- Tytuł:
- Difference methods for the Darboux problem for functional partial differential equations
- Autorzy:
- Człapiński, Tomasz
- Powiązania:
- https://bibliotekanauki.pl/articles/1294108.pdf
- Data publikacji:
- 1999
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
functional differential equation
Darboux problem
classical - Opis:
- 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.
- Źródło:
-
Annales Polonici Mathematici; 1999, 71, 2; 171-193
0066-2216 - Pojawia się w:
- Annales Polonici Mathematici
- Dostawca treści:
- Biblioteka Nauki
Artykuł