Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications,... more Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317-328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Numerical Functional Analysis and Optimization, 2007
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf ... more Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
The aim of the present paper is to obtain the solution of certain integral equations by using P α... more The aim of the present paper is to obtain the solution of certain integral equations by using P α-transform. The concept of P α-transform is introduced by Kumar [11]. The P α-transform is binomial type containing many classes of transforms including the Laplace transform. We have found the solution of fractional Volterra equation with Caputo fractional derivative using P α-transform. Also the solution of non-homogeneous time fractional heat equation in spherical domain with Caputo derivative has been found. The results for the classical Laplace transform are retrieved by letting α → 1.
Possible modification in the velocity distribution in the non-resonant reaction rates leads to an... more Possible modification in the velocity distribution in the non-resonant reaction rates leads to an extended reaction rate probability integral. The closed form representation for these thermonuclear functions are used to obtain the stellar luminosity and neutrino emission rates. The composite parameter C that determines the standard nuclear reaction rate through the Maxwell-Boltzmann energy distribution is extended to C * by the extended reaction rates through a more general distribution than the Maxwell-Boltzmann distribution. The new distribution is obtained by the pathway model introduced by Mathai in 2005 [Linear Algebra and Its Applications, 396, 317-328]. Simple analytic models considered by various authors are utilized for evaluating stellar luminosity and neutrino emission rates and are obtained in generalized special functions such as Meijer's G-function and Fox's H-function. The standard and extended non-resonant thermonuclear functions are compared by plotting them. Behavior of the new energy distribution, more general than Maxwell-Boltzmann is also studied.
The method for the evaluation of the non-resonant thermonuclear function in the Maxwell-Boltzmann... more The method for the evaluation of the non-resonant thermonuclear function in the Maxwell-Boltzmann case with depleted tail is discussed. Closed forms of the analytical results are obtained in computational format, and written in terms of the H-function in two variables. The standard non-resonant cases are extended to Tsallis reaction rates through the pathway model. Behavior of the depleted non-resonant thermonuclear function is studied. A comparison of the Maxwell-Boltzmann energy distribution with a more general energy distribution called pathway energy distribution is also done.
In this paper we investigate the scaling behavior, based on Diffusion Entropy Analysis and Standa... more In this paper we investigate the scaling behavior, based on Diffusion Entropy Analysis and Standard Deviation Analysis, of the magnetic field strength fluctuations recorded by Voyager-I in the heliosphere. The Voyager-I data set exhibits scaling behavior and may follow Lévy-type probability distribution. A general fractional-order spatial and temporal diffusion model could be utilized for the interpretation of this Lévy-type behavior in comparison to Gaussian behavior. This result confirms earlier studies of scaling behavior of the heliospheric magnetic field strength fluctuations based on non-extensive statistical mechanics leading to the determination of the nonextensivity q-triplet.
The reaction rate probability integral is extended from Maxwell–Boltzmann approach to a more gene... more The reaction rate probability integral is extended from Maxwell–Boltzmann approach to a more general approach by using the pathway model introduced by Mathai in 2005 (A pathway to matrix-variate gamma and normal densities.Linear Algebr. Appl.396, 317–328). The extended thermonuclear reaction rate is obtained in the closed form via a Meijer'sG-function and the so-obtainedG-function is represented as a solution of a homogeneous linear differential equation. A physical model for the hydrodynamical process in a fusion plasma-compressed and laser-driven spherical shock wave is used for evaluating the fusion energy integral by integrating the extended thermonuclear reaction rate integral over the temperature. The result obtained is compared with the standard fusion yield obtained by Haubold and John in 1981 (Analytical representation of the thermonuclear reaction rate and fusion energy production in a spherical plasma shock wave.Plasma Phys.23, 399–411). An interpretation for the path...
