scholarly journals The Exponential Dichotomy under Discretization on General Approximation Scheme

2011 ◽  
Vol 2011 ◽  
pp. 1-22 ◽  
Author(s):  
Javier Pastor ◽  
Sergey Piskarev
Keyword(s):  
Initial Value Problems ◽  
Approximation Scheme ◽  
Finite Difference Methods ◽  
General Setting ◽  
Approximation Schemes ◽  
Initial Value ◽  
General Approximation ◽  
Compact Approximation ◽  
Hyperbolic Equilibrium ◽  
Compact Resolvents

This paper is devoted to the numerical analysis of abstract parabolic problem ; , with hyperbolic generator . We are developing a general approach to establish a discrete dichotomy in a very general setting in case of discrete approximation in space and time. It is a well-known fact that the phase space in the neighborhood of the hyperbolic equilibrium can be split in a such way that the original initial value problem is reduced to initial value problems with exponential decaying solutions in opposite time direction. We use the theory of compact approximation principle and collectively condensing approximation to show that such a decomposition of the flow persists under rather general approximation schemes. The main assumption of our results is naturally satisfied, in particular, for operators with compact resolvents and condensing semigroups and can be verified for finite element as well as finite difference methods.

scholarly journals A Step Block Algorithm for Solving First Order Initial Value Problems in Ordinary Differential Equations

2017 ◽  
Vol 2 (1) ◽  
Author(s):  
Emmanuel O Adeyefa ◽  
Oluwatosin Fadaka
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problems ◽  
Finite Difference Methods ◽  
Initial Value ◽  
Step Method ◽  
Block Algorithm ◽  
One Step ◽  
Difference Methods

The implementation of the newly formulated polynomials, ADEM-B orthogonal polynomials,  valid in the interval [-1, 1] with respect to weight function is our major focus in this work. The polynomials, which serve as basis function are employed to develop finite difference methods. Varying off-step points are considered for only One-Step method for the solution of the initial value problems of Ordinary Differential Equations (ODEs).  By selection of points for both interpolation and collocation, threeimportant class of block finite difference methods are produced. The methods are analyzed for their basic properties and findings show that they are accurate and convergent.

Mathematical Models of Papermaking

2001 ◽  
Vol 6 (1) ◽  
pp. 9-19 ◽  
Author(s):  
A. Buikis ◽  
J. Cepitis ◽  
H. Kalis ◽  
A. Reinfelds ◽  
A. Ancitis ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Initial Value Problems ◽  
Moisture Transfer ◽  
Bound Water ◽  
Drying Process ◽  
Initial Value ◽  
First Order ◽  
Heat And Moisture ◽  
The Mathematical Model ◽  
The Web

The mathematical model of wood drying based on detailed transport phenomena considering both heat and moisture transfer have been offered in article. The adjustment of this model to the drying process of papermaking is carried out for the range of moisture content corresponding to the period of drying in which vapour movement and bound water diffusion in the web are possible. By averaging as the desired models are obtained sequence of the initial value problems for systems of two nonlinear first order ordinary differential equations. 

scholarly journals Intrinsic Discontinuities in Solutions of Evolution Equations Involving Fractional Caputo–Fabrizio and Atangana–Baleanu Operators

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2023
Author(s):  
Christopher Nicholas Angstmann ◽  
Byron Alexander Jacobs ◽  
Bruce Ian Henry ◽  
Zhuang Xu
Keyword(s):  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Evolution Equations ◽  
Integral Transform ◽  
Initial Value ◽  
Intrinsic Nature ◽  
Recent Interest ◽  
Special Cases ◽  
Singular Kernels

There has been considerable recent interest in certain integral transform operators with non-singular kernels and their ability to be considered as fractional derivatives. Two such operators are the Caputo–Fabrizio operator and the Atangana–Baleanu operator. Here we present solutions to simple initial value problems involving these two operators and show that, apart from some special cases, the solutions have an intrinsic discontinuity at the origin. The intrinsic nature of the discontinuity in the solution raises concerns about using such operators in modelling. Solutions to initial value problems involving the traditional Caputo operator, which has a singularity inits kernel, do not have these intrinsic discontinuities.

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