Multistep Method for Solving Degenerate Integral-Differential Equations

In this work we consider the linear integral-differential equations of the fist order with the identically singular matrix at the derivative. For these systems, the initial conditions is given and assumed consistent with the right part. Considered tasking in this paper arise in the mathematical modeling of complex electric circuits. By using the apparatus of matrix polynomials a class of problems, which having a unique solution, is marked out. The difficulties of the numerical solution of such problems, in particular the instability of many implicit methods is considered. For numerical solution of this class of problems we have suggested multistep methods, which are based on an explicit quadrature formula for the integral term Adams and extrapolation formulas. Sufficient conditions for the convergence of such algorithms to the exact solution is formulated.

Keywords

integral-differential equations; multistep methods; matrix polynomials.

References

1. Usakov E.I. Staticheskaya ustoychivost' elektricheskikh sistem [Static Stability of Electrical Systems]. Novosibirsk, Nauka, 1988.
2. Scendy K. Sovremennyye metody analiza elektricheskikh tsepey [Modern Methods of Analysis of Electric Circuits]. Moscow, Energiya, 1971. 360 p.
3. Bulatov M.V., Chistyakova E.V. On a Family of Singylar Integro-Differential Equations. Computational Mathematics and Mathematical Physics, 2011, vol. 51, issue 9, pp. 1558-1566. DOI: 10.1134/S0965542511090065
4. Budnikova O.S., Bulatov M.V. Numerical Solution of Integral-algebraic Equations for Multistep Methods. Computational Mathematics and Mathematical Physics, 2012, vol. 52, issue 5, pp. 691-701. DOI: 10.1134/S0965542512050041
5. Bakhvalov N.S. Chislennye metody [Numeriсal Methods]. Moscow, Nauka, 1975.
6. Ten Men Yan Priblizhennoye resheniye lineynykh integral'nykh uravneniy Vol'terra I roda [Approximate Solution of Linear Volterra Integral Equations of the First Kind. Candidate's Dissertation in Mathematics and Physics]. Irkutsk, 1985.
7. Brunner H., van der Houwen P.J. The Numerical Solution of Volterra Equations. Amsterdam: North-Holland, CWI Monographs 3, 1986.
8. Linz P. Analytical and Numerical Methods for Volterra Equations. SIAM, Philadelphia, 1985. DOI: 10.1137/1.9781611970852
9. Bulatov M.V. Regularization of Singular Systems of Volterra Integral Equations. Computational Mathematics and Mathematical Physics, 2002, vol. 42, issue 3, pp. 315-320.
10. Bulatov M.V., Chistyakova E.V. Numerical Solution of Integro-Differential Systems with a Degenerate Matrix Multiplying the Derivative by Multistep Methods. Differential Equations, 2006, vol. 42, issue 9, pp. 1317-1325. DOI: 10.1134/S0012266106090102
11. Bulatov M.V., Lee M.-G. Applications of Matrix Polynomials to the Analysis of Linear Differential-Algegraic Equations of Higher Order. Differential Equations, 2008, vol. 44, issue 10, pp. 1353-1360. DOI: 10.1134/S0012266108100017