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On stable implicit difference scheme for hyperbolic-parabolic equations in a Hilbert space. (English) Zbl 1175.65103
For a self-adjoint, positive-definite operator \(A\) the authors consider the differential equation \(u' + Au = g\) for \(-1 < t < 0\) followed by \(u'' + Au = f\) for \(0 < t < 1\). The initial condition at \(t = -1\) depends linearly on future values of \(u\) for \(0 < t < 1\). The authors analyze the stability of an implicit discretization.

MSC:
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35M13 Initial-boundary value problems for PDEs of mixed type
34G10 Linear differential equations in abstract spaces
65L12 Finite difference and finite volume methods for ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
65J08 Numerical solutions to abstract evolution equations
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[1] Ashyralyev, On stability estimation of difference scheme of a first order of accuracy for hyperbolic-parabolic equations, Izv Akad Nauk Turkmenistan Ser Fiz-Tekhn Khim Geol Nauk 1 pp 35– (1996)
[2] Ashyralyev, Stability of difference schemes for hyperbolic-parabolic equations, Comput Math Appl 50 pp 1443– (2005) · Zbl 1088.65082
[3] Ashyralyev, On nonlocal boundary value problems for hyperbolic-parabolic equations, Taiwanese J Math 11 pp 1077– (2007) · Zbl 1140.65039
[4] Ashyralyev, A note on the difference schemes for hyperbolic-elliptic equations, Abstract Appl Anal 2006 pp 1– (2006) · Zbl 1133.65058
[5] Bazarov, Some local and nonlocal boundary value problems for equations of mixed and mixed-composite types (1995)
[6] Dzhuraev, Boundary value problems for equations of mixed and mixed-composite types (1979) · Zbl 0487.35068
[7] Glazatov, Nonlocal boundary value problems for linear and nonlinear equations of variable type (1998) · Zbl 0917.35080
[8] Krein, Linear differential equations in a Banach space (1966)
[9] Nakhushev, Equations of mathematical biology (1995)
[10] Salakhitdinov, Boundary value problems for equations of mixed type with a spectral parameter (1997) · Zbl 0993.35002
[11] Berdyshev, Some non-local problems for the parabolic-hyperbolic type equation with non-characteristic line of changing type, Cent Eur J Math 4 pp 183– (2006) · Zbl 1098.35116
[12] Vragov, Boundary value problems for nonclassical equations of mathematical physics (1983)
[13] Piskarev, On certain operator families related to cosine operator function, Taiwanese J Math 1 pp 3585– (1997) · Zbl 0906.47030
[14] Sobolevskii, Difference methods for the approximate solution of differential equations (1975)
[15] Fattorini, Second order linear differential equations in Banach spaces (1985) · Zbl 0564.34063
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