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Tytuł artykułu

Difference methods for the Darboux problem for functional partial differential equations

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EN
Abstrakty
EN
We consider the following Darboux problem:
(1) $D_{xy}z(x,y) = f(x,y,z_{(x,y)},(D_xz)_{(x,y)},(D_yz)_{(x,y)})$,
(2) z(x,y) = ϕ(x,y) on [-a₀,a] × [-b₀,b] \ (0,a] × (0,b],
where $a₀,b₀ ∈ ℝ₊, a,b > 0. The operator $[0,a] × [0,b] ∋ (x,y) ↦ ω_{(x,y)} ∈ C([-a₀,0] × [-b₀,0],ℝ)$ defined by $ω_{(x,y)}(t,s) = ω(t+x,s+y)$ represents the functional dependence on the unknown function and its derivatives. We construct a wide class of difference methods for problem (1),(2). We prove the existence of solutions of implicit functional systems by means of a comparative method. We get two convergence theorems for implicit and explicit schemes, in the latter case with a nonlinear estimate with respect to the third variable. We give numerical examples to illustrate these results.
Twórcy
  • Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
Bibliografia
  • [1] P. Brandi, Z. Kamont and A. Salvadori, Approximate solutions of mixed problems for first order partial differential-functional equations, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 277-302.
  • [2] T. Człapiński, Existence of solutions of the Darboux problem for partial differen- tial-functional equations with infinite delay in a Banach space, Comment. Math. 35 (1995), 111-122.
  • [3] Z. Denkowski and A. Pelczar, On the existence and uniqueness of solutions of some partial differential functional equations, Ann. Polon. Math. 35 (1978), 261-304.
  • [4] Z. Kamont, Finite difference approximations for first-order partial differential-functional equations, Ukrain. Math. J. 46 (1994), 265-287.
  • [5] Z. Kamont and M. Kwapisz, On the Cauchy problem for differential-delay equations in a Banach space, Math. Nachr. 74 (1976), 173-190.
  • [6] Z. Kamont and M. Kwapisz, On non-linear Volterra integral-functional equations in several variables, Ann. Polon. Math. 40 (1981), 1-29.
  • [7] Z. Kamont and H. Leszczyński, Stability of difference equations generated by parabolic differential-functional problems, Rend. Mat. 16 (1996), 265-287.
  • [8] Z. Kamont and H. Leszczyński, Numerical solutions to the Darboux problem with the functional dependence, Georgian Math. J. 5 (1998), 71-90.
  • [9] H. Leszczyński, Convergence of one-step difference methods for nonlinear para- bolic differential-functional systems with initial-boundary conditions of the Dirichlet type, Comment. Math. 30 (1990), 357-375.
  • [10] H. Leszczyński, Convergence of difference analogues to the Darboux problem with functional dependence, Bull. Belgian Math. Soc. 5 (1998), 39-57.
  • [11] M. Malec, C. Mączka and W. Voigt, Weak difference-functional inequalities and their applications to the difference analogue of non-linear parabolic differential-functional equations, Beiträge Numer. Math. 11 (1983), 69-79.
  • [12] M. Malec and M. Rosati, Weak monotonicity for non linear systems of functional-finite difference inequalities of parabolic type, Rend. Mat. 3 (1983), 157-170.
  • [13] M. Malec et A. Schiaffino, Méthode aux différence finies pour une équation non-linéaire différentielle fonctionnelle du type parabolique avec une condition initiale de Cauchy, Boll. Un. Mat. Ital. 7B (1987), 99-109.
  • [14] A. Pelczar, Some functional-differential equations, Dissertationes Math. 100 (1973).
  • [15] T. Ważewski, Sur une extension du procédé de I. Jungermann pour établir la convergence des approximations successives au cas des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8 (1960), 43-46.
  • [16] M. Zennaro, Delay differential equations: theory and numerics, in: Theory and Numerics of Ordinary and Partial Differential Equations, Clarendon Press, Oxford, 1995, 291-333.
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Bibliografia
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bwmeta1.element.bwnjournal-article-apmv71z2p171bwm
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