Difference methods for the Darboux problem for functional partial differential equations

We consider the following Darboux problem: (1) ${\displaystyle D_{xy}z(x,y)=f(x,y,z_{(x,y)},{\left(D_{x}z\right)}_{(x,y)},{\left(D_{y}z\right)}_{(x,y)})}$, (2) z(x,y) = ϕ(x,y) on [-a₀,a] × [-b₀,b] \ (0,a] × (0,b], where ${\displaystyle a₀,b₀\in ℝ₊,a,b>0.Theopera\to r}$[0,a] × [0,b] ∋ (x,y) ↦ ω_{(x,y)} ∈ C([-a₀,0] × [-b₀,0],ℝ)${\displaystyle def\in edby}$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.