Abstract
The normal spline method is developed for the initial and boundary-value problems for linear integro-differential equations, probably being unresolved with respect to the derivatives, in Sobolev spaces of the arbitrary smoothness. It allows to solve a high-order systems without the reduction to first-order ones. The solving system can be arbitrary degenerate (with high differentiation index or irreducible to normal form). The method of nonuniform collocation grid creation for stiff problems is offered. Results of numerical solution to test problems are demonstrated.
Supported by Russian Foundation for Basic Research, project N o 01-01-00731
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Keywords
- Adaptive Grid
- Arbitrary Degenerate
- Functional Analysis Result
- Collocation Grid
- Apply Functional Analysis
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References
Hairer, E., Wanner, G.: Solving Ordinary Differential equations II. Stiff and Differential-algebraic Problems. Springer-Verlag, Berlin (1996)
März, R., Weinmüller, E.B.: Solvability of boundary value problems for systems of singular differential-algebraic equations. SIAM J. Math. Anal. Vol. 24. 1 (1993) 200–215
Bulatov, M.V., Chistyakov, V.F.: About the numerical method of differantial-algebraic equations solving. Comput. Math. Math. Phys. Vol. 42.4 (2002) 459–470
Gorbunov, V.K.: The method of normal spline-collocation. Comput. Math. Math. Phys. Vol. 29.2 (1989) 212–224
Gorbunov, V.K.: Ekstremalnie zadachi obrabotki rezul’tatov izmerenei. Ilim, Frunze (1990)
Sobolev, S.L.: Applications of functional analysis to mathematical physics. Amer. Math. Soc. Providence RI (1963)
Balakrishnan, A.: Applied Functional Analysis. Springer-Verlag, New York (1976)
Aronszajn, N.: Theory of reproducing kernels. Tranzactions of the AMS 68 (1950) 337–404
Gorbunov, V.K., Petrischev, V.V.: Developing of the normal spline-collocation method for linear differential equations. Comput. Math. Math. Phys. (2003) (to appear)
Gorobetz, A.S.: Metod normal’nih splainov dlya system ODU vtorogo poryadka i zadach matematicheskoi phiziki. Differencial’nie uravneniya i ih prilogeniya: Sbornik trudov mejdunarodnoy konferencii. Samara (2002) 99–104
Kohanovsky, I.I.: Normalnije splaini v vichislitel’noi tomografii. Autometria 2 (1995) 84–89
Sviridov, V. Yu.: Optimizaciya setok metoda normal’nih splainov dlya integro-differencial’nih uravnenei. Trudi Srednevolgskogo matematicheskogo obschestva. SVMO, Saransk 3–4 (2002) 236–245
Himmelblau, D.M.: Applied nonlinear programming. McGraw-Hill Book Company, Texas (1972)
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Gorbunov, V.K., Petrischev, V.V., Sviridov, V.Y. (2003). Development of the Normal Spline Method for Linear Integro-Differential Equations. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J.J., Zomaya, A.Y. (eds) Computational Science — ICCS 2003. ICCS 2003. Lecture Notes in Computer Science, vol 2658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44862-4_52
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DOI: https://doi.org/10.1007/3-540-44862-4_52
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