scholarly journals Implicit difference methods for nonlinear first order partial functional differential systems

2010 ◽  
Vol 37 (4) ◽  
pp. 459-482 ◽  
Author(s):  
Elżbieta Puźniakowska-Gałuch
Keyword(s):  
Differential Systems ◽  
First Order ◽  
Functional Differential Systems ◽  
Implicit Difference Methods ◽  
Difference Methods ◽  
Functional Differential

Asymptotic analysis of positive solutions of first-order cyclic functional differential systems

2017 ◽  
Vol 24 (1) ◽  
pp. 63-80
Author(s):  
Jaroslav Jaroš ◽  
Kusano Takaŝi
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Positive Solutions ◽  
Continuous Functions ◽  
Regularly Varying Functions ◽  
Differential Systems ◽  
First Order ◽  
Functional Differential Systems ◽  
Functional Differential ◽  
Scalar Equations

AbstractThe structure and the asymptotic behavior of positive increasing solutions of functional differential systems of the form$x^{\prime}(t)=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(t)=% q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}$are investigated in detail, where α and β are positive constants,${p(t)}$and${q(t)}$are positive continuous functions on${[0,\infty)}$,${k(t)}$and${l(t)}$are positive continuous functions on${[0,\infty)}$tending to${\infty}$witht, and${\varphi_{\gamma}(u)=\lvert u\rvert^{\gamma}\operatorname{sgn}u}$,${\gamma>0}$,${u\in\mathbb{R}}$. An extreme class of positive increasing solutions, calledrapidly increasing solutions, of the system above is analyzed by means of regularly varying functions. The results obtained find applications to systems of the form$x^{\prime}(g(t))=p(t)\varphi_{\alpha}\bigl{(}y(k(t))\bigr{)},\quad y^{\prime}(% h(t))=q(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)},$and to scalar equations of the type$\Bigl{(}p(t)\varphi_{\alpha}\bigl{(}x^{\prime}(g(t))\bigr{)}\Bigr{)}^{\prime}=% p(t)\varphi_{\beta}\bigl{(}x(l(t))\bigr{)}.$

scholarly journals Solvability of discontinuous functional differential systems in l∞(M)

2006 ◽  
Vol 48 (2) ◽  
pp. 237-243
Author(s):  
A. Cabada ◽  
J. Ángel Cid ◽  
S. Heikkilä
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Infinite System ◽  
Functional Differential Equations ◽  
Extremal Solutions ◽  
Differential Systems ◽  
First Order ◽  
Functional Differential Systems ◽  
Bounded Functions ◽  
Functional Differential

AbstractWe study the existence of extremal solutions for an infinite system of first-order discontinuous functional differential equations in the Banach space of the bounded functions I∞(M).

Stability of Implicit Difference Equations Generated by Parabolic Functional Differential Problems

2007 ◽  
Vol 7 (1) ◽  
pp. 68-82
Author(s):  
K. Kropielnicka
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Initial Boundary ◽  
Numerical Examples ◽  
Convergence Results ◽  
Nonlinear Parabolic ◽  
Implicit Difference Methods ◽  
Neumann Type ◽  
Difference Methods ◽  
Functional Differential

AbstractA general class of implicit difference methods for nonlinear parabolic functional differential equations with initial boundary conditions of the Neumann type is constructed. Convergence results are proved by means of consistency and stability arguments. It is assumed that given functions satisfy nonlinear estimates of Perron type with respect to functional variables. Differential equations with deviated variables and differential integral problems can be obtained from a general model by specializing given operators. The results are illustrated by numerical examples.

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