Semerdzhieva, R. I. On a difference method for solving a class of hyperbolic equations with degeneration of the order. (Russian) Zbl 0685.65084 Differ. Uravn. 25, No. 10, 1755-1766 (1989). 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 Cited in 1 Review 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 Keywords:degeneration of the order; finite difference method; singular hyperbolic differential equations; maximum principle; difference equation; convergence PDFBibTeX XMLCite \textit{R. I. Semerdzhieva}, Differ. Uravn. 25, No. 10, 1755--1766 (1989; Zbl 0685.65084)