BSU Bulletin. Mathematics, Informatics
Bibliographic description:
ON SELF-REGULATING PROPERTIES OF COLLOCATION VARIATIONAL METHOD FOR LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS // BSU Bulletin. Mathematics, Informatics. - 2020. №3. . - С. 12-18.
Title:
ON SELF-REGULATING PROPERTIES OF COLLOCATION VARIATIONAL METHOD FOR LINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS
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The article deals with linear interrelated systems of algebraic and ordinary differential equations, which are commonly occurring in important applied problems of power
engineering, kinetic chemistry, biology, and other areas. In the literature, they are usually
called differential-algebraic equations. We have emphasized the difficulties arising in the
numerical solution of such problems and their characteristic properties, in particular, illposedness. The article described in detail the construction of one particular case of the collocation variational approach, which have given a good account of solving various classes
of differential-algebraic equations. This approach is based on solving a specific problem of
mathematical programming. Using a test example, we have shown that this difference
scheme can generate a regularization algorithm (the so-called self-regulation property) with
a regularization parameter — a grid step.
engineering, kinetic chemistry, biology, and other areas. In the literature, they are usually
called differential-algebraic equations. We have emphasized the difficulties arising in the
numerical solution of such problems and their characteristic properties, in particular, illposedness. The article described in detail the construction of one particular case of the collocation variational approach, which have given a good account of solving various classes
of differential-algebraic equations. This approach is based on solving a specific problem of
mathematical programming. Using a test example, we have shown that this difference
scheme can generate a regularization algorithm (the so-called self-regulation property) with
a regularization parameter — a grid step.
Keywords:
algebraic-differential equations; difference schemes; numerical technique; regularizing algorithm; ill-posed problems; regularization algorithm; operational margin; quasioptimal step; ordinary differential equations; integral equations.
List of references:
1. Apartsin A. S., Bakushinskii A. B. Priblizhennoe reshenie integralnykh uravnenii Volterra 1 roda metodom kvadratur [Approximate Solution of Volterra Integral Equations of the First Kind by the Quadrature Method]. Differentsialnye i integralnye uravneniya. 1972. Vol. 1. Pp. 248–258.
2. Boyarintsev Yu. E. Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii [Regular and Singular Systems of Ordinary Differential Equations]. Novosibirsk: Nauka Publ., 1980. 222 p.
3. Bulatov M. V., Gorbunov V. K., Martynenko Yu. V., Nguen Din Kong. Variatsionnye podkhody k chislennomu resheniyu differentsialno-algebraicheskikh uravnenii [Variational Approaches to the Numerical Solution of Differential-Algebraic Equations]. Vychislitelnye tekhnologii. 2010. Vol. 15. No. 5. Pp. 3–14.
4. Bulatov M. V., Rakhvalov N. P., Solovarova L. S. Chislennoe reshenie differentsialno-algebraicheskikh uravnenii metodom kollokatsionno-variatsionnykh splainov [Numerical Solution of Differential-Algebraic Equations using the Spline Collocation Variational Method]. Computational Mathematics and Mathematical Physics. 2013. Vol. 53. No. 3. Pp. 46–58. DOI: 10.7868/S0044466913030046.
5. Solovarova L. S. O kollokatsionno-variatsionnoi raznostnoi skheme dlya differentsialno-algebraicheskikh uravnenii [On Collocation Variational Difference Scheme for Differential-Algebraic Equations]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2017. No. 3. Pp. 3–9. DOI: 10.18101/2304-5728-2017-3-3-9.
6. Tikhonov A. N., Leonov A. S., Yagola A. G. Nelineinye nekorrektnye zadachi. [Nonlinear Ill-Posed Problems]. Moscow: Kurs Publ., 2017. 400 p.
7. Chistyakov V. F. Algebro-differentsialnye operatory s konechnomernym yadrom [Algebraic Differential Operators with a Finite-Dimensional Kernel]. Novosibirsk: Nauka Publ., 1996. 280 p.
8. Bulatov M., Solovarova L. On Self-Regularization Properties of a Difference Scheme for Linear Differential-Algebraic Equations. Applied Numerical Mathematics. 2018. Vol. 130. P. 86–94. DOI:10.1016/j.apnum.2018.03.015
9. Lamour R., März R., Tischendorf C. Differential-Algebraic Equations: A Projector Based Analysis. Berlin: Springer, 2013. 649 p.
10. Apartsyn A. S. Nonclassical Linear Volterra Equations of the First Kind. Utrecht; Boston: VSP, 2003. 168 p.
2. Boyarintsev Yu. E. Regulyarnye i singulyarnye sistemy lineinykh obyknovennykh differentsialnykh uravnenii [Regular and Singular Systems of Ordinary Differential Equations]. Novosibirsk: Nauka Publ., 1980. 222 p.
3. Bulatov M. V., Gorbunov V. K., Martynenko Yu. V., Nguen Din Kong. Variatsionnye podkhody k chislennomu resheniyu differentsialno-algebraicheskikh uravnenii [Variational Approaches to the Numerical Solution of Differential-Algebraic Equations]. Vychislitelnye tekhnologii. 2010. Vol. 15. No. 5. Pp. 3–14.
4. Bulatov M. V., Rakhvalov N. P., Solovarova L. S. Chislennoe reshenie differentsialno-algebraicheskikh uravnenii metodom kollokatsionno-variatsionnykh splainov [Numerical Solution of Differential-Algebraic Equations using the Spline Collocation Variational Method]. Computational Mathematics and Mathematical Physics. 2013. Vol. 53. No. 3. Pp. 46–58. DOI: 10.7868/S0044466913030046.
5. Solovarova L. S. O kollokatsionno-variatsionnoi raznostnoi skheme dlya differentsialno-algebraicheskikh uravnenii [On Collocation Variational Difference Scheme for Differential-Algebraic Equations]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2017. No. 3. Pp. 3–9. DOI: 10.18101/2304-5728-2017-3-3-9.
6. Tikhonov A. N., Leonov A. S., Yagola A. G. Nelineinye nekorrektnye zadachi. [Nonlinear Ill-Posed Problems]. Moscow: Kurs Publ., 2017. 400 p.
7. Chistyakov V. F. Algebro-differentsialnye operatory s konechnomernym yadrom [Algebraic Differential Operators with a Finite-Dimensional Kernel]. Novosibirsk: Nauka Publ., 1996. 280 p.
8. Bulatov M., Solovarova L. On Self-Regularization Properties of a Difference Scheme for Linear Differential-Algebraic Equations. Applied Numerical Mathematics. 2018. Vol. 130. P. 86–94. DOI:10.1016/j.apnum.2018.03.015
9. Lamour R., März R., Tischendorf C. Differential-Algebraic Equations: A Projector Based Analysis. Berlin: Springer, 2013. 649 p.
10. Apartsyn A. S. Nonclassical Linear Volterra Equations of the First Kind. Utrecht; Boston: VSP, 2003. 168 p.
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