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
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Author
Ali Filiz, Department of Mathematics, Adnan Menderes University, 09010 AYDIN-TURKEY
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Abstract
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.
Keywords
A fourth and Fifth-Order Accuracy, Lagrange Polynomial Interpolating, Newton-Cotes Formulas, Runge-Kutta Methods, Linear Volterra Integro-Differential Equation
To cite this article
Ali Filiz, 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. doi: 10.11648/j.acm.20140301.12
References
[1]
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs and Mathematical Tables, New York: Dover, 1972, pp. 885–887.
[2]
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.
[3]
R. L. Burden and J. D. Faires, Numerical Analysis, New York: Brooks/Cole Publishing Company, USA, 1997, ch.5.
[4]
C. T. H. Baker, The Numerical Treatment of Integral Equations, Clarendon Press; Oxford University Press, 1977.
[5]
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.
[6]
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.
[7]
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.
[8]
K. L. Cooke, "Functional differential equations close to cifferential equation," Amer. Math. Soc., 1966, Vol.72, pp. 285-288.
[9]
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.
[10]
A. Filiz, "Numerical Solution of Some Volterra Integral Equations," PhD Thesis, The University of Manchester, 2000.
[11]
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.
[12]
P. Linz, Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia, 1985.
[13]
C. W. Ueberhuber, Numerical Computation 2: Methods, Software and analysis, Berlin: Springer-Verlag, 1997.
[14]
V. Volterra, Leçons Sur la Theorie Mathematique de la Lutte Pour La Vie, Gauthier-villars, Paris, 1931.
[15]
V. Volterra, Theory of Functional and of Integro-Differential Equations. Dover, New York, 1959.
[16]
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.
[17]
Wolfram MathWorld, Newton-Cotes Formulas, http://mathworld.wolfram.com/Newton-CotesFormulas.html
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