The method of normal splines for linear implicit differential equations of second order

V. K. Gorbunov ; A. Gorobetz ; V. Sviridov

Lobachevskii journal of mathematics, Tome 20 (2005), pp. 59-75.

Résumé

The method of normal splines is specified for the initial and boundary-value problems for systems of linear ordinary differential equations of second order, possible being stiff or unresolved with respect to derivatives (differential-algebraic equations), without their reduction to first order ones. The algorithm of nonuniform collocation grid creation for stiff problems is described. Results of numerical solution to test problems, including linear mathematical physics boundary-value problem of the second order are given. Numerical schemes for the last case are based on the method of lines.

Détail

Keywords: normal splines, singular differential-algebraic equations, adaptive grids, partial differential equations, method of lines.

Bibliographie

[1] Aronszajn N., “Theory of reproducing kernels”, Transactions of the AMS, 68 (1950), 337–404 | DOI | MR | Zbl

[2] Balakrishnan A., Applied Functional Analysis, Springer-Verlag, New York, 1976 | MR | Zbl

[3] Comput. Math. Math. Phys., 42:4 (2002) | MR | Zbl

[4] Doolan E., Miller J., Schilders W., Uniform Numerical Methods for Problems with Initial and Boundary Layers, Bool Press, Dublin, 1980 | MR | Zbl

[5] Comput. Math. Math. Phys., 19:2 (1979) | DOI | MR | Zbl

[6] Comput. Math. Math. Phys., 29:2 (1989) | MR

[7] Gorbunov V. K., Extremum Problems of Measurements Data Processing, Ilim, Frunze, 1990 (Russian)

[8] J. of Computer and Systems Sciences International, 2004, no. 5 | MR

[9] Gorbunov V. K., Lutoshkin I. V., “The parameterization method in optimal control problems and differential-algebraic equations”, J. of Comput. and Applied Math., 185:2 (2006), 377–390 | DOI | MR | Zbl

[10] Comput. Math. Math. Phys., 43:8 (2003), 1099–1108 | MR | Zbl

[11] Gorbunov V. K., Petrischev V. V., Sviridov V. Yu., “Development of the normal spline method for linear integro-differential equations”, Computational Science - ICCS 2003, LNCS 2658, eds. P. Slot et al., Springer, Berlin, 2003, 492–499 | MR

[12] Gorobetz A., “Normal spline-collocation's method for linear differential equations of second order”, Differential Equations and Applications, Proceedings of the International Scientific Conference (26–31 May, 2002), Samara State Architectural - Building Academy, Samara, 2002, 99–104 (Russian)

[13] Hairer E., Wanner G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1996 | MR | Zbl

[14] Himmelblau D., Applied Nonlinear Programming, McGraw-Hill Book Company, Texas, 1972 | Zbl

[15] Krasnoselsky M. A. et al., Approximate Solution to Operator Equations, Nauka, Moskow, 1969 (Russian) | MR

[16] März R., Weinmüller E., “Solvability of boundary value problems for systems of singular differential-algebraic equations”, J. Math. Anal., SIAM, 24 (1993), 200–215 | DOI | MR | Zbl

[17] Comput. Math. Math. Phys., 42 (2002) | MR | Zbl

[18] Comput. Math. Math. Phys., 36 (1996)

[19] Sobolev S. L., Applications of Functional Analysis to Mathematical Physics, Amer. Math. Soc., Providence RI, 1963 | MR | Zbl

[20] Sviridov V. Yu., “Grid optimization in normal spline's method for integro-differential equations”, Proceedings of the Middle Volga Mathematical Society, vol. 3-4, No 1, SVMO, Saransk, 2002, 236–245 (Russian)