Introduction to the Theory and Application of the Laplace Transformation

Doetsch, Gustav; Debnath, Lokenath

doi:10.1007/978-3-642-65690-3

Cited by 477 publications

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“…De®nition 5.2. d n 0 will stand for the tempered distribution whose Laplace transform is z n with n e R. Some known properties of this family of distributions are summarized in the following proposition [13]. It can be proved [25] that in this de®nition it is su½cient to consider any family ff s g sb0 r D such that y 0 f s t dt 3 1 when s 3 0.…”

Section: An Application: Fractional Powersmentioning

confidence: 97%

“…d n 0 with n e N f0g is a distribution with compact support and Ld n 0 z hd n 0 Y e z i hd 0 Y À1 n Àz n e z i z n X For n e R, we denote by d n 0 the tempered distribution such that Ld n 0 z z n [13]. Now we recall certain basic properties of the vector-valued Riemann-Liouville fractional derivation and integration, and the vector-valued Laplace transform.…”

Section: Fractional Derivation and The Laplace Transformmentioning

confidence: 99%

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a-times integrated semigroups and fractional derivation

Miana¹

2002

Forum Mathematicum

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By applying fractional integration and derivation to the vector-valued Laplace transform, a functional calculus for -times integrated semigroups is obtained. This functional calculus is related to smooth distribution semigroups. As application, fractional powers of its in®nitesimal generator are de®ned.2000 Mathematics Subject Classi®cation: 47D62, 26A33. IntroductionOne of the concepts used in the last ®fteen years to study the Abstract Cauchy Problemis the n-times integrated semigroup with n e N [32], [19]. These semigroups are related to other notions: C-semigroups [9], spectral distributions [4], holomorphic semigroups [2]. They have been used with C-semigroups [39], cosine functions [26], [15] or sine functions [23]. In 1991 Hieber [18] introduced -times integrated semigroups with e R . In an informal way, the``formal solution'' of the Cauchy Problem is smoothed integrating it times in the fractional sense of Riemann-Liouville. Some works about these semigroups were published later on, see for instance [29], [8] (about spectral mapping theorem), [37] (in locally convex spaces).In this paper, we investigate the deep relationship between these semigroups and

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Section: An Application: Fractional Powersmentioning

confidence: 97%

Section: Fractional Derivation and The Laplace Transformmentioning

confidence: 99%

a-times integrated semigroups and fractional derivation

Miana¹

2002

Forum Mathematicum

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“…Through a differentiation theorem [5], it can be shown that these moments can also be expanded in terms of the Laplace transform of the temperature profile, DT(x,s), through…”

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confidence: 99%

Determination of thermal parameters of one-dimensional nanostructures through a thermal transient method

Arriagada

Bandaru

2009

J Therm Anal Calorim

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We present an improved methodology for a thermal transient method enabling simultaneous measurement of thermal conductivity and specific heat of nanoscale structures with one-dimensional heat flow. The temporal response of a sample to finite duration heat pulse inputs for both short (1 ns) and long (5 ls) pulses is analyzed and exploited to deduce the thermal properties. Excellent agreement has been obtained between the recovered physical parameters and computational simulations through choosing an optimized pulse width.

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“…Decomposing 1/p(λ) according to (3.4) and taking the inverse Laplace transform gives the result (see [1], Equation (21) p. 81). See also [2], p. 141-142, for another proof using distribution theory in the convolution algebra D + .…”

Section: Downloaded By [University Of Lethbridge] At 05:51 09 Octobermentioning

confidence: 99%

“…This method is generally regarded as too difficult to implement in a first course on differential equations. Students become aware of it only later, as an application of the theory of the Laplace transform [1] or of distribution theory. [2] An alternative approach which is sometimes used consists in developing the theory of linear systems first, considering then linear equations of order n as a particular case of this theory.…”

Section: Introductionmentioning

confidence: 99%

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

Camporesi

2015

International Journal of Mathematical Education in Science and

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We present an approach to the impulsive response method for solving linear constantcoefficient ordinary differential equations of any order based on the factorization of the differential operator. The approach is elementary, we only assume a basic knowledge of calculus and linear algebra. In particular, we avoid the use of distribution theory, as well as of the other more advanced approaches: Laplace transform, linear systems, the general theory of linear equations with variable coefficients and variation of parameters. The approach presented here can be used in a first course on differential equations for science and engineering majors.

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Introduction to the Theory and Application of the Laplace Transformation

Cited by 477 publications

References 0 publications

a-times integrated semigroups and fractional derivation

a-times integrated semigroups and fractional derivation

Determination of thermal parameters of one-dimensional nanostructures through a thermal transient method

A fresh look at linear ordinary differential equations with constant coefficients. Revisiting the impulsive response method using factorization

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