The Cauchy problem for a degenerate differential equation

1976 ◽  
Vol 15 (3) ◽  
pp. 457-463 ◽  
Author(s):  
B. Kvedaras ◽  
J. Macionis
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Degenerate Differential Equation ◽  
The Cauchy Problem

scholarly journals Optimal time and space regularity for solutions of degenerate differential equations

2009 ◽  
Vol 7 (2) ◽  
Author(s):  
Alberto Favaron
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Optimal Time ◽  
Time And Space ◽  
Optimal Regularity ◽  
Degenerate Differential Equation ◽  
The Cauchy Problem ◽  
Degenerate Differential Equations

AbstractWe derive optimal regularity, in both time and space, for solutions of the Cauchy problem related to a degenerate differential equation in a Banach space X. Our results exhibit a sort of prevalence for space regularity, in the sense that the higher is the order of regularity with respect to space, the lower is the corresponding order of regularity with respect to time.

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.

Solving a second-order algebro-differential equation with respect to the derivative

2021 ◽  
pp. 414-420
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Rigid Body ◽  
Fredholm Operator ◽  
Arbitrary Dimension ◽  
Second Order ◽  
Index Zero ◽  
Second Order Differential Equations ◽  
The Cauchy Problem ◽  
One Step

We consider a second-order algebro-differential equation. Equations and systems of second-order differential equations describe the operation of an electronic triode circuit with feedback, rotation of a rigid body with a cavity, reading information from a disk, etc. The highest derivative is preceded by an irreversible operator. This is a Fredholm operator with index zero, kernel of arbitrary dimension, and Jordan chains of arbitrary lengths. Equations with irreversible operators at the highest derivative are called algebro-differential. In this case, the solution to the problem exists under certain conditions on the components of the desired function. To solve the equation with respect to the derivative, the method of cascade splitting of the equation is used, which consists in the stepwise splitting of the equation into equations in subspaces of decreasing dimensions. Cases of one-step and two-step splitting are considered. The splitting uses the result on the solution of a linear equation with Fredholm operator. In each case, the corresponding result is formulated as a theorem. To illustrate the result obtained in the case of one-step splitting, an illustrative example of the Cauchy problem is given.

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.

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