On a differential-algebraic problem. (English) Zbl 1058.34005

A method for solving the following system \[ x'(t) = f(t,x(t),y(t)),\;t\in J= [0,b],\;x(0) = k_0,\;y(t) = g(t,x(t),y(t)), \;t\in J, \] where \(f\in C(J,\mathbb R\times \mathbb R,\mathbb R,\;g\in C(J,\mathbb R\times \mathbb R,\mathbb R)\) and \(k_0\in \mathbb R\) are given, of differential-algebraic equations is suggested. The method is based on the idea of quasilinearisation known as a useful instrument for constructing approximate solutions of ordinary differential equations. In the present case, this approach leads to a sequence of approximate solutions which converges – under some natural assumptions – to the unique solution of the original problem. Moreover, the convergence is shown to be monotone and quadratic and hence, the method may be very efficient when solving practical problems.


34A34 Nonlinear ordinary differential equations and systems
34A45 Theoretical approximation of solutions to ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
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[1] J. Bahi, E. Griepentrog and J. C. Miellou: Parallel treatment of a class of differential-algebraic systems. SIAM J. Numer. Anal. 33 (1996), 1969-1980. · Zbl 0859.65073 · doi:10.1137/S0036142993258105
[2] R. Bellman, R. Kalaba: Quasilinearization and Nonlinear Boundary Value Problems. American Elsevier, New York, 1965. · Zbl 0139.10702
[3] Z. Jackiewicz, M. Kwapisz: Convergence of waveform relaxation methods for differential-algebraic systems. SIAM J. Numer. Anal. 33 (1996), 2303-2317. · Zbl 0889.34064 · doi:10.1137/S0036142992233098
[4] T. Jankowski, V. Lakshmikantham: Monotone iterations for differential eguations with a parameter. J. Appl. Math. Stoch. Anal. 10 (1997), 273-278. · Zbl 0992.34010 · doi:10.1155/S1048953397000348
[5] T. Jankowski, M. Kwapisz: Convergence of numerical methods for systems of neutral functional-differential-algebraic eguations. Appl. Math. 40 (1995), 457-472. · Zbl 0853.65077
[6] M. Kwapisz: On solving systems of differential algebraic eguations. Appl. Math. 37 (1992), 257-264. · Zbl 0764.65038
[7] G. S. Ladde, V. Lakshmikantham and A. S. Vatsala: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston, 1985. · Zbl 0658.35003
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