BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, MULTISTEP METHODS FOR NUMERICAL SOLUTION OF INTEGRAL ALGEBRAIC EQUATIONS OF INDEX-2 // BSU Bulletin. Mathematics, Informatics. - 2019. №2. . - С. 3-15.
Title:
MULTISTEP METHODS FOR NUMERICAL SOLUTION OF INTEGRAL ALGEBRAIC EQUATIONS OF INDEX-2
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Many processes in various eco-biological and physical systems are described by in- terconnected Volterra integral equations of the first and the second kinds. Such equations may be written as a system with an identically singular principal part, in other words, in the form of integral algebraic equation. The article studies the class of linear integral algebraic equations, for which the sufficient conditions for exis- tence of a unique continuous solution are formulated in terms of matrix polynomi- als. The fundamental difference between the problems under consideration and the Volterra integral equations of the first and the second kinds is noted. There are few works on the qualitative theory of integral algebraic equations, and numerical methods for their solution are underdeveloped. Many methods for numerical solu- tion of Volterra integral equations are not applicable or lead to divergent processes. We have proposed one- and two-step methods based on modifications of Adams– Bashforth and Adams–Moulton formulas for the selected class of problems. The re- sults of numerical experiments confirm the efficiency of the algorithms.
Keywords:
multistep methods; integral algebraic equations; quadrature rule; ap- proximation; index; matrix polynomial.
List of references:
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Glushkov V. M., Ivanov V. V., Yanenko V. M. Modelirovanie razviva- yushchikhsya sistem [Modeling of Evolutionary Systems]. Moscow: Nauka Publ., 1983. 350 p.
Ten Men Yan. Priblizhennoe reshenie lineinykh integralnykh uravnenii Volterra I roda. Dis. … kand. fiz.-mat. nauk [Approximate Solution of Linear Volterra Integral Equations of the First Kind. Cand. math. and phys. sci. diss.]. Irkutsk, 1985. 215 p.
Chistyakov V. F. Chistyakov V. F. O singulyarnykh sistemakh obyknovennykh differentsialnykh uravnenii i ikh integralnykh analogakh [On Singular Systems of Ordinary Differential Equations and Their Integral Analogues]. Funktsii Lyapunova i ikh primeneniya. Novosibirsk: Nauka Publ., 1987. Pp. 231–239.
Algebro-differentsialnye operatory s konechnomernym yadrom [Algebraic Differential Operators with Finite-Dimensional Kernel]. Novosibirsk: Nauka Publ., 1996. 280 p.
О. С. Будникова, М. Н. Ботороева. Многошаговые методы для численного реше- ния интегро-алгебраических уравнений индекса два
image
Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Unversity Press, 2004. 612 p.
Bulatov M. V., Lee M. G. Application of Matrix Polynomials to the Analysis of Linear Differential Algebraic Equations of Higher Order. Differential Equations. 2008. V. 44. Pp. 1353–1360.
Bulatov M. V., Lima P. M. Two-Dimensional Integral-Algebraic Systems: Analysis and Computational Methods. Journal of Computational and Applied Mathe- matics. 2011. V. 236. No. 2. Pp. 132–140. DOI: 10.1016/j.cam.2011.06.001.
Bulatov M. V., Lima P. M., Wienmüller E. B. Existence and Uniqueness of So- lutions to Weakly Singular Integral-Algebraic and Integro-Differential Equations. Cen- tral European Journal of Mathematics. 2014. V. 12. No. 2. Pp. 308–321. DOI: 10.2478/s11533-013-0334-5.
Gear C. W. Differential-Algebraic Equations, Indices, and Integral Algebraic Equations. SIAM Journal on Numerical Analysis. 1990. V. 27. No. 6. Pp. 1527–1534.
Hadizadeh M., Ghoreishi F., Pishbin S. Jacobi Spectral Solution for Integral Algebraic Equations of Index-2. Applied Numerical Mathematics. 2011. V. 61. No. 1. Pp. 131–148. DOI: 10.1016/j.apnum.2010.08.009.
Kauthen J. P. The Numerical Solution of Integral-Algebraic Equations of Index- 1 by Pollinomial Spline Collocation Methods. Mathematics of Computation. 2000. V. 236. Pp. 1503–1514.
Linz P. Analytical and Numerical Methods for Volterra Equations. Philadel- phia: SIAM, 1985. 227 p. DOI: 10.1137/1.9781611970852.
Pishbin S. Ghoreishi F., Hadizadeh M. The Semi-Explicit Volterra Integral Al- gebraic Equations with Weakly Singular Kernel: The Numerical Treatments. Journal of Computational and Applied Mathematics. 2013. V. 245. No. 1. Pp. 121–132. DOI: 10.1016/j.cam.2012.12.012.
Pishbin S. Numerical Solution and Structural Analysis of Two-Dimensional In- tegral-Algebraic Equations. Numerical Algorithms. 2016. V. 73. No. 2. Pp. 305–322. DOI: 10.1007/s11075-016-0096-9.
Apartsin A. S. Neklassicheskie uravneniya Volterra I roda: teoriya i chis-lennye metody [Nonclassical Linear Volterra Equations of the First Kind: Theory and Numeri- cal Methods]. Novosibirsk: Nauka Publ., 1999. 193 p.
Botoroeva M. N., Bulatov M. V. Prilozheniya i metody chislennogo resheniya odnogo klassa integro-algebraicheskikh uravnenii s peremennymi predelami integriro- vaniya [Applications and Methods for Numerical Solution of a Particular Class of Inte- gral Algebraic Equations with Variable Limits of Integration]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika. 2017. No. 20. Pp. 3–16. DOI: 10.26516/1997-7670.2017.20.3.
