scholarly journals LIOUVILLE AND BOGOLIUBOV EQUATIONS WITH FRACTIONAL DERIVATIVES

2007 ◽  
Vol 21 (05) ◽  
pp. 237-248 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Liouville Equation ◽  
Fractional Derivatives ◽  
Kinetic Equations ◽  
Charged Particles ◽  
Volume Element ◽  
Vlasov Equations ◽  
Fractional Kinetic Equation ◽  
Derivatives Of ◽  
Bogoliubov Equations ◽  
Fractional Kinetic Equations

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fractional generalization of the Hamiltonian systems is discussed. Fractional kinetic equation for the system of charged particles are considered.

FRACTIONAL KINETIC EQUATIONS INVOLVING GENERALIZED V-FUNCTION VIA LAPLACE TRANSFORM

2021 ◽  
Vol 10 (5) ◽  
pp. 2593-2610
Author(s):  
Wagdi F.S. Ahmed ◽  
D.D. Pawar ◽  
W.D. Patil
Keyword(s):  
Kinetic Equation ◽  
Laplace Transform ◽  
Kinetic Equations ◽  
Graphical Interpretation ◽  
Fractional Kinetic Equation ◽  
Fractional Kinetic Equations

In this study, a new and further generalized form of the fractional kinetic equation involving the generalized V$-$function has been developed. We have discussed the manifold generality of the generalized V$-$function in terms of the solution of the fractional kinetic equation. Also, the graphical interpretation of the solutions by employing MATLAB is given. The results are very general in nature, and they can be used to generate a large number of known and novel results.

scholarly journals Fractional kinetic equations driven by Gaussian or infinitely divisible noise

2005 ◽  
Vol 37 (2) ◽  
pp. 366-392 ◽  
Author(s):  
J. M. Angulo ◽  
V. V. Anh ◽  
R. McVinish ◽  
M. D. Ruiz-Medina
Keyword(s):  
Kinetic Equation ◽  
Kinetic Equations ◽  
Unbounded Domains ◽  
Long Range Dependence ◽  
Infinitely Divisible ◽  
Fractional Kinetic Equation ◽  
Noise Input ◽  
Spatial Operators ◽  
Fractional Kinetic Equations

In this paper, we consider a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The solutions to the equation are provided in the cases of both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second- and higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operators.

scholarly journals Fractional Calculus involving (p, q)-Mathieu Type Series

2020 ◽  
Vol 5 (2) ◽  
pp. 15-34 ◽  
Author(s):  
Daljeet Kaur ◽  
Praveen Agarwal ◽  
Madhuchanda Rakshit ◽  
Mehar Chand
Keyword(s):  
Kinetic Equation ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Kinetic Equations ◽  
Integral Transforms ◽  
Type Series ◽  
Integral Formulas ◽  
Fractional Kinetic Equation ◽  
Fractional Calculus Operators ◽  
Fractional Kinetic Equations

AbstractAim of the present paper is to establish fractional integral formulas by using fractional calculus operators involving the generalized (p, q)-Mathieu type series. Then, their composition formulas by using the integral transforms are introduced. Further, a new generalized form of the fractional kinetic equation involving the series is also developed. The solutions of fractional kinetic equations are presented in terms of the Mittag-Leffler function. The results established here are quite general in nature and capable of yielding both known and new results.

scholarly journals Fractional kinetic equations driven by Gaussian or infinitely divisible noise

2005 ◽  
Vol 37 (02) ◽  
pp. 366-392 ◽  
Author(s):  
J. M. Angulo ◽  
V. V. Anh ◽  
R. McVinish ◽  
M. D. Ruiz-Medina
Keyword(s):  
Kinetic Equation ◽  
Kinetic Equations ◽  
Unbounded Domains ◽  
Long Range Dependence ◽  
Infinitely Divisible ◽  
Fractional Kinetic Equation ◽  
Noise Input ◽  
Spatial Operators ◽  
Fractional Kinetic Equations

In this paper, we consider a certain type of space- and time-fractional kinetic equation with Gaussian or infinitely divisible noise input. The solutions to the equation are provided in the cases of both bounded and unbounded domains, in conjunction with bounds for the variances of the increments. The role of each of the parameters in the equation is investigated with respect to second- and higher-order properties. In particular, it is shown that long-range dependence may arise in the temporal solution under certain conditions on the spatial operators.

scholarly journals On a class of stochastic fractional kinetic equation with fractional noise

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Min Lu ◽  
Junfeng Liu
Keyword(s):  
Kinetic Equations ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Riesz Potentials ◽  
Diffusion Operator ◽  
Fractional Noise ◽  
Mild Conditions ◽  
Fractional Kinetic Equation ◽  
Time Variables ◽  
Fractional Kinetic Equations

AbstractIn this article we study a class of stochastic fractional kinetic equations with fractional noise which are spatially homogeneous and are fractional in time with $H>1/2$ H > 1 / 2 . The diffusion operator involved in the equation is the composition of the Bessel and Riesz potentials with any fractional parameters. We prove the existence of the solution under some mild conditions which generalized some results obtained by Dalang (Electron. J. Probab. 4(6):1–29, 1999) and Balan and Tudor (Stoch. Process. Appl. 120:2468–2494 , 2010). We study also its Hölder continuity with respect to space and time variables with $b=0$ b = 0 . Moreover, we prove the existence for the density of the solution and establish the Gaussian-type lower and upper bounds for the density by the techniques of Malliavin calculus.

