Evolutionary Integral Equations and Applications

Prüß, Jan

doi:10.1007/978-3-0348-0499-8

Cited by 426 publications

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“…For analysis of the evolution equations of the type mentioned above, methods and tools of fractional calculus, integral transforms, and higher transcendental functions have been employed in the pioneering papers [Wyss (1986)], [Schneider and Wyss (1989)], [Fujita (1990)], [Gorenflo and Rutman (1994)], [Kochubei (1989), Kochubei (1990)], and in the book [Prüss (1993)]. We also mention the papers [Mainardi (1994), Mainardi (1996a), Mainardi (1996b)] and [Mainardi and Tomirotti (1995)], where fundamental solutions of the evolution equations related to phenomena intermediate between diffusion and wave propagation have been expressed in terms of some auxiliary functions of the Wright type that sometimes are referred to as Mainardi functions, see i.e.…”

Section: Introductionmentioning

confidence: 99%

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Luchko

Mainardi

Povstenko

2013

Computers & Mathematics with Applications

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In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order α, 1 < α < 2 is revisited. This equation interpolates between the diffusion and the wave equations that behave quite differently regarding their response to a localized disturbance: whereas the diffusion equation describes a process, where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. For the timefractional diffusion-wave equation, the propagation speed of a disturbance is infinite, but its fundamental solution possesses a maximum that disperses with a finite speed. In this paper, the fundamental solution of the Cauchy problem for the time-fractional diffusion-wave equation, its maximum location, maximum value, and other important characteristics are investigated in detail. To illustrate analytical formulas, results of numerical calculations and plots are presented. Numerical algorithms and programs used to produce plots are discussed.

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Section: Introductionmentioning

confidence: 99%

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Luchko

Mainardi

Povstenko

2013

Computers & Mathematics with Applications

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“…Here, we recall some definitions and basic theorems, given in [15], concerning abstract Volterra integral equations.…”

Section: Preliminariesmentioning

confidence: 99%

“…The generation theorem ( [15], Theorem 1.3) states that Problem (15) is well-posed with solution operator S(t) satisfying S(t) ≤ M, t ≥ 0, if and only if:…”

Section: Preliminariesmentioning

confidence: 99%

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Subordination Principle for a Class of Fractional Order Differential Equations

Bazhlekova

2015

Mathematics

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The fractional order differential equationis studied, where A is an operator generating a strongly continuous one-parameter semigroup on a Banach space X, D α t is the Riemann-Liouville fractional derivative of order α ∈ (0, 1), γ > 0 and f is an X-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the C 0 -semigroup, generated by the operator A. As an example, the Rayleigh-Stokes problem for a generalized second-grade fluid is considered.

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“…In section 3, we show a general result on existence and uniqueness of solutions for (1.1). In section 4, we relate equation (1.1) to an integral Volterra equation of convolution type to finally link the (α, µ)-regularized families with the theory of resolvent families for evolutionary integral equations developed in reference [20].…”

Section: Introductionmentioning

confidence: 99%

An operator theoretical approach to a class of fractional order differential equations

Lizama

2011

Applied Mathematics Letters

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Abstract. We propose a general method to obtain the representation of solutions for linear fractional order differential equations based on the theory of (a, k)-regularized families of operators. We illustrate the method in case of the fractional order differential equationwhere A is an unbounded closed operator defined on a Banach space X and f is a X-valued function. IntroductionWe study in this paper existence of solutions for fractional order differential equations of the form Fractional order differential equations is a subject of increasing interest in different contexts and areas of research, see e. Note that when A = 2∆− 2 ∆ 2 (where ∆ is the Laplace operator) α = 1 and µ = 1/ 2 the above equation was recently considered by Nane [18, Theorem 2.2]. In particular, in case µ = 0, α = 1, A = −∆ 2 and u (0) = − 2 π ∆u 0 with u 0 ∈ D(∆), the equation, t > 0, has been studied in [18, Theorem 2.1] in connection with PDE's and iterated processes. A precise interplay between entire and fractional order differential equations was investigated in reference [10].Observe that one cannot apply semigroup theory directly to solve problem (1.1) in terms of a variation of constant formula. However, our methods based on the theory of (a, k)-regularized families allows us to construct a solution. In fact, we will show that it is possible to give an abstract operator approach to equation (1.1) by defining an ad-hoc family of strongly continuous operators. Then, we are able to show that the solution of equation (1.1) can be written in terms of a kind of variation of constants formula (cf. Theorem 3.1 below). We believe that the method indicated in this paper can be used to handle many classes of linear fractional order differential equations. Our method can be viewed as an extension of the ideas in reference [4] to state the existence of solutions for the abstract fractional order Cauchy problem.Our plan is as follows: In section 2, we introduce some preliminaries on fractional order derivatives, the Mittag-Leffler function and the concept of (α, µ)-regularized families, which give us the necessary 2000 Mathematics Subject Classification. 26A33, 47D06, 45N05.

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Evolutionary Integral Equations and Applications

Cited by 426 publications

References 0 publications

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Subordination Principle for a Class of Fractional Order Differential Equations

An operator theoretical approach to a class of fractional order differential equations

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