Numerical Method for a Linear Volterra Integro-differential Equation with Cash-Karp Method
Keywords:Volterra integro-differential (integral) equation, Cash-Karp method, Runge-Kutta Method, Truncation error, quadrature rule, fifth order.
In this paper a linear Volterra integro-differential equation is studied. Example of this question has been solved numerically using Cash-Karp method for ODE (Ordinary Differential Equation) parts and Newton-Cotes formulae (quadrature rules) for integral part. Finally, a new fifth order routine is used for the numerical solution of the linear Volterra integro-differential equation.
Baker, C. T. H., The Numerical Treatment of Integral Equations, Clarendon Press; Oxford University Press, 1977.
Baker, C. T. H., G A Bochorov, A Filiz, N J Ford, C A H Paul, F A Rihan, A Tang, R M Thomas, H Tian and D R Wille., Numerical Modelling By Retarded Functional Differential Equations, Numerical analysis report, Manchester Centre for Computational Mathematics, No:335, ISSN 1360-1725, 1998.
Baker, C. T. H., G A Bochorov, A Filiz, N J Ford, C A H Paul, F A Rihan, A Tang, R M Thomas, H Tian and D R Wille., Numerical modelling by delay and Volterra functional differential equations, In: Computer Mathematics and its Applications- Advances and Developments (1994-2005), Editor: Elias A. Lipitakis, LEA Publishers, Athens, Greece, 2006.
Bellman, R., A survey of the theory of the boundedness stability and asymptotic behaviour of solutions of linear and non-linear differential and difference equations, Washington, 1949.
Cooke, K. L., Functional differential equations close to differential equation, Amer. Math. Soc., 72, 285-288, 1966.
Filiz, A., On the solution of Volterra and Lotka-Volterra type equations, LMS supported One Day Meeting in Delayed Differential Equation (Liverpool, UK), 12th March 2000.
Filiz, A., Numerical solution of some Volterra integral equations, PhD Thesis, University of Manchester, 2000.
Linz, P., Analytical and Numerical Methods for Volterra Equations, SIAM, Philadelphia,1985.
Volterra, V. (1931)., LeÂ¸cons sur la ThÂ´eorie MathÂ´ematique de la Lutte Pour la Vie, Gauthier-Villars, Paris, 1931.
Volterra, V., Theory of Functionals and of Integro-Differential Equations, Dover, New York, 1959.
Volterra, V., Sulle equazioni integro-differenziali della teoria dellâ€™elastica, Atti della Reale Accademia dei Lincei 18 (1909), Reprinted in Vito Volterra, Opera Mathematiche; Memorie e Note Vol. 3. Accademia dei Lincei Rome, pp. 288-293, 1957.
Bogacki, Przemyslaw and Shampine, Lawrence F., A 3(2) pair of Rungeâ€“Kutta formulas, Applied Mathematics Letters, 2 (4), pp. 321â€“325, 1989.
Ralston, Anthony, A First Course in Numerical Analysis, New York: McGraw-Hill, 1965.
Dormand, J. R.; Prince, P. J., A family of embedded Runge-Kutta formulae, Journal of Computational and Applied Mathematics, 6 (1): 19â€“26, 1980.
Hairer, Ernst; NÃ¸rsett, Syvert Paul; Wanner, Gerhard, Solving ordinary differential equations I: Nonstiff problems, Berlin, New York: Springer-Verlag, 2008.
Erwin Fehlberg, Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems, NASA Technical Report 315, 1969.
Cash, J. R. and A. H. Karp., A variable order Runge-Kutta method for initial value problems with rapidly varying right-hand sides, ACM Transactions on Mathematical Software, 16: 201-222, 1990.
Filiz, A., Fourth-Order Robust Numerical Method for Integro-differential Equations, Asian Journal of Fuzzy and Applied Mathematics, 2013, Vol 1, pp. 28-33.
Filiz, A., Numerical Solution of Linear Volterra Integro-differential Equation using Runge-Kutta-Fehlberg Method, Applied and Computational Mathematics (ACM), submitted.
Burden, R. L. and J. D. Faires, Numerical Analysis. New York: Brooks/Cole Publishing Company, USA, 1997, ch.5.
Kutta, W., Beitrag zur nÃ¤herungsweisen Integration totaler Differentialgleichungen, Z. Math. Phys., 435-453, 1901.
Henrici, P., Discrete variable methods in ordinary differential equation, John Wiley and Sons, New York, 1962.
Abraham, O. and G. Bolarin, On error estimation in Runge-Kutta Methods, Leonardo J. Sci., 2011
ijs.academicdirect.org / A18/ 001_010.htm.
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