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On a difference method for solving a class of hyperbolic equations with degeneration of the order. (Russian) Zbl 0685.65084

The numerical solution for boundary value problems of singular hyperbolic differential equations is discussed. The equation considered is as follows: \(LU=k(y)U_{xx}-\partial_ y(l(y)U_ y)+a(x,y)U_ x+c(x,y)U=f(x,y),\) where \(k(y)>0\) if \(y>0,\) \(k(0)=0;\) \(l(y)>0\) if \(y>0,\) \(l(0)=0;\) \(l'(0)=\tau =const.>0,\) in the finite one-connected region G with partial-smooth boundaries.
The problem is to find a solution in G, where \(U|_{AC}=\phi\), \(\phi\) is a continuous function, determined on the characteristic line AC. A difference operator \(L_ h\) is involved instead of the differential operator L on the net boundary \(\bar G_ h\) \((G_ h=\bar G_ h\Gamma_ h\), where \(\Gamma_ h\) are the set points on AC). A theorem for \(\sup_{G_ h}| L_ hU-LU| \to 0\) for \(U(x,y)\in C^ 2(\bar G)\) is proved. A maximum principle for the difference equation \(L_ hU_ h=f\) is also proved and sufficient conditions for the coefficients are presented. An a priori evaluation of the solution of the difference equations and a convergence theorem are derived.
Reviewer: S.Spassov

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L80 Degenerate hyperbolic equations
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