Convergence results for unbounded solutions of first order non-linear differential-functional equations

Henryk Leszczyński

Annales Polonici Mathematici (1996)

  • Volume: 64, Issue: 1, page 1-16
  • ISSN: 0066-2216

Abstract

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We consider the Cauchy problem in an unbounded region for equations of the type either D t z ( t , x ) = f ( t , x , z ( t , x ) , z ( t , x ) , D x z ( t , x ) ) or D t z ( t , x ) = f ( t , x , z ( t , x ) , z , D x z ( t , x ) ) . We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.

How to cite

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Henryk Leszczyński. "Convergence results for unbounded solutions of first order non-linear differential-functional equations." Annales Polonici Mathematici 64.1 (1996): 1-16. <http://eudml.org/doc/269997>.

@article{HenrykLeszczyński1996,
abstract = {We consider the Cauchy problem in an unbounded region for equations of the type either $D_\{t\}z(t,x) = f(t,x,z(t,x),z_\{(t,x)\},D_\{x\}z(t,x))$ or $D_\{t\}z(t,x)= f(t,x,z(t,x),z,D_\{x\}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.},
author = {Henryk Leszczyński},
journal = {Annales Polonici Mathematici},
keywords = {error estimates; recurrence inequalities; difference scheme; Cauchy problem},
language = {eng},
number = {1},
pages = {1-16},
title = {Convergence results for unbounded solutions of first order non-linear differential-functional equations},
url = {http://eudml.org/doc/269997},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Henryk Leszczyński
TI - Convergence results for unbounded solutions of first order non-linear differential-functional equations
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 1
SP - 1
EP - 16
AB - We consider the Cauchy problem in an unbounded region for equations of the type either $D_{t}z(t,x) = f(t,x,z(t,x),z_{(t,x)},D_{x}z(t,x))$ or $D_{t}z(t,x)= f(t,x,z(t,x),z,D_{x}z(t,x))$. We prove convergence of their difference analogues by means of recurrence inequalities in some wide classes of unbounded functions.
LA - eng
KW - error estimates; recurrence inequalities; difference scheme; Cauchy problem
UR - http://eudml.org/doc/269997
ER -

References

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  1. [1] P. Besala, On solutions of first order partial differential equations defined in an unbounded zone, Bull. Acad. Polon. Sci. 12 (1964), 95-99. Zbl0151.14304
  2. [2] P. Besala, Finite difference approximation to the Cauchy problem for non-linear parabolic differential equations, Ann. Polon. Math. 46 (1985), 19-26. Zbl0601.65073
  3. [3] Z. Kamont, On the Cauchy problem for system of first order partial differential equations, Serdica 5 (1979), 327-339. Zbl0446.35029
  4. [4] M. Krzyżański, Partial Differential Equations of Second Order, PWN, Warszawa, 1971. 
  5. [5] H. Leszczyński, General finite difference approximation to the Cauchy problem for non-linear parabolic differential-functional equations, Ann. Polon. Math. 53 (1991), 15-28. Zbl0731.65079
  6. [6] H. Leszczyński, Uniqueness results for unbounded solutions of first order non-linear differential-functional equations, Acta Math. Hungar. 64 (1994), 75-92. Zbl0804.35137
  7. [7] M. Malec et A. Schiaffino, Méthode aux différences finies pour une équation non-linéaire différentielle fonctionnelle du type parabolique avec une condition initiale de Cauchy, Boll. Un. Mat. Ital. (7) 1-B (1987), 99-109. Zbl0617.65083
  8. [8] K. Prządka, Difference methods for non-linear partial differential functional equations of the first order, Math. Nachr. 138 (1988), 105-123. Zbl0668.65093
  9. [9] J. Szarski, Differential Inequalities, PWN, Warszawa, 1967. 

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