On the numerical solution of a quasilinear algebraic-differential system
The author proposes a spline-collocation method for the initial value problem for the quasilinear differential-algebraic equation \[ A(t)\dot{x}(t) + F(x,t) = 0, \quad t\in{I}=[t_0,T], \quad x(t_0)=x_0, \] where \(A(t)\) is a square \((n\times{n})\) matrix with \(\det A(t)=0\) for all \(t\in{I}\), and \(F(x,t)\) is a given sufficient smooth function. In view of the stiffness of the problem, the author suggests to use the next variant of spline-collocation methods. For a uniform grid \(I_{\tau}=\{t_j=t_0+j\tau\), \(j=0,1,\dots,N\}\) the solution \(x(t)\) is being approximated on the sequence of overlapped intervals \(I_{k}=[t_k, \, t_k + m\tau]\), \(k=0,1,\dots,(N-1)\), by polynomials \(L_k^m(t)\) of degree \(m\) with indefinite coefficients. Having the representation of the polynomials through the Newton interpolation formula, the author obtains a finite-difference scheme which is a set of \(m\) of \(n\)-dimensional systems of nonlinear equations. The convergence of the proposed method is proved under some rank-type conditions, the main of those is the Chistyakov's ``rank-degree'' condition: \[ \operatorname {rank} A(t_0)= \deg\{\det(\lambda A(t_0)+\partial{F(x_0,t_0)}/\partial{x})\}. \] This corresponds to the ``index-one'' property of the DAE. A few of numerical results for test examples are presented.
quasilinear differential-algebraic equations
scientific article; zbMATH DE number 5611140