scholarly journals Existence and uniqueness of analytic solutions of the Shabat equation

2005 ◽  
Vol 2005 (8) ◽  
pp. 855-862 ◽  
Author(s):  
Eugenia N. Petropoulou
Keyword(s):  
Upper Bound ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Analytic Solution ◽  
Sufficient Conditions ◽  
Analytic Solutions ◽  
Initial Condition ◽  
Initial Value ◽  
First Derivative

Sufficient conditions are given so that the initial value problem for the Shabat equation has a unique analytic solution, which, together with its first derivative, converges absolutely forz∈ℂ:|z|<T,T>0. Moreover, a bound of this solution is given. The sufficient conditions involve only the initial condition, the parameters of the equation, andT. Furthermore, from these conditions, one can obtain an upper bound forT. Our results are in consistence with some recently found results.

Bounded solutions of functional differential equations of the neutral type with infinite delays

1982 ◽  
Vol 93 (1-2) ◽  
pp. 33-39 ◽  
Author(s):  
V. G. Angelov ◽  
D. D. Bainov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Functional Differential Equations ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Bounded Solutions ◽  
Initial Value ◽  
Infinite Delays ◽  
Functional Differential

SynopsisIn this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

scholarly journals New properties of the time-scale fractional operators with application to dynamic equations

2021 ◽  
Vol 25 (1) ◽  
pp. 123-136
Author(s):  
Cherif Benaissa ◽  
Ladrani Zohra
Keyword(s):  
Differential Equations ◽  
Time Scales ◽  
Initial Value Problem ◽  
Time Scale ◽  
Existence And Uniqueness ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Fractional Operators ◽  
Initial Value

We introduce new properties of Riemann-Liouville fractional integral and derivative on time scales. As well as sufficient conditions for existence and uniqueness of solution to an initial value problem for a class differential equations on time scales.

ORDER AND CONVERGENCE OF THE ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING INITIAL VALUE PROBLEM

2021 ◽  
Vol 5 (2) ◽  
pp. 442-446
Author(s):  
Muhammad Abdullahi ◽  
Hamisu Musa
Keyword(s):  
Initial Value Problem ◽  
Initial Value Problems ◽  
Sufficient Conditions ◽  
Initial Value ◽  
Differentiation Formula ◽  
Stiff Initial Value Problems ◽  
First Order ◽  
Backward Differentiation Formula ◽  
Fully Implicit ◽  
Necessary And Sufficient

This paper studied an enhanced 3-point fully implicit super class of block backward differentiation formula for solving stiff initial value problems developed by Abdullahi & Musa and go further to established the necessary and sufficient conditions for the convergence of the method. The method is zero stable, A-stable and it is of order 5. The method is found to be suitable for solving first order stiff initial value problems

scholarly journals Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

Universe ◽  
2019 ◽  
Vol 5 (12) ◽  
pp. 224 ◽  
Author(s):  
Stefano Vignolo
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Well Posedness ◽  
Junction Conditions ◽  
Field Equations ◽  
Initial Value ◽  
The Cauchy Problem ◽  
General Conditions

We discuss the Cauchy problem and the junction conditions within the framework of f ( R ) -gravity with torsion. We derive sufficient conditions to ensure the well-posedness of the initial value problem, as well as general conditions to join together on a given hypersurface two different solutions of the field equations. The stated results can be useful to distinguish viable from nonviable f ( R ) -models with torsion.

scholarly journals Existence, uniqueness and continuability of solutions of impulsive differential-difference equations

1999 ◽  
Vol 12 (3) ◽  
pp. 293-300 ◽  
Author(s):  
D. D. Bainov ◽  
I. M. Stamova
Keyword(s):  
Difference Equations ◽  
Initial Value Problem ◽  
Sufficient Conditions ◽  
Initial Value

We consider an initial value problem for impulsive differential-difference equations, and obtain sufficient conditions for the existence, uniqueness, and continuability of solutions of such problem.

THE INITIAL VALUE PROBLEM FOR THE DEEP BÉNARD CONVECTION EQUATIONS WITH DATA IN Lq

1996 ◽  
Vol 06 (02) ◽  
pp. 269-277 ◽  
Author(s):  
Z. CHARKI
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Bénard Convection ◽  
Benard Convection ◽  
Point Argument

A fixed point argument is used to prove the existence and uniqueness of solutions for the unsteady deep Bénard convection equations in [Formula: see text] for [Formula: see text].

The Cauchy problem for the system of Zakharov equations arising from ion-acoustic modes

1996 ◽  
Vol 126 (4) ◽  
pp. 811-820 ◽  
Author(s):  
Guo Boling ◽  
Yuan Guangwei
Keyword(s):  
Cauchy Problem ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
A Priori ◽  
Cauchy Data ◽  
Continuous Method ◽  
Initial Value ◽  
Acoustic Modes ◽  
Ion Acoustic ◽  
The Cauchy Problem

In this paper the initial value problem for a class of Zakharov equations arising from ion-acoustic modes is discussed. Without assuming the Cauchy data are small, we prove the existence and uniqueness of the global smooth solution for the problem via the so-called continuous method and delicate a priori estimates.

York’s Solution to the Initial-Value Problem

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Hamiltonian Constraint ◽  
Conformal Transformations ◽  
Initial Value ◽  
Spatial Metrics ◽  
Conformal Method ◽  
Conformal Equivalence

In this chapter I briefly review York’s method (or the conformal method) for solving the initial value problem of (GR). This method, developed initially by Lichnerowicz and then generalized by Choquet-Bruhat and York, allows to find solutions of the constraints of (GR) (in particular the Hamiltonian, or refoliation constraint) by scanning the conformal equivalence class of spatial metrics for a solution of the Hamiltonian constraint, exploiting the fact that, in a particular foliation (CMC), the transverse nature of the momentum field is preserved under conformal transformations. This method allows to transform the initial value problem into an elliptic problem for the solution for which good existence and uniqueness theorems are available. Moreover this method allows to identify the reduced phase space of (GR) with the cotangent bundle to conformal superspace (the space of conformal 3-geometries), when the CMC foliation is valid. SD essentially amounts to taking this phase space as fundamental and renouncing the spacetime description when the CMC foliation is not available.

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