Existence, uniqueness and non-uniqueness of weak solutions of parabolic initial-value problems with discontinuous nonlinearities

2005 ◽  
Vol 135 (6) ◽  
pp. 1139-1167 ◽  
Author(s):  
Hideo Deguchi
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Weak Solutions ◽  
Initial Value Problem ◽  
Parabolic Equations ◽  
Initial Value Problems ◽  
Initial Value ◽  
Discontinuous Nonlinearities

We deal with the initial-value problem for parabolic equations with discontinuous nonlinearities and establish the existence of its weak solution. Next, we show that for a suitable class of initial data, the weak solution is locally or globally unique in time. Lastly, we prove that there exist at least two different weak solutions in general if initial data do not belong to this class.

Numerical relativity. I. The characteristic initial value problem

1982 ◽  
Vol 384 (1787) ◽  
pp. 427-454 ◽  
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Numerical Relativity ◽  
Hyperbolic Systems ◽  
Systems Of Equations ◽  
Initial Value ◽  
Hyperbolic Systems Of Equations ◽  
Characteristic Initial Value Problem

In this, the first of a series of papers on numerical relativity, the characteristic initial value problem is posed in a form suitable for numerical integration. It can be reduced to the solution of two initial value problems for sets of ordinary differential equations (on the initial surfaces) and the solution of two initial value problems for hyperbolic systems of equations, one linear, one quasilinear. The initial data may be specified freely. Subsequent papers will develop numerical solutions of Einstein’s equations with use of this formalism.

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 Nonanalytic solutions of the KdV equation

2004 ◽  
Vol 2004 (6) ◽  
pp. 453-460 ◽  
Author(s):  
Peter Byers ◽  
A. Alexandrou Himonas
Keyword(s):  
Initial Data ◽  
Initial Value Problem ◽  
Kdv Equation ◽  
Initial Value ◽  
The Kdv Equation

We construct nonanalytic solutions to the initial value problem for the KdV equation with analytic initial data in both the periodic and the nonperiodic cases.

Dual extremum principles for the heat equation

1977 ◽  
Vol 77 (3-4) ◽  
pp. 273-292 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Initial Value Problem ◽  
General Class ◽  
Initial Value Problems ◽  
Linear Operators ◽  
Point Boundary ◽  
Initial Value ◽  
Quantities Of Interest

SynopsisDual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

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