A Certain Class of Laplace Transforms with Applications to Reaction and Reaction-Diffusion Equations

Mathai, A. M.; Saxena, R. K.; Haubold, H. J.

doi:10.1007/s10509-006-9188-7

Cited by 25 publications

(23 citation statements)

References 19 publications

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“…Analytical solutions are highly appreciated, but only a few types of DE allow explicit, if any, exact analytical solutions. Recently some fractional ordinary DE and partial differential equations (PDE) were analyzed and analytically solved in [24][25][26][27][28][29][30][31][32][33][34][35][36][37]. They benefit from the use of special functions [38][39][40].…”

Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Zhukovsky

2016

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Abstract:We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power-exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.

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Section: Introductionmentioning

confidence: 99%

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Zhukovsky

2016

Axioms

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“…The solution of the previous Fractional Differential Equation (FDE) is given by: [42,43]. This solution is interesting as it shows that that scale factor increases in time and depends on the parameter α.…”

Section: Frw Cosmology With Constant Hubble Parametermentioning

confidence: 99%

“…is the generalized Mittag-Leffler function [42,43]. In that aspect, the following relation is useful:…”

Section: 2mentioning

confidence: 99%

A Cosmology Governed by a Fractional Differential Equation and the Generalized Kilbas-Saigo-Mittag-Leffler Function

El-Nabulsi

2015

Int J Theor Phys

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In this paper we discussed the FRW cosmology characterized by a scale factor obeying different independent types of fractional differential equations with solutions givens in terms of Mittag-Leffler and generalized Kilbas-Saigo-Mittag-Leffler functions. Both types of fractional operators: the Riemann-Liouville fractional integral and the Caputo fractional derivative were considered independently. Some new cosmological features were observed and discussed accordingly.

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“…This is a special case of a general class of Laplace transforms considered in Mathai, Saxena, and Haubold [35], Saxena, Mathai, and Haubold [40], and Haubold, Mathai, and Saxena [12].…”

Section: −[φ(T)]mentioning

confidence: 99%

Matrix-variate statistical distributions and fractional calculus

Mathai

Haubold

2011

fcaa

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Dedicated to Professor R. Gorenflo on the occasion of his 80th birthdayA connection between fractional calculus and statistical distribution theory has been established by the authors recently. Some extensions of the results to matrix-variate functions were also considered. In the present article, more results on matrix-variate statistical densities and their connections to fractional calculus will be established. When considering solutions of fractional differential equations, Mittag-Leffler functions and Fox Hfunction appear naturally. Some results connected with generalized MittagLeffler density and their asymptotic behavior will be considered. Reference is made to applications in physics, particularly superstatistics and nonextensive statistical mechanics.MSC 2010 : 15A15, 15A52, 33C60, 33E12, 44A20, 62E15

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A Certain Class of Laplace Transforms with Applications to Reaction and Reaction-Diffusion Equations

Cited by 25 publications

References 19 publications

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

Operational Approach and Solutions of Hyperbolic Heat Conduction Equations

A Cosmology Governed by a Fractional Differential Equation and the Generalized Kilbas-Saigo-Mittag-Leffler Function

Matrix-variate statistical distributions and fractional calculus

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