Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method
Applied and Computational Mathematics
Volume 3, Issue 1, February 2014, Pages: 9-14
Received: Aug. 18, 2013;
Published: Feb. 20, 2014
Views 3835 Downloads 559
Ali Filiz, Department of Mathematics, Adnan Menderes University, 09010 AYDIN-TURKEY
In this paper a new fourth and fifth-order numerical solution of linear Volterra integro-differential equation is discussed. One popular technique that uses here for error control is called the Runge-Kutta-Fehlberg method for Ordinary Differential Equation (ODE) part and Newton-Cotes formulae for integral parts.
Numerical Solution of Linear Volterra Integro-Differential Equation using Runge-Kutta-Fehlberg Method, Applied and Computational Mathematics.
Vol. 3, No. 1,
2014, pp. 9-14.
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, New York: Dover, 1972, pp. 885–887.
A. Asanov, Uniqueness of the solution of systems of convolution-type Volterra integral equations of the first kind, In: Inverse problems for differential equations of the mathematical physics (Russian), Novasibirsk: Akad. Nauk SSSR Sibirsk. Otdel. Vychil. Tsentr, 1978, Vol 155, pp. 2–34.
R. L. Burden and J. D. Faires, Numerical Analysis, New York: Brooks/Cole Publishing Company, USA, 1997, ch.5.
C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press; Oxford University Press, 1977.
C. T. H. Baker, G. A. Bochorov, A. Filiz, N. J. Ford, C. A. H. Paul, F. A. Rihan, A. Tang, R. M. Thomas, H. Tian, D. R. Wille "Numerical Modelling by Retarded Functional Differential Equations," Numerical Analysis Report, Manchester Center for Computational Mathematics, No:335, ISS 130-1725,1998.
C. T. H. Baker, G. A. Bochorov, A. Filiz, N. J. Ford, C. A. H. Paul, F. A. Rihan, A. Tang, R. M. Thomas, H. Tian, D. R. Wille "Numerical Modelling by Delay and Volterra Functional Differential Equations," Numerical Analysis Report, In: Computer Mathematics and its Aplications-Advances & Developments (1994-2005), Elias A. Lipitakis (Editor), LEA Publishers, Athens, Greece, 2006, pp. 233-256.
R. Bellman, A Survey of the Theory of the Boundedness Stability and Asymptotic Behaviour of Solutions of Linear and Non-linear differential and difference equations, Washington, D. C., 1949.
K. L. Cooke, "Functional differential equations close to cifferential equation," Amer. Math. Soc., 1966, Vol.72, pp. 285-288.
A. Filiz, "On the solution of Volterra and Lotka-Volterra Type Equations," LMS supported One Day Meeting in Delayed Differential equation (Liverpool, UK), 12th March 2000.
A. Filiz, "Numerical Solution of Some Volterra Integral Equations," PhD Thesis, The University of Manchester, 2000.
A. Filiz, "Fourth-order robust numerical method for integro-differential equations," Asian Journal of Fuzzy and Applied Mathematics, 2013, Vol. 1 I, pp. 28-33.
P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
C. W. Ueberhuber, Numerical Computation 2: Methods, Software and analysis, Berlin: Springer-Verlag, 1997.
V. Volterra, Leçons Sur la Theorie Mathematique de la Lutte Pour La Vie, Gauthier-villars, Paris, 1931.
V. Volterra, Theory of Functional and of Integro-Differential Equations. Dover, New York, 1959.
V. Volterra, "Sulle Equazioni Integro-differenziali Della Teoria Dell’elastica," Atti Della Reale Accademia dei Lincei 18 (1909), Reprinted in Vito Volterra, Opera Mathematiche; Memorie e Note, Vol. 3, Accademia dei Lincei Rome, 1957.
Wolfram MathWorld, Newton-Cotes Formulas, http://mathworld.wolfram.com/Newton-CotesFormulas.html