The laplace transform of the bessel functions $$K_\mu \left[ {xt^{ \pm \frac{1}{n}} } \right]$$ and $$J_u \left[ {xt^{ \pm \frac{1}{n}} } \right]$$ wheren=1, 2, 3, ...

1963 ◽  
Vol 62 (1) ◽  
pp. 317-335 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Laplace Transform ◽  
Bessel Functions ◽  
The Laplace Transform

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

scholarly journals Laplace transform of certain functions with applications

2000 ◽  
Vol 23 (2) ◽  
pp. 99-102
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Laplace Transform ◽  
Bessel Functions ◽  
Whittaker Functions ◽  
Special Cases ◽  
The Laplace Transform ◽  
Special Case

The Laplace transform of the functionstν(1+t)β,Reν>−1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity∫0∞sin (ax)/x dx=π/2is a special case of our main result.

Application of q-Bessel Functions in the Solution of Generalized Fractional Kinetic Equations

2021 ◽  
Vol 88 (1-2) ◽  
pp. 01
Author(s):  
Garima Agarwal ◽  
Sunil Joshi ◽  
Kottakkaran Sooppy Nisar
Keyword(s):  
Laplace Transform ◽  
Bessel Function ◽  
Bessel Functions ◽  
Kinetic Equations ◽  
Mathematical Physics ◽  
The Laplace Transform ◽  
Fractional Kinetic Equations

The present investigation aims to extract a solution from the generalized fractional kinetic equations involving the generalized q-Bessel function by applying the Laplace transform. Methodology and results can be adopted and extended to a variety of related fractional problems in mathematical physics.

scholarly journals Dirichlet Random Walks

2014 ◽  
Vol 51 (4) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk in Rd. We compute explicitly, for a number of cases, the distribution of the random variable W using a form of Stieltjes transform of W instead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere of Rd.

scholarly journals Propagation of non-stationary antisymmetric kinematic perturbations from a spherical cavity in Cosserat medium

2020 ◽  
pp. 201-210
Author(s):  
D. V Tarlakovskii ◽  
Van Lam Nguyen
Keyword(s):  
Laplace Transform ◽  
Spherical Cavity ◽  
Bessel Functions ◽  
Initial Conditions ◽  
Displacement Vector ◽  
Elementary Functions ◽  
Linear Combinations ◽  
Spherical Coordinate ◽  
Cosserat Medium ◽  
The Laplace Transform

We consider a space filled with a linearly elastic Cosserat medium with a spherical cavity under given nonstationary antisymmetric surface perturbations, which are understood as the corresponding analogue of classical antiplane deformations. The motion of a medium is described by a system of three equations with respect to nonzero components of the displacement vector and potentials of the rotation field, written in a spherical coordinate system with the origin at its center of the cavity. The initial conditions are assumed to be zero. To solve the problem, we use decomposition of functions to Legendre and Gegenbauer polynomials, as well as the Laplace transform in time. As a result, the problem is reduced to independent systems of ordinary differential equations with the Laplace operator for the coefficients of the series. A statement about the structure of the general solution of this system is formulated. Images of the series coefficients are presented in the form of linear combinations of boundary conditions with coefficients - transformants of surface influence functions, the explicit formulas for which include the Bessel functions of a half-integer index. Due to the complexity of these expressions, to determine the originals in the linear approximation, the method of a small parameter is used, which is taken as a coefficient characterizing the relationship between the displacement and rotation fields. Then, taking into account the connection between the Bessel functions and elementary functions, the images are written in the form of linear combinations of exponentials with coefficients - rational functions of the transformation parameter. The further procedure for inverting the Laplace transform is carried out using residues. It is shown that there are three wave fronts corresponding to a shear wave modified with allowance for free rotation and two rotation waves. Examples of calculations for a granular composite of aluminum shot in an epoxy matrix are presented.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

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