Su un problema di Abel

Tonelli, Leonida

doi:10.1007/bf01459094

Cited by 26 publications

(7 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Tonelli [Ton28] sowed that if h is absolutely continuous on [0, T ], then a unique solution to J α u(t) = h(t) is given by (6.37) and u ∈ L 1 (0, T ). Tonelli [Ton28] sowed that if h is absolutely continuous on [0, T ], then a unique solution to J α u(t) = h(t) is given by (6.37) and u ∈ L 1 (0, T ).…”

Section: Lemma 68 Let H(t) Be a Continuously Differentiable Functiomentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

View full text Add to dashboard Cite

Fractional order differential equations are an efficient tool to model various processes arising in science and engineering. Fractional models adequately reflect subtle internal properties, such as memory or hereditary properties, of complex processes that the classical integer order models neglect. In this chapter we will discuss the theoretical background of fractional modeling, that is the fractional calculus, including recent developments -distributed and variable fractional order differential operators.Fractional order derivatives interpolate integer order derivatives to real (not necessarily fractional) or complex order derivatives. There are different types of fractional derivatives not always equivalent. The first attempt to develop the fractional calculus systematically was taken by Liouville (1832) and Riemann (1847) in the first half of the nineteenth century, even though discussions on non-integer order derivatives had been started long ago. 1 In the 1870s Letnikov and Grünwald independently used an approach for the definition of the fractional order derivative and integral different from that of Riemann and Liouville. The Cauchy problem for fractional order differential equations with the Riemann-Liouville derivative is not well posed (Section 3.3), that is the Cauchy problem in this case is unphysical. In the 1960s Caputo and Djrbashian introduced independently, so-called, a regularization of the Riemann-Liouville fractional derivative, which was later named a fractional derivative in the sense of Caputo-Djrbashian (Section 3.5). The usefulness of the Caputo-Djrbashian derivative is that the Cauchy problem for fractional order differential equations with the Caputo-Djrbashian derivative is well posed. In the operators language one can write the latter in the form DJ = I, where I is the identity operator, which means that the operator D is a left inverse to the operator J. One can easily check that D is not a right inverse to J, since, according to the same fundamental theorem of calculus, for any differentiable function f the equality JD f (t) = f (t) − f (0) holds. The similar relations are true by induction for operators D n and J n , where D n = d n dt n , "n-th derivative," and J n is the n-fold integration operator. Namely,andThus D n is the left inverse to J n , and is the right inverse to J n in the class of functions satisfying additional conditions: f (k) (0) = 0, k = 0,..., n − 1. These relations between "differentiation" and "integration" operators valid for n = 1, 2, . . . , form the basis for the definitions of fractional derivatives in the sense of RiemannLiouville and Caputo-Djrbashian, as soon as the fractional order integration operator is defined. Fractional order integration operator 123 Fractional order integration operatorIn this section we introduce the fractional order integration operator of order α > 0 (ℜ(α) > 0). One can verify (by changing order of integration) that the n-fold integration operatorTaking into account the relationship Γ (n) = (n − 1)!, whereis the Euler's ...

show abstract

Section: Lemma 68 Let H(t) Be a Continuously Differentiable Functiomentioning

confidence: 99%

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Umarov

2015

Developments in Mathematics

View full text Add to dashboard Cite

show abstract

“…) is related to Abel's integral equation [26,27]. We use again the Laplace transform in time t, as is defined in (3.10).…”

Section: Preliminary Resultsmentioning

confidence: 99%

Asymptotic stability of viscous shocks in the modular Burgers equation

Le,

Pelinovsky,

Poullet

2020

Preprint

View full text Add to dashboard Cite

Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface position. Exponential weights in space are imposed on the initial data of general perturbations in order to gain the asymptotic decay of perturbations in time. We give numerical illustrations of asymptotic stability of the viscous shocks under general perturbations.

show abstract

“…we have In this case we arrive at the classical Abel equation. Assuming F(x) to be of class C 1 , we have from [9], since F(0) = 0,σ…”

Section: Determination Of the Electric Conductivity In The Thermistormentioning

confidence: 99%

Application of the Abel integral equation to an inverse problem in thermoelectricity

Cimatti

2009

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

Su un problema di Abel

Cited by 26 publications

References 0 publications

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols

Asymptotic stability of viscous shocks in the modular Burgers equation

Application of the Abel integral equation to an inverse problem in thermoelectricity

Contact Info

Product

Resources

About