The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis st... more The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis statistics (Tsallis, 1988) and more general cases of distribution functions. An analytical study of respective thermonuclear functions is being conducted with the help of statistical techniques. The pathway model, recently introduced by Mathai (2005), is utilized for thermonuclear functions and closed-form representations are obtained in terms of H-functions and G-functions. Maxwell-Boltzmannian thermonuclear functions become particular cases of the extended thermonuclear functions. A brief review on the development of the theory of analytic representations of nuclear reaction rates is given.
ABSTRACT In this paper, the type-1 and type-2 P-transform or pathway transform, which is a genera... more ABSTRACT In this paper, the type-1 and type-2 P-transform or pathway transform, which is a generalization of many existing integral transforms, is considered. This P-transform is obtained by using the pathway model introduced by A.M. Mathai [Linear Algebra Appl. 396, 317–328 (2005: Zbl 1084.62044)]. First, the linearity and shifting property of the P-transform of both types are proven and their composition with differential operators is considered. Moreover, the connection of the type-2 P-transform with the Mellin transform and the H-transform is established. As example, the type-2 P-transform of some generalized special functions, namely an H-function and a generalized hypergeometric function, is obtained. Finally, an application of P-transforms in reaction rate theory in astrophysics is explained.
The fractional calculus of the P-transform or pathway transform which is a generalization of many... more The fractional calculus of the P-transform or pathway transform which is a generalization of many well known integral transforms is studied. The Mellin and Laplace transforms of a P-transform are obtained. The composition formulae for the various ...
The paper provides a review of A.M. Mathai's applications of the theory of special functions, par... more The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathais research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics.
... Anatoly A. Kilbas a * & Dilip Kumar b pages 835-846. ... Mathai, AM and Saxena, RK 1978. ... more ... Anatoly A. Kilbas a * & Dilip Kumar b pages 835-846. ... Mathai, AM and Saxena, RK 1978. The H-function with Applications in Statistics and Other Disciplines , New York, London, Sydney: Halsted Press (Wiley). View all references, Chapter 2], Srivastava et al. [2222. ...
Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications,... more Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317-328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Numerical Functional Analysis and Optimization, 2007
Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf ... more Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
The aim of the present paper is to obtain the solution of certain integral equations by using P α... more The aim of the present paper is to obtain the solution of certain integral equations by using P α-transform. The concept of P α-transform is introduced by Kumar [11]. The P α-transform is binomial type containing many classes of transforms including the Laplace transform. We have found the solution of fractional Volterra equation with Caputo fractional derivative using P α-transform. Also the solution of non-homogeneous time fractional heat equation in spherical domain with Caputo derivative has been found. The results for the classical Laplace transform are retrieved by letting α → 1.
Possible modification in the velocity distribution in the non-resonant reaction rates leads to an... more Possible modification in the velocity distribution in the non-resonant reaction rates leads to an extended reaction rate probability integral. The closed form representation for these thermonuclear functions are used to obtain the stellar luminosity and neutrino emission rates. The composite parameter C that determines the standard nuclear reaction rate through the Maxwell-Boltzmann energy distribution is extended to C * by the extended reaction rates through a more general distribution than the Maxwell-Boltzmann distribution. The new distribution is obtained by the pathway model introduced by Mathai in 2005 [Linear Algebra and Its Applications, 396, 317-328]. Simple analytic models considered by various authors are utilized for evaluating stellar luminosity and neutrino emission rates and are obtained in generalized special functions such as Meijer's G-function and Fox's H-function. The standard and extended non-resonant thermonuclear functions are compared by plotting them. Behavior of the new energy distribution, more general than Maxwell-Boltzmann is also studied.
The method for the evaluation of the non-resonant thermonuclear function in the Maxwell-Boltzmann... more The method for the evaluation of the non-resonant thermonuclear function in the Maxwell-Boltzmann case with depleted tail is discussed. Closed forms of the analytical results are obtained in computational format, and written in terms of the H-function in two variables. The standard non-resonant cases are extended to Tsallis reaction rates through the pathway model. Behavior of the depleted non-resonant thermonuclear function is studied. A comparison of the Maxwell-Boltzmann energy distribution with a more general energy distribution called pathway energy distribution is also done.
