UNIQUE SOLUTIONS OF DISCONTINUOUS O.D.E.'S IN BANACH SPACES

2006 ◽  
Vol 04 (03) ◽  
pp. 247-262 ◽  
Author(s):  
ALBERTO BRESSAN ◽  
WEN SHEN
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Right Hand ◽  
Directional Variation ◽  
The Cauchy Problem

We consider the Cauchy problem for an ordinary differential equation with discontinuous right-hand side in an L∞ space. Under the assumptions that the vector field is directionally continuous with bounded directional variation, we prove that the O.D.E. has a unique Carathéodory solution, which depends Lipschitz continuously on the data.

scholarly journals The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2113
Author(s):  
Alla A. Yurova ◽  
Artyom V. Yurov ◽  
Valerian A. Yurov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Exact Solutions ◽  
Airy Function ◽  
The Real ◽  
The Cauchy Problem

We begin by introducing a new procedure for construction of the exact solutions to Cauchy problem of the real-valued (hyperbolic) Novikov–Veselov equation which is based on the Moutard symmetry. The procedure shown therein utilizes the well-known Airy function Ai(ξ) which in turn serves as a solution to the ordinary differential equation d2zdξ2=ξz. In the second part of the article we show that the aforementioned procedure can also work for the n-th order generalizations of the Novikov–Veselov equation, provided that one replaces the Airy function with the appropriate solution of the ordinary differential equation dn−1zdξn−1=ξz.

scholarly journals Continuity equations and ODE flows with non-smooth velocity

2014 ◽  
Vol 144 (6) ◽  
pp. 1191-1244 ◽  
Author(s):  
Luigi Ambrosio ◽  
Gianluca Crippa
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Cauchy Problem ◽  
Bounded Variation ◽  
Velocity Fields ◽  
Well Posedness ◽  
Continuity Equations ◽  
Quantitative Estimates ◽  
The Cauchy Problem

In this paper we review many aspects of the well-posedness theory for the Cauchy problem for the continuity and transport equations and for the ordinary differential equation (ODE). In this framework, we deal with velocity fields that are not smooth, but enjoy suitable ‘weak differentiability’ assumptions. We first explore the connection between the partial differential equation (PDE) and the ODE in a very general non-smooth setting. Then we address the renormalization property for the PDE and prove that such a property holds for Sobolev velocity fields and for bounded variation velocity fields. Finally, we present an approach to the ODE theory based on quantitative estimates.

scholarly journals Weak solutions of differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 407-418 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Banach Spaces ◽  
Existence Result ◽  
Measure Of Noncompactness ◽  
Continuous Vector ◽  
Weakly Continuous ◽  
Continuous Vector Field ◽  
The Cauchy Problem

We consider the Cauchy problem x (t) = f (t,x (t)), x (0) = x0 in a nonreflexive Banach space X and for f: T × X → X a weakly continuous vector field. Using a compactness hypothesis involving a weak measure of noncompactness we prove an existence result that generalizes earlier theorems by Chow-Shur, Kato and Cramer-Lakshmikantham-Mitchell.

scholarly journals A Study of a Nonlinear Ordinary Differential Equation in Modular Function Spaces Endowed with a Graph

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jaauad Jeddi ◽  
Mustapha Kabil ◽  
Samih Lazaiz
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fixed Point ◽  
Existence Result ◽  
Modular Function ◽  
Nonlinear Ordinary Differential Equation ◽  
Function Spaces ◽  
The Cauchy Problem ◽  
Modular Function Spaces

In this paper, we prove by means of a fixed-point theorem an existence result of the Cauchy problem associated to an ordinary differential equation in modular function spaces endowed with a reflexive convex digraph.

Cauchy problem for second-order ordinary differential equation with continuously distributed differentiation operator

2020 ◽  
Vol 20 (1) ◽  
pp. 15-20
Author(s):  
B.I. Efendiev ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Order Differential Equation ◽  
Differentiation Operator ◽  
Second Order ◽  
Order Ordinary Differential Equation ◽  
Second Order Differential Equation ◽  
The Cauchy Problem

In this paper, we construct the fundamental solution for ordinary second-order differential equation with continuously distributed differentiation operator. With the help of fundamental solution the solution of the Cauchy problem is written out.

scholarly journals ROBUST DIFFERENCE SCHEME FOR THE CAUCHY PROBLEM FOR A SINGULARLY PERTURBED ORDINARY DIFFERENTIAL EQUATION

2018 ◽  
Vol 23 (4) ◽  
pp. 527-537 ◽  
Author(s):  
Lidia Pavlovna Shishkina ◽  
Grigorii Ivanovich Shishkin
Keyword(s):  
Differential Equation ◽  
Boundary Layer ◽  
Ordinary Differential Equation ◽  
Cauchy Problem ◽  
Difference Scheme ◽  
Singularly Perturbed ◽  
Uniform Grid ◽  
Piecewise Uniform Grid ◽  
The Cauchy Problem ◽  
Standard Difference

Grid approximation of the Cauchy problem on the interval D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N −1 , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N −1 lnN). Such a scheme is called a robust one. For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

scholarly journals Inverse problem for $2b$-order differential equation with a time-fractional derivative

2019 ◽  
Vol 11 (1) ◽  
pp. 107-118 ◽  
Author(s):  
A.O. Lopushansky ◽  
H.P. Lopushanska
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Cauchy Problem ◽  
Fractional Derivative ◽  
Sufficient Conditions ◽  
Integral Condition ◽  
Generalized Solution ◽  
Right Hand ◽  
The Cauchy Problem ◽  
The Right

We study the inverse problem for a differential equation of order $2b$ with the Riemann-Liouville fractional derivative of order $\beta\in (0,1)$ in time and given Schwartz type distributions in the right-hand sides of the equation and the initial condition. The problem is to find the pair of functions $(u, g)$: a generalized solution $u$ to the Cauchy problem for such equation and the time dependent multiplier $g$ in the right-hand side of the equation. As an additional condition, we use an analog of the integral condition $$(u(\cdot,t),\varphi_0(\cdot))=F(t), \;\;\; t\in [0,T],$$ where the symbol $(u(\cdot,t),\varphi_0(\cdot))$ stands for the value of an unknown distribution $u$ on the given test function $\varphi_0$ for every $t\in [0,T]$, $F$ is a given continuous function. We prove a theorem for the existence and uniqueness of a generalized solution of the Cauchy problem, obtain its representation using the Green's vector-function. The proof of the theorem is based on the properties of conjugate Green's operators of the Cauchy problem on spaces of the Schwartz type test functions and on the structure of the Schwartz type distributions. We establish sufficient conditions for a unique solvability of the inverse problem and find a representation of anunknown function $g$ by means of a solution of a certain Volterra integral equation of the second kind with an integrable kernel.

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