scholarly journals Convergence of a Time Discretization for a Nonlinear Second-Order Inclusion

2017 ◽  
Vol 61 (1) ◽  
pp. 93-120
Author(s):  
Krzysztof Bartosz ◽  
Leszek Gasiński ◽  
Zhenhai Liu ◽  
Paweł Szafraniec
Keyword(s):  
Source Term ◽  
Existence Of Solutions ◽  
Time Discretization ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Smooth Potential ◽  
Locally Lipschitz ◽  
Time Differential ◽  
Second Order In Time ◽  
The Clarke Subdifferential

AbstractWe study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

Existence of solutions for a second order boundary value problem with the Clarke subdifferential

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2763-2771 ◽  
Author(s):  
Dalila Azzam-Laouir ◽  
Samira Melit
Keyword(s):  
Boundary Value Problem ◽  
Differential Inclusion ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Clarke Subdifferential ◽  
Lipschitzian Function ◽  
The Clarke Subdifferential

In this paper, we prove a theorem on the existence of solutions for a second order differential inclusion governed by the Clarke subdifferential of a Lipschitzian function and by a mixed semicontinuous perturbation.

scholarly journals Relaxed submonotone mappings

2003 ◽  
Vol 2003 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Tzanko Donchev ◽  
Pando Georgiev
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Lipschitz Function ◽  
Clarke Subdifferential ◽  
Monotone Mappings ◽  
Locally Lipschitz Function ◽  
Residual Subset ◽  
Locally Lipschitz ◽  
Whole Space ◽  
The Clarke Subdifferential

The notions ofrelaxed submonotoneandrelaxed monotonemappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hypotheses, the surjectivity of the relaxed monotone mappings.

scholarly journals Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3050
Author(s):  
Sarita Nandal ◽  
Mahmoud A. Zaky ◽  
Rob H. De Staelen ◽  
Ahmed S. Hendy
Keyword(s):  
Fourth Order ◽  
Numerical Scheme ◽  
Source Term ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Accuracy ◽  
Nonlinear Source ◽  
Second Order In Time ◽  
Convergence Orders ◽  
Temporal Direction

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

ON FIRST-ORDER NECESSARY CONDITIONS FOR OPTIMALITY OF REGULAR GENERALIZED SEMI-INFINITE PROGRAMMING

2012 ◽  
Vol 05 (04) ◽  
pp. 1250054
Author(s):  
Nader Kanzi
Keyword(s):  
Constraint Qualification ◽  
Necessary Conditions ◽  
Natural Extension ◽  
Clarke Subdifferential ◽  
Optimal Solutions ◽  
First Order ◽  
Locally Lipschitz ◽  
Necessary Conditions For Optimality ◽  
Infinite Programming ◽  
The Clarke Subdifferential

This paper is concerned with the optimality for generalized semi-infinite programming (GSIP) with nondifferentiable and nonconvex (but being regular in Clarke sense) constraint functions. The objective function is only locally Lipschitz. We consider a lower level constraint qualification which is based on the Clarke subdifferential. This constraint qualification is a natural extension of Mangasarian–Fromovitz one to the differentiable GSIP. The main results are Fritz-John type necessary conditions for optimal solutions.

A MATHEMATICAL MODEL OF ANEURYSM OF CIRCLE OF WILLIS

1995 ◽  
Vol 03 (03) ◽  
pp. 653-659 ◽  
Author(s):  
J. J. NIETO ◽  
A. TORRES
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Ordinary Differential Equation ◽  
Blood Flow ◽  
Nonlinear Analysis ◽  
Existence Of Solutions ◽  
Circle Of Willis ◽  
Second Order ◽  
Qualitative Properties

We introduce a new mathematical model of aneurysm of the circle of Willis. It is an ordinary differential equation of second order that regulates the velocity of blood flow inside the aneurysm. By using some recent methods of nonlinear analysis, we prove the existence of solutions with some qualitative properties that give information on the causes of rupture of the aneurysm.

scholarly journals On second order nonlocal boundary value problem at resonance

2016 ◽  
Vol 56 (1) ◽  
pp. 143-153 ◽  
Author(s):  
Katarzyna Szymańska-Dębowska
Keyword(s):  
Boundary Value Problem ◽  
Bounded Variation ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Second Order ◽  
Function Of Bounded Variation ◽  
Nonlocal Boundary Value Problem ◽  
At Resonance ◽  
Problem At Resonance

Abstract This work is devoted to the existence of solutions for a system of nonlocal resonant boundary value problem $$\matrix{{x'' = f(t,x),} \hfill & {x'(0) = 0,} \hfill & {x'(1) = {\int_0^1 {x(s)dg(s)},} }} $$ where f : [0, 1] × ℝk → ℝk is continuous and g : [0, 1] → ℝk is a function of bounded variation.

scholarly journals On a Differential Equation Involving Hilfer-Hadamard Fractional Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
M. D. Qassim ◽  
K. M. Furati ◽  
N.-E. Tatar
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Global Solution ◽  
Differential Inequality ◽  
Source Term ◽  
Existence Of Solutions ◽  
Hadamard Fractional Derivative ◽  
Fractional Differential

This paper studies a fractional differential inequality involving a new fractional derivative (Hilfer-Hadamard type) with a polynomial source term. We obtain an exponent for which there does not exist any global solution for the problem. We also provide an example to show the existence of solutions in a wider space for some exponents.

An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data

2021 ◽  
pp. 2150056
Author(s):  
Sheetal Chawla ◽  
Jagbir Singh ◽  
Urmil
Keyword(s):  
Uniform Convergence ◽  
Order Parameter ◽  
Source Term ◽  
Reaction Diffusion ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Diffusion Type ◽  
Bakhvalov Mesh ◽  
Second Order Parameter

In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.

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