scholarly journals The probabilistic approach to the analysis of the limiting behavior of an integro-diffebential equation depending on a small parameter, and its application to stochastic processes

1994 ◽  
Vol 7 (1) ◽  
pp. 25-31
Author(s):  
O. V. Borisenko ◽  
A. D. Borisenko ◽  
I. G. Malyshev
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Stochastic Differential Equation ◽  
Small Parameter ◽  
Stochastic Processes ◽  
Probabilistic Approach ◽  
Problem Solution ◽  
Poisson Measure ◽  
Limiting Behavior ◽  
The Cauchy Problem

Using connection between stochastic differential equation with Poisson measure term and its Kolmogorov's equation, we investigate the limiting behavior of the Cauchy problem solution of the integro differential equation with coefficients depending on a small parameter. We also study the dependence of the limiting equation on the order of the parameter.

scholarly journals Semiclassical Approach to the Nonlocal Kinetic Model of Metal Vapor Active Media

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2995
Author(s):  
Alexander V. Shapovalov ◽  
Anton E. Kulagin
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Kinetic Equation ◽  
Linear Partial Differential Equation ◽  
Metal Vapor ◽  
Problem Solution ◽  
Semiclassical Approach ◽  
Active Media ◽  
The Cauchy Problem

A semiclassical approach based on the WKB–Maslov method is developed for the kinetic ionization equation in dense plasma with approximations characteristic of metal vapor active media excited by a contracted discharge. We develop the technique for constructing the leading term of the semiclassical asymptotics of the Cauchy problem solution for the kinetic equation under the supposition of weak diffusion. In terms of the approach developed, the local cubic nonlinear term in the original kinetic equation is considered in a nonlocal form. This allows one to transform the nonlinear nonlocal kinetic equation to an associated linear partial differential equation with a given accuracy of the asymptotic parameter using the dynamical system of moments of the desired solution of the equation. The Cauchy problem solution for the nonlinear nonlocal kinetic equation can be obtained from the solution of the associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation. Within the developed approach, the plasma relaxation in metal vapor active media is studied with asymptotic solutions expressed in terms of higher transcendental functions. The qualitative analysis of such the solutions is given.

ASYMPTOTIC SOLUTION OF FIRST-ORDER EQUATION WITH SMALL PARAMETER UNDER THE DERIVATIVE WITH PERTURBED OPERATOR

2018 ◽  
pp. 784-796
Author(s):  
Vladimir Igorevich Uskov
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Asymptotic Expansion ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Decomposition Method ◽  
Order Equation ◽  
First Order ◽  
The Cauchy Problem

The paper is devoted to the Cauchy problem for a differential equation with a small parameter when using a Fredholm operator in a Banach space with a certain method. The investigated effect of this parameter. The solution is in the form of an asymptotic expansion. When solving the problems of using the cascade decomposition method for equations, which allows us to split the equation into equations in subspaces.

scholarly journals Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

2021 ◽  
Vol 5 (3) ◽  
pp. 66
Author(s):  
Azmat Ullah Khan Niazi ◽  
Jiawei He ◽  
Ramsha Shafqat ◽  
Bilal Ahmed
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
The Cauchy Problem ◽  
Fractional Differential ◽  
Ulam Type Stability ◽  
Fuzzy Fractional Differential Equations ◽  
Fuzzy Fractional Differential Equation

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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