Budnikova O. S., Bulatov M. V. Chislennoe reshenie integro-algebraicheskikh uravnenii mnogoshagovymi metodami [Numerical Solution of Integral Algebraic Equations by Multistep Methods]. Computational Mathematics and Mathematical Physics. 2012. V. 52. No. 5. Pp. 691–701. DOI: 10/1134/S0965542512050041.
Budnikova O. S. O modifitsirovannykh mnogoshagovykh metodakh dlya chis- lennogo resheniya lineinykh integro-algebraicheskikh uravnenii indeksa dva [On Modi- fied Multistep Methods for Numerical Solution of Linear Integral Algebraic Equations of Index-2]. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 2014. V. 16. No. 1. Pp. 45–54.
Bulatov M. V. O preobrazovanii vyrozhdennykh sistem integralnykh uravnenii tipa Volterra [Transformation of Singular Systems of Volterra-Type Integral Equa- tions]. Vychislitelnye tekhnologii. 2000. V. 5. No. 4. Pp. 22–30.
Bulatov M. V., Chistyakova E. V. Ob odnom semeistve vyrozhdennykh integ- rodifferentsialnykh uravnenii [On a Certain Family of Singular Integro-Differential Equations]. Computational Mathematics and Mathematical Physics. 2011. V. 51. No. 9. Pp. 1558–1566. DOI: 10.1134/S0965542511090065.
Bulatov M. V., Budnikova O. S. Chislennoe reshenie integro-algebraicheskikh uravnenii so slaboi osobennostyu v yadre k-shagovymi metodami [Numerical Solution of Integral Algebraic Equations with Weakly Singular Kernels by k-step Methods]. Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika. 2015. No. 13. Pp. 3–15.
Glushkov V. M., Ivanov V. V., Yanenko V. M. Modelirovanie razviva- yushchikhsya sistem [Modeling of Evolutionary Systems]. Moscow: Nauka Publ., 1983. 350 p.
Ten Men Yan. Priblizhennoe reshenie lineinykh integralnykh uravnenii Volterra I roda. Dis. … kand. fiz.-mat. nauk [Approximate Solution of Linear Volterra Integral Equations of the First Kind. Cand. math. and phys. sci. diss.]. Irkutsk, 1985. 215 p.
Chistyakov V. F. Chistyakov V. F. O singulyarnykh sistemakh obyknovennykh differentsialnykh uravnenii i ikh integralnykh analogakh [On Singular Systems of Ordinary Differential Equations and Their Integral Analogues]. Funktsii Lyapunova i ikh primeneniya. Novosibirsk: Nauka Publ., 1987. Pp. 231–239.
Algebro-differentsialnye operatory s konechnomernym yadrom [Algebraic Differential Operators with Finite-Dimensional Kernel]. Novosibirsk: Nauka Publ., 1996. 280 p.
О. С. Будникова, М. Н. Ботороева. Многошаговые методы для численного реше- ния интегро-алгебраических уравнений индекса два
image
Brunner H. Collocation Methods for Volterra Integral and Related Functional Equations. Cambridge: Unversity Press, 2004. 612 p.
Bulatov M. V., Lee M. G. Application of Matrix Polynomials to the Analysis of Linear Differential Algebraic Equations of Higher Order. Differential Equations. 2008. V. 44. Pp. 1353–1360.
Bulatov M. V., Lima P. M. Two-Dimensional Integral-Algebraic Systems: Analysis and Computational Methods. Journal of Computational and Applied Mathe- matics. 2011. V. 236. No. 2. Pp. 132–140. DOI: 10.1016/j.cam.2011.06.001.
Bulatov M. V., Lima P. M., Wienmüller E. B. Existence and Uniqueness of So- lutions to Weakly Singular Integral-Algebraic and Integro-Differential Equations. Cen- tral European Journal of Mathematics. 2014. V. 12. No. 2. Pp. 308–321. DOI: 10.2478/s11533-013-0334-5.
Gear C. W. Differential-Algebraic Equations, Indices, and Integral Algebraic Equations. SIAM Journal on Numerical Analysis. 1990. V. 27. No. 6. Pp. 1527–1534.
Hadizadeh M., Ghoreishi F., Pishbin S. Jacobi Spectral Solution for Integral Algebraic Equations of Index-2. Applied Numerical Mathematics. 2011. V. 61. No. 1. Pp. 131–148. DOI: 10.1016/j.apnum.2010.08.009.
Kauthen J. P. The Numerical Solution of Integral-Algebraic Equations of Index- 1 by Pollinomial Spline Collocation Methods. Mathematics of Computation. 2000. V. 236. Pp. 1503–1514.
Linz P. Analytical and Numerical Methods for Volterra Equations. Philadel- phia: SIAM, 1985. 227 p. DOI: 10.1137/1.9781611970852.
Pishbin S. Ghoreishi F., Hadizadeh M. The Semi-Explicit Volterra Integral Al- gebraic Equations with Weakly Singular Kernel: The Numerical Treatments. Journal of Computational and Applied Mathematics. 2013. V. 245. No. 1. Pp. 121–132. DOI: 10.1016/j.cam.2012.12.012.
Pishbin S. Numerical Solution and Structural Analysis of Two-Dimensional In- tegral-Algebraic Equations. Numerical Algorithms. 2016. V. 73. No. 2. Pp. 305–322. DOI: 10.1007/s11075-016-0096-9.
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