ef93aad3-cb17-4ec9-9870-25c0298947f920210209083409729naunmdt@crossref.orgMDT DepositInternational Journal of Circuits, Systems and Signal Processing1998-446410.46300/9106http://www.naun.org/cms.action?id=3029118202111820211510.46300/9106.2021.15https://naun.org/cms.action?id=23283Application of Splines of the Second Order Approximation to Volterra Integral Equations of the Second Kind. Applications in Systems Theory and Dynamical SystemsI. G.BurovaSt. Petersburg State University 7/9 Universitetskaya nab., St.Petersburg, 199034 RussiaG. O.AlcybeevSt. Petersburg State University 7/9 Universitetskaya nab., St.Petersburg, 199034 RussiaThis paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented2920212920216371https://www.naun.org/main/NAUN/circuitssystemssignal/2021/a162005-008(2021).pdf10.46300/9106.2021.15.8https://www.naun.org/main/NAUN/circuitssystemssignal/2021/a162005-008(2021).pdfRonald A. Devore, Karl Scherer, Quantitative Approximation, Academic Press, New York, London, Sydney, Toronto, San Francisco 1980, 324 p. Zdeněk Kopal, Numerical Analysis with emphasis on the applications of numerical techniques to problems of infinitesimal calculus in single variable, Wiley, New York, 1955, 556 p. Walter Gautschi, Numerical Analysis. An Introduction, Boston, Basel, Berlin, 1997, 506 p. 10.37394/23202.2020.19.1Galina Mehdiyeva, Vagif Ibrahimov, Mehrib Imanova, “Application of the Finite Differences Methods to Computation of Definite Integrals, WSEAS Transactions on System,” vol. 19, pp 1-6, 2020. 10.37394/23202.2020.19.20Boris Shumilov, Shifted Cubic Spline Wavelets with Two Vanishing Moments on the Interval and a Splitting Algorithm, WSEAS Transactions On Systems, vol. 19, pp.149-158, 2020. I.G. Burova, V.M. Ryabov, M.A.Kalnitskaia, A.V. Malevich, “The interpolation method for calculating eigenvalues of matrices,” WSEAS Transactions on Systems and Control, vol.14, pp. 104-111, 2019. N.Onn, M.Hussein, C.H.H.Tang, M.Z.M.Zain, M.Mohamad, W.Y.Lai, “Motion control of human bipedal model in sagittal planem,” WSEAS Transactions on Systems and Control, vol.10, pp. 160-171, 2015. 10.1016/j.aml.2019.106117H. Du, Z.Chen, “A new reproducing kernel method with higher convergence order for solving a Volterra– Fredholm integral equation,” Applied Mathematics Letters, vol.102, paper 106117, 2020. 10.1063/1.4992712Galina Mehdiyeva, Vagif Ibrahimov, Mehriban Imanova, “On the Construction of the Advanced Hybrid Methods and Application to Solving Volterra Integral Equation,” WSEAS Transactions on Systems and Control, vol.14, pp 183-189, 2019. R.I. Esa, A.J. Saleh, “Numerical treatment of first order Volterra integro-differential equation using nonpolynomial spline functions,” Iraqi Journal of Science, Special Issue, pp. 114-121, 2020. 10.5269/bspm.v38i2.38043M.N.Sahlan, “Convergence of approximate solution of mixed Hammerstein type integral equations,” Boletim da Sociedade Paranaense de Matematica, vol.38 (2), pp. 61- 74, 2020. 10.2298/fil1811947mK.Maleknejad, J.Rashidinia, H. Jalilian, “Non-polynomial spline functions and quasi-linearization to approximate nonlinear volterra integral equation,” \ Filomat, vol.32 (11), pp. 3947-3956, 2018. 10.1016/j.aej.2020.09.040M. Asif, I. Khan, N. Haider, Q. Al-Mdallal, “Legendre multi-wavelets collocation method for numerical solution of linear and nonlinear integral equations.” Alexandria Engineering Journal, vol. 59, pp.5099-5109, 2020. I.G.Burova, N.S. Domnin, “On the solution of the Fredholm equation with the use of quadratic integrodifferential splines,” Lecture Notes in Electrical Engineering, vol.574, pp. 35-41, 2019. I.G.Burova, N.S. Domnin, A.E.Vezhlev, A.V.Lebedeva, A.N.Pakulina, “On the solution of the Fredholm equation of the second kind,” WSEAS Transactions on Mathematics, vol.17, pp. 319-328, 2018. I.G.Burova, E.F. Muzafarova, I.I. Narbutovskikh, “Local splines of the second and third order, complex-valued splines and image processing,” International Journal of Circuits, Systems and Signal Processing, vol.13, pp. 419- 429, 2019. I.G.Burova, “On left integro-differential splines and Cauchy problem,” International Journal of Mathematical Models and Methods in Applied Sciences, vol.9, pp. 683- 690, 2015. 10.1063/1.4964969V. Myrhorod, and I. Hvozdeva, “On one solution of Volterra integral equations of second kind,” AIP Conference Proceedings 1773, 040006, 2016, doi: 10.1063/1.4964969, Published by the American Institute of Physics, View online: http://dx.doi.org/10.1063/1.4964969 Nicholas D. Assimakis, "Kalman Filter Gain Elimination in Linear Estimation", Engineering World, Published by International Academic Publishers, pp. 183-188, Volume 2, 2020. Nicholas Assimakis, Maria Adam, Grigorios Tziallas, Lainiotis Information Filter, Engineering World, Published by International Academic Publishers, pp. 270-273, Volume 2, 2020.