Time distributed-order diffusion-wave equation. I. Volterra-type equation

Atanacković, Teodor M.; Pilipović, Stevan; Zorica, Dušan

doi:10.1098/rspa.2008.0445

Cited by 80 publications

(43 citation statements)

References 17 publications

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“…Recall the known results concerning the two forms of distributional fractional derivatives from [1], [2]…”

Section: From Classical To Distributional Approach To the Two Forms Omentioning

confidence: 99%

“…c 2011 Diogenes Co., Sofia pp. 125-137 , DOI: 10.2478/s13540-011-0009-5 FC in spaces of distributions S + and D + as well as the distributions with compact support E is employed in papers [1], [2]. Two forms of fractional derivatives are embedded into the space of tempered distributions to obtain a framework for solving distributional diffusion-wave phenomena.…”

Section: Introductionmentioning

confidence: 99%

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Fractional derivatives in spaces of generalized functions

Stojanović

2011

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Dedicated to 80-th birthday of Prof. Rudolf GorenfloWe generalize the two forms of the fractional derivatives (in RiemannLiouville 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.MSC 2010 : 26A33, 46Fxx, 58C05

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“…Recall the known results concerning the two forms of distributional fractional derivatives from [1], [2]…”

Section: From Classical To Distributional Approach To the Two Forms Omentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional derivatives in spaces of generalized functions

Stojanović

2011

fcaa

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“…To show the uniqueness of the solution of the problem, an appropriate maximum principle for the generalized time-fractional diffusion equation of distributed order was formulated and proved there. In [1] and [2], time-fractional distributed order diffusion-wave equations have been considered in spaces of generalized functions.…”

Section: Introductionmentioning

confidence: 99%

Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density

Gorenflo

Luchko

Stojanović

2013

fcaa

128

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In this paper, the Cauchy problem for the spatially one-dimensional distributed order diffusion-wave equationis considered. Here, the time-fractional derivative D β t is understood in the Caputo sense and p(β) is a non-negative weight function with support somewhere in the interval [0,2]. By employing the technique of the Fourier and Laplace transforms, a representation of the fundamental solution of the Cauchy problem in the transform domain is obtained. The main focus is on the interpretation of the fundamental solution as a probability density function of the space variable x evolving in time t. In particular, the fundamental solution of the time-fractional distributed order wave equation (p(β) ≡ 0, 0 ≤ β < 1) is shown to be non-negative and normalized. In the proof, properties of the completely monotone functions, the Bernstein functions, and the Stieltjes functions are used.

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“…An abstract theory of integral equations, including equations of this type, can be found in [30]. This paper is devoted to the study of systems of the type (1 ≤ i ≤ n, q ≥ 1): (especially smoothness results), but we rather refer to [2][3][4]6,19,30]. However, for n ≥ 1, we obtain an estimate of the type: |R(ξ, λ)| ≤ K/|λ| 1−σ (0 ≤ σ ≤ (1/2) sup[α i+1 − α i ]) on the resolvent matrix R of system (1.1).…”

Section: Introductionmentioning

confidence: 99%