33D15, 44A10, 44A20Fopid Controller Design for Robust Performance Using Particle Swarm Optimization
http://hdl.handle.net/10525/1315
Title: Fopid Controller Design for Robust Performance Using Particle Swarm Optimization<br/><br/>Authors: Zamani, Majid; Karimi-Ghartemani, Masoud; Sadati, Nasser<br/><br/>Abstract: This paper proposes a novel method to design an H∞ -optimal fractional order PID (FOPID) controller with ability to control the transient,steady-state response and stability margins characteristics. The method uses particle swarm optimization algorithm and operates based on minimizing a general cost function. Minimization of the cost function is carried outsubject to the H∞ -norm; this norm is also included in the cost function toachieve its lower value. The method is applied to a phase-locked-loop motorspeed system and an electromagnetic suspension system as two examples toillustrate the design procedure and verify performance of the proposed controller. The results show that the proposed method is capable of improving system responses as compared to the conventional H∞ -optimal controller while still maintains the H∞ -optimality of the solutions.<br/><br/>Description: Mathematics Subject Classification: 26A33; 93C15, 93C55, 93B36, 93B35,93B51; 03B42; 70Q05; 49N05Representations of Inverse Functions by the Integral Transform with the Sign Kernel
http://hdl.handle.net/10525/1314
Title: Representations of Inverse Functions by the Integral Transform with the Sign Kernel<br/><br/>Authors: Yamada, Masato; Matsuura, Tsutomu; Saitoh, Saburou<br/><br/>Abstract: In this paper we give practical and numerical representations of inversefunctions by using the integral transform with the sign kernel, and showcorresponding numerical experiments by using computers. We derive a verysimple formula from a general idea for the representation of the inversefunctions, based on the theory of reproducing kernels.<br/><br/>Description: Mathematics Subject Classification: Primary 30C40Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations
http://hdl.handle.net/10525/1313
Title: Smoothness Properties of Solutions of Caputo-Type Fractional Differential Equations<br/><br/>Authors: Diethelm, Kai<br/><br/>Abstract: We consider ordinary fractional differential equations with Caputo-typedifferential operators with smooth right-hand sides. In various places inthe literature one can find the statement that such equations cannot havesmooth solutions. We prove that this is wrong, and we give a full characterization of the situations where smooth solutions exist. The results canbe extended to a class of weakly singular Volterra integral equations.<br/><br/>Description: Mathematics Subject Classification: 26A33, 34A25, 45D05, 45E10On a Differential Equation with Left and Right Fractional Derivatives
http://hdl.handle.net/10525/1312
Title: On a Differential Equation with Left and Right Fractional Derivatives<br/><br/>Authors: Atanackovic, Teodor; Stankovic, Bogoljub<br/><br/>Abstract: We treat the fractional order differential equation that contains the leftand right Riemann-Liouville fractional derivatives. Such equations arise asthe Euler-Lagrange equation in variational principles with fractional derivatives.We reduce the problem to a Fredholm integral equation and constructa solution in the space of continuous functions. Two competing approachesin formulating differential equations of fractional order in Mechanics andPhysics are compared in a specific example. It is concluded that only thephysical interpretation of the problem can give a clue which approach shouldbe taken.<br/><br/>Description: Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05Integral Representations of Generalized Mathieu Series Via Mittag-Leffler Type Functions
http://hdl.handle.net/10525/1311
Title: Integral Representations of Generalized Mathieu Series Via Mittag-Leffler Type Functions<br/><br/>Authors: Tomovski, Živorad<br/><br/>Abstract: The main purpose of this paper is to present a number of potentiallyuseful integral representations for the generalized Mathieu series as well asfor its alternating versions via Mittag-Leffler type functions.<br/><br/>Description: Mathematics Subject Classification: 33C05, 33C10, 33C20, 33C60, 33E12,33E20, 40A30On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials
http://hdl.handle.net/10525/1310
Title: On the Equivalence of the Riemann-Liouville and the Caputo Fractional Order Derivatives in Modeling of Linear Viscoelastic Materials<br/><br/>Authors: Bagley, Ron<br/><br/>Abstract: In the process of constructing empirical mathematical models of physical phenomena using the fractional calculus, investigators are usually faced with the choice of which definition of the fractional derivative to use, theRiemann-Liouville definition or the Caputo definition. This investigationpresents the case that, with some minimal restrictions, the two definitionsproduce completely equivalent mathematical models of the linear viscoelastic phenomenon.<br/><br/>Description: Mathematics Subject Classification: 26A33Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders
http://hdl.handle.net/10525/1309
Title: Comparative Analysis of Viscoelastic Models Involving Fractional Derivatives of Different Orders<br/><br/>Authors: Rossikhin, Yuriy; Shitikova, Marina<br/><br/>Abstract: In this paper, a comparative analysis of the models involving fractionalderivatives of di®erent orders is given. Such models of viscoelastic materialsare widely used in various problems of mechanics and rheology. Rabotnov'shereditarily elastic rheological model is considered in detail. It is shownthat this model is equivalent to the rheological model involving fractionalderivatives in the stress and strain with the orders proportional to a certainpositive value less than unit. In the scienti¯c literature such a model isreferred to as Koeller's model. Inversion of Rabotnov's model developedby himself based on algebra of operators results in similar rheological dependences. Inversion of Koeller's model carried out using Miller's theoremcoincides inherently with Rabotnov's inversion procedure.<br/><br/>Description: Mathematics Subject Classification: 74D05, 26A33Some Fractional Extensions of the Temperature Field Problem in Oil Strata
http://hdl.handle.net/10525/1294
Title: Some Fractional Extensions of the Temperature Field Problem in Oil Strata<br/><br/>Authors: Boyadjiev, Lyubomir<br/><br/>Abstract: This survey is devoted to some fractional extensions of the incompletelumped formulation, the lumped formulation and the formulation of Lauwerier of the temperature field problem in oil strata. The method of integral transforms is used to solve the corresponding boundary value problems forthe fractional heat equation. By using Caputo’s differintegration operatorand the Laplace transform, new integral forms of the solutions are obtained.In each of the different cases the integrands are expressed in terms of a convolution of two special functions of Wright’s type.