Abstract
In a finite-dimensional Hilbert space, we consider the Cauchy problem for a second-order integro-differential evolution equation with memory where the integrand is the product of a difference kernel by a linear operator of the time derivative of the solution. The main difficulties in finding the approximate value of the solution of such nonlocal problems at a given point in time are due to the need to work with approximate values of the solution for all previous points in time. A transformation of the integro-differential equation in question to a system of weakly coupled local evolution equations is proposed. It is based on the approximation of the difference kernel by a sum of exponentials. We state a local problem for a weakly coupled system of equations with additional ordinary differential equations. To solve the corresponding Cauchy problem, stability estimates of the solution with respect to the initial data and the right-hand side are given. The main attention is paid to the construction and stability analysis of three-level difference schemes and their computational implementation.
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REFERENCES
Dautray, R. and Lions, J.-L., Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 1 , Berlin: Springer-Verlag, 2000.
Evans, L.C., Partial Differential Equations, Providence: Am. Math. Soc., 2010.
Gripenberg, G., Londen, S.-O., and Staffans, O., Volterra Integral and Functional Equations, Cambridge: Cambridge Univ. Press, 1990.
Prüss, J., Evolutionary Integral Equations and Applications, Basel: Birkhäuser, 1993.
Christensen, R.M., Theory of Viscoelasticity: an Introduction, New York: Academic Press, 1982.
Marques, S.P. and Creus, G.J., Computational Viscoelasticity, Berlin: Springer, 2012.
Knabner, P. and Angermann, L., Numerical Methods for Elliptic and Parabolic Partial Differential Equations, New York: Springer-Verlag, 2003.
Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Berlin: Springer, 1994.
Chen, C. and Shih, T., Finite Element Methods for Integrodifferential Equations, Singapore: World Sci., 1998.
McLean, W. and Thomee, V., Numerical solution of an evolution equation with a positive-type memory term, ANZIAM J., 1993, vol. 35, no. 1, pp. 23–70.
McLean, W., Thomee, V., and Wahlbin, L.B., Discretization with variable time steps of an evolution equation with a positive-type memory term, J. Comput. Appl. Math., 1996, vol. 69, no. 1, pp. 49–69.
Linz, P., Analytical and Numerical Methods for Volterra Equations, Philadelphia: SIAM, 1985.
Vabishchevich, P.N., Numerical solution of the Cauchy problem for Volterra integrodifferential equations with difference kernels, Appl. Numer. Math., 2022, vol. 174, pp. 177–190.
Vabishchevich, P.N., Approximate solution of the Cauchy problem for a first-order integrodifferential equation with solution derivative memory, arXiv, 2021, no. 2111.05121, pp. 1–13.
Halanay, A., On the asymptotic behavior of the solutions of an integro-differential equation, J. Math. Anal. Appl., 1965, vol. 10, no. 2, pp. 319–324.
Samarskii, A.A., The Theory of Difference Schemes, New York: Marcel Dekker, 2001.
Samarskii, A.A., Matus, P.P., and Vabishchevich, P.N., Difference Schemes with Operator Factors, Dordrecht: Springer Sci.+Bus. Media, 2002.
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Translated by V. Potapchouck
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Vabishchevich, P.N. Numerical Solution of the Cauchy Problem for a Second-Order Integro-Differential Equation. Diff Equat 58, 899–907 (2022). https://doi.org/10.1134/S0012266122070047
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DOI: https://doi.org/10.1134/S0012266122070047