Application of the Hybrid Methods to Solving Volterra Integro-Differential Equations
Beginning from the creator of integro-differential
equations Volterra, many scientists have investigated these
equations. Classic method for solving integro-differential
equations is the quadratures method that is successfully applied up
today. Unlike these methods, Makroglou applied hybrid methods
that are modified and generalized in this paper and applied to the
numerical solution of Volterra integro-differential equations. The
way for defining the coefficients of the suggested method is also
given.
[1] V.Volterra. Theory of functional and of integral and integrodifferential
equations. Moscow, Nauka, 1982, 306 p.
[2] M. James, G. Ortega William, Jr. Poole. An introduction to numerical
methods for differential equations. Moscow, Nauka, 1986, 288 p.
[3] A. Makroglou. Hybrid methods in the numerical solution of Volterra
integro-differential equations. Journal of Numerical Analysis 2, 1982,
pp.21-35
[4] Sovremenniye chislenniye metodi obiknovennix differensialnix
uravneniy. (J.Holl, J.Uatt), 1979,312 p.
[5] Butcher J.C. A modified multistep method for the numerical
integration of ordinary differential equations. J. Assoc. Comput.
Math., v.12, 1965, pp.124-135.
[6] G.K. Gupta. A polynomial representation of hybrid methods for
solving ordinary differential equations, Mathematics of comp.,
volume 33, number 148, 1979, pp.1251-1256
[7] C.S Gear. Hybrid methods for initial value problems in ordinary
differential equations. SIAM, J. Numer. Anal. v. 2, 1965, pp. 69-86.
[8] R.R. Mirzoyev, G.Yu Mehdiyeva., V.R. Ibrahimov, On an
application of a multistep method for solving Volterra integral
equations of the second kind, Proceeding the 2010 International
Conference on Theoretical and Mathematical Foundations of
Computer Science, Orlando, USA, 2010, pp. 46-50.
[9] G.Dahlquist Convergence and stability in the numerical integration
of ordinary differential equations. Math. Scand. 1956, Ôäû4, p.33-53.
[10] V.R. Ibrahimov. On a nonlinear method for numerical calculation of
the Cauchy problem for ordinary differential equation, Diff. equation
and applications. Pron. of II International Conference Russe.
Bulgarian, 1982, pp. 310-319.
[11] Duglas J.F., Burden R.L. Numerical analysis, 7 edition Cengage
Learning 2001,850 pp.
[12] M.N. Imanova, G.Yu. Mehdiyeva, V.R. Ibrahimov. Research of a
multistep method applied to numerical solution of Volterra integrodifferential
equation, World Academy of Science, Engineering and
Technology, Amsrterdam, 2010, pp.349-352
[1] V.Volterra. Theory of functional and of integral and integrodifferential
equations. Moscow, Nauka, 1982, 306 p.
[2] M. James, G. Ortega William, Jr. Poole. An introduction to numerical
methods for differential equations. Moscow, Nauka, 1986, 288 p.
[3] A. Makroglou. Hybrid methods in the numerical solution of Volterra
integro-differential equations. Journal of Numerical Analysis 2, 1982,
pp.21-35
[4] Sovremenniye chislenniye metodi obiknovennix differensialnix
uravneniy. (J.Holl, J.Uatt), 1979,312 p.
[5] Butcher J.C. A modified multistep method for the numerical
integration of ordinary differential equations. J. Assoc. Comput.
Math., v.12, 1965, pp.124-135.
[6] G.K. Gupta. A polynomial representation of hybrid methods for
solving ordinary differential equations, Mathematics of comp.,
volume 33, number 148, 1979, pp.1251-1256
[7] C.S Gear. Hybrid methods for initial value problems in ordinary
differential equations. SIAM, J. Numer. Anal. v. 2, 1965, pp. 69-86.
[8] R.R. Mirzoyev, G.Yu Mehdiyeva., V.R. Ibrahimov, On an
application of a multistep method for solving Volterra integral
equations of the second kind, Proceeding the 2010 International
Conference on Theoretical and Mathematical Foundations of
Computer Science, Orlando, USA, 2010, pp. 46-50.
[9] G.Dahlquist Convergence and stability in the numerical integration
of ordinary differential equations. Math. Scand. 1956, Ôäû4, p.33-53.
[10] V.R. Ibrahimov. On a nonlinear method for numerical calculation of
the Cauchy problem for ordinary differential equation, Diff. equation
and applications. Pron. of II International Conference Russe.
Bulgarian, 1982, pp. 310-319.
[11] Duglas J.F., Burden R.L. Numerical analysis, 7 edition Cengage
Learning 2001,850 pp.
[12] M.N. Imanova, G.Yu. Mehdiyeva, V.R. Ibrahimov. Research of a
multistep method applied to numerical solution of Volterra integrodifferential
equation, World Academy of Science, Engineering and
Technology, Amsrterdam, 2010, pp.349-352
@article{"International Journal of Engineering, Mathematical and Physical Sciences:60243", author = "G.Mehdiyeva and M.Imanova and V.Ibrahimov", title = "Application of the Hybrid Methods to Solving Volterra Integro-Differential Equations", abstract = "Beginning from the creator of integro-differential
equations Volterra, many scientists have investigated these
equations. Classic method for solving integro-differential
equations is the quadratures method that is successfully applied up
today. Unlike these methods, Makroglou applied hybrid methods
that are modified and generalized in this paper and applied to the
numerical solution of Volterra integro-differential equations. The
way for defining the coefficients of the suggested method is also
given.", keywords = "Integro-differential equations, initial value
problem, hybrid methods, predictor-corrector method", volume = "5", number = "5", pages = "774-5", }