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Indexes and special discretization methods for linear partial differential algebraic equations. (English) Zbl 0941.65096
The authors consider linear partial differential algebraic equations with constant matrix coefficients of the form $$Au_t (t,x)+Bu_{xx} (t,x)+Cu(t,x)=f(x,t)\tag 1$$ where $A, B, C \in R^{n\times n}$, at least one of the matrices $A$ and $B$ is singular and none of the matrices $A$ or $B$ is the zero matrix. The equation (1) is supplemented by the boundary (BCs) and initial conditions (ICs) at that, in general, any ICs and BCs cannot be prescribed for all components of the solution vector $u(t,x)$. Two numerical schemes by means of the method of lines and the scheme of the full discretization (BTCS) of initial boundary value problems for (1) are constructed. The influence of the differential spatial and the uniform differential time index on the convergence of the numerical solution to the exact solution is studied. Numerical examples using the BTCS scheme are presented.

65M20Method of lines (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
35R10Partial functional-differential equations
35K15Second order parabolic equations, initial value problems
65M12Stability and convergence of numerical methods (IVP of PDE)
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