scholarly journals Analysis of Multiterm Initial Value Problems with Caputo–Fabrizio Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Mohammed Al-Refai ◽  
Muhammed Syam
Keyword(s):  
Laplace Transform ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Initial Value ◽  
The Laplace Transform ◽  
Necessary And Sufficient

In this paper, we discuss the solvability of a class of multiterm initial value problems involving the Caputo–Fabrizio fractional derivative via the Laplace transform. We derive necessary and sufficient conditions to guarantee the existence of solutions to the problem. We also obtain the solutions in closed forms. We present two examples to illustrate the validity of the obtained results.

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.

scholarly journals Weighted Exponential Inequalities

2001 ◽  
Vol 8 (1) ◽  
pp. 69-86
Author(s):  
H. P. Heinig ◽  
R. Kerman ◽  
M. Krbec
Keyword(s):  
Laplace Transform ◽  
Sufficient Conditions ◽  
Two Dimensions ◽  
Necessary And Sufficient Conditions ◽  
Exponential Inequalities ◽  
Averaging Operator ◽  
The Laplace Transform ◽  
Discrete Analogues ◽  
Carleman’S Inequality ◽  
Necessary And Sufficient

Abstract Necessary and sufficient conditions on weight pairs are found for the validity of a class of weighted exponential inequalities involving certain classical operators. Among the operators considered are the Hardy averaging operator and its variants in one and two dimensions, as well as the Laplace transform. Discrete analogues yield characterizations of weighted forms of Carleman's inequality.

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

1986 ◽  
Vol 23 (4) ◽  
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 satisfy λ0 = 0, λ j > 0 for each j > 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 (λ j = jλ, µj = jμ) with catastrophes of several different types.

Necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and solution representation

2021 ◽  
pp. 1-16
Author(s):  
Sunae Pak ◽  
Huichol Choe ◽  
Kinam Sin ◽  
Sunghyok Kwon
Keyword(s):  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Inequality Constraints ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Initial Value ◽  
Solution Representation ◽  
Necessary And Sufficient

In this paper, we investigate the necessary and sufficient conditions for existence of solutions for initial value problem of fuzzy Bagley-Torvik equation and the solution representation by using the multivariate Mittag-Leffler function. First we convert fuzzy initial value problem into the cut problem (system of fractional differential equations with inequality constraints) and obtain existence results for the solution of the cut problem under (1,1)- differentiability. Next we study the conditions for the solutions of the cut problem to constitute the solution of a fuzzy initial value problem and suggest a necessary and sufficient condition for the (1,1)-solution. Also, some examples are given to verify the effectiveness of our proposed method. The necessary and sufficient condition, solution representation for (1,2)-solution of initial value problem of fuzzy fractional Bagley-Torvik equation are shown in Appendix.

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 satisfy λ 0 = 0, λ j > 0 for each j > 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 (λ j = jλ, µj = jμ) with catastrophes of several different types.

scholarly journals Laplace Transforms and Generalized Laguerre Polynomials

1958 ◽  
Vol 10 ◽  
pp. 177-182 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Laplace Transform ◽  
Laguerre Polynomials ◽  
Sufficient Conditions ◽  
Laplace Transforms ◽  
Necessary And Sufficient Conditions ◽  
Representation Theorems ◽  
Generalized Laguerre Polynomials ◽  
Inversion Operator ◽  
The Laplace Transform ◽  
Necessary And Sufficient

Various sets of necessary and sufficient conditions are known in order that a function ƒ(s), analytic for Re s > 0, be represented as the Laplace transform of a function in L p(0,∞), 1 < p ⩽ ∞ . Most of these theories are based on the properties of some inversion operator for the transformation— see, for example, (7, chap. 7). However in the case p = 2 a number of representation theorems of a much simpler type are available.

scholarly journals Variational Problems with Partial Fractional Derivative: Optimal Conditions and Noether’s Theorem

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Jun Jiang ◽  
Yuqiang Feng ◽  
Shougui Li
Keyword(s):  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Variational Problems ◽  
Necessary And Sufficient Conditions ◽  
Optimal Conditions ◽  
Lagrange Equations ◽  
Noether’S Theorem ◽  
Noether's Theorem ◽  
Necessary And Sufficient

In this paper, the necessary and sufficient conditions of optimality for variational problems with Caputo partial fractional derivative are established. Fractional Euler-Lagrange equations are obtained. The Legendre condition and Noether’s theorem are also presented.

scholarly journals Fractional order differential inclusions on an unbounded domain with infinite delay

Mathematica ◽  
2020 ◽  
Vol 62 (85) (2) ◽  
pp. 167-178
Author(s):  
Mohamed Helal
Keyword(s):  
Fractional Order ◽  
Initial Value Problems ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Infinite Delay ◽  
Sufficient Conditions ◽  
Fréchet Spaces ◽  
Initial Value ◽  
Nonlinear Alternative ◽  
Hyperbolic Differential Inclusions

We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

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