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Numerical solutions to the Darboux problem with functional dependence. (English) Zbl 0955.65076

Summary: The paper deals with the Darboux problem for the equation \[ D_{xy}z(x, y)= f(x,y,z_{(x,y)}), \] where \(z_{(x,y)}\) is a function defined by \(z_{(z,y)}(t,s)= z(x+ t,y+s)\), \((t,s)\in [-a_0,0]\times [- b_0,0]\). We construct a general class of difference methods for this problem. We prove the existence and uniqueness of solutions to implicit functional difference equations by means of a comparison method; moreover, we give an error estimate. The convergence of explicit difference schemes is proved under a general assumptions that given functions satisfy nonlinear estimates of the Perron type. Our results are illustrated by a numerical example.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
35R10 Partial functional-differential equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
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