On a Class of Non-Markovian Langevin Equations

2013 ◽  
Vol 20 (04) ◽  
pp. 1350016
Author(s):  
Carlos Lizama ◽  
Rolando Rebolledo
Keyword(s):  
Differential Equation ◽  
Finite Number ◽  
Mechanical System ◽  
Unique Solution ◽  
Gaussian Processes ◽  
Langevin Equations ◽  
Memory Kernel ◽  
Small System ◽  
Main System ◽  
Gaussian Transform

This paper proposes a generalized Langevin's equation for a small classical mechanical system embedded in a reservoir. The interaction of the main system with the reservoir is given by a Gaussian transform as introduced in our previous paper [8]. Thus, a first result proves the existence of a strong solution to this equation in the space where the Gaussian transform (or non-Markovian noise) is defined. The interpretation of the noise is obtained by considering a finite number n of oscillating particles with discrete frequencies in the reservoir. The action of this discrete reservoir on the small system is described by a memory kernel and a sequence of zero-mean Gaussian processes. So, an integro-differential equation for the evolution of a generic particle in the main system arises for each n. This equation has a unique solution Xn which converges in distribution towards the solution of the initial non-Markovian Langevin's equation.

A unique solution for the exact differential equation of nonuniform lines

1968 ◽  
Vol 56 (12) ◽  
pp. 2181-2182 ◽  
Author(s):  
A.G.J. Holt ◽  
K.U. Ahmed
Keyword(s):  
Differential Equation ◽  
Unique Solution

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

Mechanical Modeling and Design for Reduction of Parkinsonian Hand Tremor

2010 ◽  
Author(s):  
Timothy A. Doughty ◽  
Nicholas Bankus
Keyword(s):  
Differential Equation ◽  
Mechanical System ◽  
Model Identification ◽  
Equation Model ◽  
Vibration Absorber ◽  
Differential Equation Model ◽  
Hand Tremor ◽  
Test Fixture ◽  
Parkinsonian Tremor ◽  
Active Research

Most of the active research in reducing Parkinsonian tremor involves invasive surgeries or medical treatment. In this paper hand tremors associated with Parkinson’s Disease (PD) are studied and passive vibration control methods are developed and tested. Patients with PD are surveyed regarding difficulties with hand tremor during the act of eating. The result leads to design criteria for an enhanced eating utensil and the establishment of meaningful testing methods for measuring hand tremor. Tremor data collected from several PD patients provides insight into the nature of the motion and allows for the development of test fixture and prototypes. This experimental data is coupled with linear model identification testing for the free response of a “healthy” hand undergoing the same motions. The resulting differential equation model, where the system input is realized as actuation through the biomechanics of the forearm and wrist, is used in the design of an eating utensil for vibration reduction. With self-excitation and the existence of harmonics, the tremor data is also used to develop a nonlinear differential equation model, where the complete neurological/mechanical system is realized with an equivalent mechanical system. This nonlinear model is shown to mimic the tremor data and is used to enhance the development of the vibration absorber. A prototype of the vibration absorber is built, validated on the test fixture, and tremor reduction data is collected again with PD patients.

scholarly journals The Iterative Scheme and the Convergence Analysis of Unique Solution for a Singular Fractional Differential Equation from the Eco-Economic Complex System’s Co-Evolution Process

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Teng Ren ◽  
Helu Xiao ◽  
Zhongbao Zhou ◽  
Xinguang Zhang ◽  
Lining Xing ◽  
...  
Keyword(s):  
Differential Equation ◽  
Convergence Analysis ◽  
Unique Solution ◽  
Fractional Differential Equation ◽  
Iterative Process ◽  
Iterative Scheme ◽  
Singular Fractional Differential Equation ◽  
Complex Processes ◽  
Fractional Differential ◽  
And Diffusion

In this paper, we focus on a class of singular fractional differential equation, which arises from many complex processes such as the phenomenon and diffusion interaction of the ecological-economic-social complex system. By means of the iterative technique, the uniqueness and nonexistence results of positive solutions are established under the condition concerning the spectral radius of the relevant linear operator. In addition, the iterative scheme that converges to the unique solution is constructed without request of any monotonicity, and the convergence analysis and error estimate of unique solution are obtained. The numerical example and simulation are also given to demonstrate the application of the main results and the effectiveness of iterative process.

scholarly journals Explicit iteration and unique solution for $ \phi $-Hilfer type fractional Langevin equations

2021 ◽  
Vol 7 (3) ◽  
pp. 3456-3476
Author(s):  
Abdulkafi M. Saeed ◽  
◽  
Mohammed A. Almalahi ◽  
Mohammed S. Abdo ◽  
◽  
...  
Keyword(s):  
Iterative Method ◽  
Unique Solution ◽  
Sufficient Conditions ◽  
Extremal Solution ◽  
Langevin Equations ◽  
Monotone Iterative Method ◽  
Hilfer Fractional Derivative ◽  
Monotone Iterative ◽  
Fixed Point Technique ◽  
Nonlinear Langevin Equation

<abstract><p>This paper proves that the monotone iterative method is an effective method to find the approximate solution of fractional nonlinear Langevin equation involving $ \phi $-Hilfer fractional derivative with multi-point boundary conditions. First, we apply a approach based on the properties of the Mittag-Leffler function to derive the formula of explicit solutions for the proposed problem. Next, by using the fixed point technique and some properties of Mittag-Leffler functions, we establish the sufficient conditions of existence of a unique solution for the considered problem. Moreover, we discuss the lower and upper explicit monotone iterative sequences that converge to the extremal solution by using the monotone iterative method. Finally, we construct a pertinent example that includes some graphics to show the applicability of our results.</p></abstract>

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