scholarly journals Differentiation of the Mittag-Leffler Functions with Respect to Parameters in the Laplace Transform Approach

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 657
Author(s):  
Alexander Apelblat
Keyword(s):  
Laplace Transform ◽  
Integral Representations ◽  
The Other ◽  
Gamma Functions ◽  
Inverse Transform ◽  
Convolution Integrals ◽  
The Laplace Transform ◽  
Infinite Power ◽  
Derivatives Of ◽  
Two Parameter

In this work, properties of one- or two-parameter Mittag-Leffler functions are derived using the Laplace transform approach. It is demonstrated that manipulations with the pair direct–inverse transform makes it far more easy than previous methods to derive known and new properties of the Mittag-Leffler functions. Moreover, it is shown that sums of infinite series of the Mittag-Leffler functions can be expressed as convolution integrals, while the derivatives of the Mittag-Leffler functions with respect to their parameters are expressible as double convolution integrals. The derivatives can also be obtained from integral representations of the Mittag-Leffler functions. On the other hand, direct differentiation of the Mittag-Leffler functions with respect to parameters produces an infinite power series, whose coefficients are quotients of the digamma and gamma functions. Closed forms of these series can be derived when the parameters are set to be integers.

scholarly journals The noise handling properties of the Talbot algorithm for numerically inverting the Laplace transform

2018 ◽  
Vol 13 ◽  
pp. 174830181879706 ◽  
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Fourier Series ◽  
Laplace Transform ◽  
Noisy Data ◽  
Standard Test ◽  
Numerical Inversion ◽  
Test Functions ◽  
Inversion Algorithm ◽  
Inverse Transform ◽  
Exact Data ◽  
The Laplace Transform

This paper examines the noise handling properties of three of the most widely used algorithms for numerically inverting the Laplace transform. After examining the genesis of the algorithms, their error handling properties are evaluated through a series of standard test functions in which noise is added to the inverse transform. Comparisons are then made with the exact data. Our main finding is that the for “noisy data”, the Talbot inversion algorithm performs with greater accuracy when compared to the Fourier series and Stehfest numerical inversion schemes as they are outlined in this paper.

scholarly journals Alenezi Transform–A New Transform to Solve Mathematical Problems

2021 ◽  
pp. 1-8
Author(s):  
Ahmad M. Alenezi
Keyword(s):  
Laplace Transform ◽  
Integral Equations ◽  
Integral Transform ◽  
The Other ◽  
The Laplace Transform ◽  
Mathematical Problems ◽  
Sumudu Transforms

In this paper, we present a new integral transform called Alenezi-transform in the category of Laplace transform. We investigate the characteristic of Alenezi-transform. We discuss this transform with the other transforms like J, Laplace, Elzaki and Sumudu transforms. We can demonstrate that Alenezi transforms are near to the condition of the Laplace transform. We can explain the new Properties of transforms using Alenezi transform. Alenezi transform can be applied to solve differential, Partial and integral equations.

scholarly journals On the integral of geometric Brownian motion

2003 ◽  
Vol 35 (1) ◽  
pp. 159-183 ◽  
Author(s):  
Michael Schröder
Keyword(s):  
Brownian Motion ◽  
Integral Representation ◽  
Laplace Transform ◽  
Finite Time ◽  
Integral Representations ◽  
Geometric Brownian Motion ◽  
Bessel Processes ◽  
Time Intervals ◽  
The Laplace Transform ◽  
The Law

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

An Asymptotic Study of the Nusselt-Graetz Problem—Part I: Large x Behavior

1974 ◽  
Vol 96 (3) ◽  
pp. 354-358 ◽  
Author(s):  
H. J. Hickman
Keyword(s):  
Nusselt Number ◽  
Laminar Flow ◽  
Laplace Transform ◽  
The Other ◽  
Circular Pipe ◽  
Parallel Plates ◽  
Newtonian Flow ◽  
Percent Error ◽  
The Laplace Transform ◽  
Analytic Expressions

A method is given for determining the large x behavior of the Nusselt number for a variety of Nusselt-Graetz problems. Exploitation of properties of the Laplace transform of the temperature yields analytic expressions for Nu as explicit functions of the other parameters of the problem. Accurate results (<1 percent error) are deduced for problems involving the laminar flow of a Newtonian flow between parallel plates and in a circular pipe (valid for all values of the wall Nusselt number).

scholarly journals Some Applications of The New Integral Transform For Partial Differential Equations

2018 ◽  
Vol 7 (1) ◽  
pp. 45-49
Author(s):  
S L Shaikh
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Integral Transform ◽  
Partial Differential ◽  
Partial Derivatives ◽  
The Laplace Transform ◽  
Sumudu Transform ◽  
Function Of Two Variables ◽  
Derivatives Of

In this paper we have derived Sadik transform of the partial derivatives of a function of two variables. We have demonstrated the applicability of the Sadik transform by solving some examples of partial differential equations. We have verified solutions of partial differential equations by Sadik transform with the Laplace transform and the Sumudu transform.

Application of the Laplace transform method to wave motions involving strong discontinuities

1964 ◽  
Vol 60 (2) ◽  
pp. 313-324 ◽  
Author(s):  
P. Chadwick ◽  
B. Powdrill
Keyword(s):  
Laplace Transform ◽  
Plane Shock ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Partial Derivatives ◽  
Strong Discontinuities ◽  
The Laplace Transform ◽  
Linear Thermoelasticity ◽  
Derivatives Of ◽  
Formulation Of Problems

AbstractFormulae are obtained for the Laplace transforms of the first and second partial derivatives of a function of time and position which, together with its first partial derivatives, is discontinuous on a moving surface. These formulae are then used in a discussion of the application of the Laplace transform method to the solution of mixed initial and boundary-value problems in linear thermoelasticity. Particular attention is given to thermoelastic disturbances involving plane shock waves, and it is pointed out that no properly posed formulation of problems of this type has yet been found.

scholarly journals On the integral of geometric Brownian motion

2003 ◽  
Vol 35 (01) ◽  
pp. 159-183 ◽  
Author(s):  
Michael Schröder
Keyword(s):  
Brownian Motion ◽  
Integral Representation ◽  
Laplace Transform ◽  
Finite Time ◽  
Integral Representations ◽  
Geometric Brownian Motion ◽  
Bessel Processes ◽  
Time Intervals ◽  
The Laplace Transform ◽  
The Law

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

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&gt; 0 for eachj&gt; 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.

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