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Difference methods for the Darboux problem for functional partial differential equations

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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.
  • Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
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