scholarly journals On Generalized Fractional Kinetic Equations Involving Generalized Bessel Function of the First Kind

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Dinesh Kumar ◽  
S. D. Purohit ◽  
A. Secer ◽  
A. Atangana
Keyword(s):  
Kinetic Equation ◽  
Bessel Function ◽  
Kinetic Equations ◽  
Generalized Bessel Function ◽  
Fractional Kinetic Equation ◽  
Fractional Kinetic Equations

We develop a new and further generalized form of the fractional kinetic equation involving generalized Bessel function of the first kind. The manifold generality of the generalized Bessel function of the first kind is discussed in terms of the solution of the fractional kinetic equation in the paper. The results obtained here are quite general in nature and capable of yielding a very large number of known and (presumably) new results.

scholarly journals Fractional Kinetic Equations Associated with Incomplete I-Functions

2020 ◽  
Vol 4 (2) ◽  
pp. 19 ◽  
Author(s):  
Manish Kumar Bansal ◽  
Devendra Kumar ◽  
Priyanka Harjule ◽  
Jagdev Singh
Keyword(s):  
Fractional Integral ◽  
Reaction Rate ◽  
Integral Transform ◽  
Special Functions ◽  
Kinetic Equations ◽  
Real Life ◽  
Great Role ◽  
Fractional Kinetic Equation ◽  
Life Problems ◽  
Fractional Kinetic Equations

In this paper, we investigate the solution of fractional kinetic equation (FKE) associated with the incomplete I-function (IIF) by using the well-known integral transform (Laplace transform). The FKE plays a great role in solving astrophysical problems. The solutions are represented in terms of IIF. Next, we present some interesting corollaries by specializing the parameters of IIF in the form of simpler special functions and also mention a few known results, which are very useful in solving physical or real-life problems. Finally, some graphical results are presented to demonstrate the influence of the order of the fractional integral operator on the reaction rate.

Time-fractional extensions of the Liouville and Zwanzig equations

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Stanislav Lukashchuk
Keyword(s):  
Liouville Equation ◽  
Fractional Derivatives ◽  
Formal Solution ◽  
Kinetic Equations ◽  
Equation Of Motion ◽  
Dynamic Equations ◽  
Cauchy Type ◽  
Operator Formalism ◽  
Long Time ◽  
Formal Solutions

AbstractThis paper presents extensions to the classical stochastic Liouville equation of motion that contain the Riemann-Liouville and Caputo time-fractional derivatives. At first, the dynamic equations with the time-fractional derivatives are formally obtained from the classical Liouville equation. A feature of these new equations is that they have the same common formal solution as the classical Liouville equation and therefore may be used for study of the Hamiltonian system dynamics. Two cases of the time-dependent and time-independent Hamiltonian are considered separately. Then, the time-fractional Liouville equations are deduced from the short- and long-time asymptotic expansions of the obtained dynamic equations. The physical meaning of the resulting equations is discussed. The statements of the Cauchy-type problems for the derived time-fractional Liouville equations are given, and the formal solutions of these problems are presented. At last, the projection operator formalism is employed to derive the time-fractional extensions of the Zwanzig kinetic equations and the corresponding formal statistical operators from the time-fractional Liouville equations.

scholarly journals Solution of Fractional Kinetic Equations Associated with the p,q-Mathieu-Type Series

2020 ◽  
Vol 2020 ◽  
pp. 1-7 ◽  
Author(s):  
D. L. Suthar ◽  
S. D. Purohit ◽  
Serkan Araci
Keyword(s):  
Kinetic Equation ◽  
Kinetic Equations ◽  
Laplace Transforms ◽  
Type Series ◽  
Fractional Kinetic Equation ◽  
Fractional Kinetic Equations

In this paper, our aim is to finding the solutions of the fractional kinetic equation related with the p,q-Mathieu-type series through the procedure of Sumudu and Laplace transforms. The outcomes of fractional kinetic equations in terms of the Mittag-Leffler function are presented.

The properties of Bessel-type fractional derivatives and integrals

2016 ◽  
pp. 3973-3982
Author(s):  
V. R. Lakshmi Gorty
Keyword(s):  
Fractional Order ◽  
Real Line ◽  
Computer Algebra System ◽  
Fractional Derivatives ◽  
Graphical Representation ◽  
Fractional Integrals ◽  
The Real ◽  
Fractional Derivatives And Integrals ◽  
Half Axis ◽  
Derivatives Of

The fractional integrals of Bessel-type Fractional Integrals from left-sided and right-sided integrals of fractional order is established on finite and infinite interval of the real-line, half axis and real axis. The Bessel-type fractional derivatives are also established. The properties of Fractional derivatives and integrals are studied. The fractional derivatives of Bessel-type of fractional order on finite of the real-line are studied by graphical representation. Results are direct output of the computer algebra system coded from MATLAB R2011b.

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