Effects of errors in the initial-time geometry on the solution of an equation from dynamo theory in an exterior domain

1995 ◽  
Vol 450 (1938) ◽  
pp. 109-121 ◽  
Keyword(s):  
Differential Equation ◽  
Continuous Dependence ◽  
Exterior Domain ◽  
Initial Time ◽  
Dynamo Theory ◽  
Compact Region ◽  
Time Problem ◽  
Constraint Set ◽  
Similar Inequality ◽  
Three Space

This paper studies the improperly posed backward-in-time problem, in addition to the forward-in-time problem, for a solution to a non-symmetric partial differential equation which arises in dynamo theory. Throughout, the spatial domain is unbounded and exterior to a compact region in three-space. Continuous dependence on changes in the initial-time geometry is established. For the forward-in-time problem, an explicit continuous dependence inequality depending solely on data is derived, while for the backward-in-time problem, a similar inequality is established but the bound depends also on a constraint set.

Spatial decay estimates and continuous dependence on modelling for an equation from dynamo theory

1994 ◽  
Vol 445 (1924) ◽  
pp. 437-451 ◽  
Keyword(s):  
Velocity Field ◽  
Continuous Dependence ◽  
Decay Estimates ◽  
Spatial Decay ◽  
Dynamo Theory ◽  
Dirichlet Integral ◽  
Time Problem ◽  
Unbounded Cylinder ◽  
Well Posed ◽  
The Magnetic Field

This paper studies the improperly posed, backward in time problem, in addition to the well posed forward in time problem, for a non-symmetric partial differential equation which describes the behaviour of the toroidal part of the magnetic field in a dynamo problem. We first show that solutions in an unbounded cylinder decay exponentially in space provided that for the backward in time problem a Dirichlet integral is bounded and provided the prescribed velocity field satisfies particular bounds; for the forward in time problem several of these constraints are relaxed. It is next shown that the solution to the same problem on a bounded spatial domain depends Hölder continuously on changes in the prescribed velocity field.

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 On the existence of positive solutions of a state-dependent neutral functional differential equation with two state-delay functions

2020 ◽  
Vol 4 (2) ◽  
pp. 132-141
Author(s):  
El-Sayed, A. M. A ◽  
◽  
Hamdallah, E. M. A ◽  
Ebead, H. R ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Functional Differential Equation ◽  
Continuous Dependence ◽  
Neutral Functional Differential Equation ◽  
Initial Value ◽  
State Delay ◽  
State Dependent ◽  
Existence Of Positive Solutions ◽  
Functional Differential

In this paper, we study the existence of positive solutions for an initial value problem of a state-dependent neutral functional differential equation with two state-delay functions. The continuous dependence of the unique solution will be proved. Some especial cases and examples will be given.

Continuous Dependence on the Initial-Time Geometry in Generalized Heat Conduction

1997 ◽  
Vol 07 (01) ◽  
pp. 125-138 ◽  
Author(s):  
L. E. Payne ◽  
J. C. Song
Keyword(s):  
Heat Conduction ◽  
Initial Data ◽  
Continuous Dependence ◽  
Conduction System ◽  
Initial Time ◽  
Generalized Heat Conduction

In this paper we investigate continuous dependence on the initial-time geometry for solutions of a generalized heat conduction system. Assuming the initial data to be measured on a surface t = εF(x), for |F| < 1, and assigned at t = 0, we examine the effects of this error in the initial-time geometry on the solution both forward and backward in time.

scholarly journals Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jun Zhou ◽  
Jun Shen
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Positive Solutions ◽  
Functional Differential Equation ◽  
Continuous Dependence ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
State Dependence ◽  
Functional Differential ◽  
Iterative Functional Differential Equation

<p style='text-indent:20px;'>In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation <inline-formula><tex-math id="M1">\begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document}</tex-math></inline-formula> As <inline-formula><tex-math id="M2">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, this equation can be regarded as a mixed-type functional differential equation with state-dependence <inline-formula><tex-math id="M3">\begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document}</tex-math></inline-formula> of a special form but, being a nonlinear operator, <inline-formula><tex-math id="M4">\begin{document}$ n $\end{document}</tex-math></inline-formula>-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.</p>

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