In this paper we investigate the scaling behavior, based on Diffusion Entropy Analysis and Standa... more In this paper we investigate the scaling behavior, based on Diffusion Entropy Analysis and Standard Deviation Analysis, of the magnetic field strength fluctuations recorded by Voyager-I in the heliosphere. The Voyager-I data set exhibits scaling behavior and may follow Lévy-type probability distribution. A general fractional-order spatial and temporal diffusion model could be utilized for the interpretation of this Lévy-type behavior in comparison to Gaussian behavior. This result confirms earlier studies of scaling behavior of the heliospheric magnetic field strength fluctuations based on non-extensive statistical mechanics leading to the determination of the nonextensivity q-triplet.
The reaction rate probability integral is extended from Maxwell–Boltzmann approach to a more gene... more The reaction rate probability integral is extended from Maxwell–Boltzmann approach to a more general approach by using the pathway model introduced by Mathai in 2005 (A pathway to matrix-variate gamma and normal densities.Linear Algebr. Appl.396, 317–328). The extended thermonuclear reaction rate is obtained in the closed form via a Meijer'sG-function and the so-obtainedG-function is represented as a solution of a homogeneous linear differential equation. A physical model for the hydrodynamical process in a fusion plasma-compressed and laser-driven spherical shock wave is used for evaluating the fusion energy integral by integrating the extended thermonuclear reaction rate integral over the temperature. The result obtained is compared with the standard fusion yield obtained by Haubold and John in 1981 (Analytical representation of the thermonuclear reaction rate and fusion energy production in a spherical plasma shock wave.Plasma Phys.23, 399–411). An interpretation for the path...
The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis st... more The Maxwell-Boltzmannian approach to nuclear reaction rate theory is extended to cover Tsallis statistics (Tsallis, 1988) and more general cases of distribution functions. An analytical study of respective thermonuclear functions is being conducted with the help of statistical techniques. The pathway model, recently introduced by Mathai (2005), is utilized for thermonuclear functions and closed-form representations are obtained in terms of H-functions and G-functions. Maxwell-Boltzmannian thermonuclear functions become particular cases of the extended thermonuclear functions. A brief review on the development of the theory of analytic representations of nuclear reaction rates is given.
ABSTRACT In this paper, the type-1 and type-2 P-transform or pathway transform, which is a genera... more ABSTRACT In this paper, the type-1 and type-2 P-transform or pathway transform, which is a generalization of many existing integral transforms, is considered. This P-transform is obtained by using the pathway model introduced by A.M. Mathai [Linear Algebra Appl. 396, 317–328 (2005: Zbl 1084.62044)]. First, the linearity and shifting property of the P-transform of both types are proven and their composition with differential operators is considered. Moreover, the connection of the type-2 P-transform with the Mellin transform and the H-transform is established. As example, the type-2 P-transform of some generalized special functions, namely an H-function and a generalized hypergeometric function, is obtained. Finally, an application of P-transforms in reaction rate theory in astrophysics is explained.
The fractional calculus of the P-transform or pathway transform which is a generalization of many... more The fractional calculus of the P-transform or pathway transform which is a generalization of many well known integral transforms is studied. The Mellin and Laplace transforms of a P-transform are obtained. The composition formulae for the various ...
The paper provides a review of A.M. Mathai's applications of the theory of special functions, par... more The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathais research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics.
... Anatoly A. Kilbas a * & Dilip Kumar b pages 835-846. ... Mathai, AM and Saxena, RK 1978. ... more ... Anatoly A. Kilbas a * & Dilip Kumar b pages 835-846. ... Mathai, AM and Saxena, RK 1978. The H-function with Applications in Statistics and Other Disciplines , New York, London, Sydney: Halsted Press (Wiley). View all references, Chapter 2], Srivastava et al. [2222. ...
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