scholarly journals THE SLICE BALANCE APPROACH USING AN ADAPTIVE-WEIGHTED CLOSURE

2021 ◽  
Vol 247 ◽  
pp. 03005
Author(s):  
Michael W. Hackemack
Keyword(s):  
Iterative Methods ◽  
Positive Solutions ◽  
Difference Method ◽  
Cartesian Grids ◽  
Closure Relation ◽  
Balance Approach ◽  
Difference Form

In this paper, we present a formulation of the slice balance approach using a nonlinear closure relation derived analogously from the adaptive-weighted diamond-difference form of the weighted diamond-difference method for Cartesian grids. The method yields strictly positive solutions that reduce to a standard diamond closure with fine-enough mesh granularity. It can be efficiently solved using Newton-like nonlinear iterative methods with diffusion preconditioning.

scholarly journals Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xiaoping Zhang ◽  
Yongping Sun
Keyword(s):  
Differential Equations ◽  
Iterative Methods ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Nonlinear Term ◽  
Interesting Point ◽  
Monotone Iterative ◽  
Zero Function ◽  
Fractional Differential ◽  
Computing Method

This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System

On Positive Solutions for Some Nonlinear Semipositone Elliptic Boundary Value

2006 ◽  
Vol 11 (4) ◽  
pp. 323-329 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Boundary Value Problems ◽  
Positive Solution ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Laplacian Operator ◽  
Smooth Bounded Domain ◽  
Elliptic Boundary Value ◽  
Existence Of Positive Solutions

This study concerns the existence of positive solutions to classes of boundary value problems of the form−∆u = g(x,u), x ∈ Ω,u(x) = 0, x ∈ ∂Ω,where ∆ denote the Laplacian operator, Ω is a smooth bounded domain in RN (N ≥ 2) with ∂Ω of class C2, and connected, and g(x, 0) < 0 for some x ∈ Ω (semipositone problems). By using the method of sub-super solutions we prove the existence of positive solution to special types of g(x,u).

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