Comparing the Efficiency of Simpson’s 1/3 and 3/8 Rules for the Numerical Solution of First Order Volterra Integro-Differential Equations
This paper compared the efficiency of Simpson’s 1/3 and 3/8 rules for the numerical solution of first order Volterra integro-differential equations. In developing the solution, collocation approximation method was adopted using the shifted Legendre polynomial as basis function. A block method approach is preferred to the predictor corrector method for being self-starting. Experimental results confirmed that the Simpson’s 3/8 rule is more efficient than the Simpson’s 1/3 rule.
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 M. H. AL-Smadi, A. Zuhier and A. G. Radwan “Approximate Solution of Second-Order Integro-differential Equation of Volterra Type in RKHS Method”, International Journal of Mathematical Analysis, 2013, 7(44): 2145 – 21.
 P. Linz “Linear Multistep methods for Volterra Integro-Differential Equations”, Journal of the Association for Computing Machinery, 1969, 16: (2) 295-301
 J. D. Lambert “Computational Methods in Ordinary Differential Equations”, John Wiley, 1973, New York B. Smith, “An approach to graphs of linear forms (Unpublished work style),” unpublished.
 P. Darania and A. Ebadian “Development of the Taylor Expansion Approach for Nonlinear Integro-Differential Equations”, International Journal of Contemporary Mathematical Sciences, 2006, 14: 651-666
 B. Sachin and P. Jayvant “A Novel Third Order Numerical Method for Solving Volterra Integro-Differential Equations, using Daftardar-Gejji and Jafari technique (DJM)”, International Journal of Computer Mathematics, 2016, 14: 93-111
 V. Volterra “Theory of Functional and of Integral and Integro-Differential Equations”, Dover publications; Ing New York, 1959, 304p
 A. Wazwaz “Linear and Nonlinear Integral Equations Methods and Applications” Higher Education Press, Beijing and Springer-Verlag Berlin Heideberg, 2011
 Yalcinbas and Sezer “The Approximate Solution of High Order Linear Volterra-Fredholm Integro-Differential Equations in Terms of Taylor polynomials”, Applied Mathematics and computation, 2000, 112: 291-308
 Al-Timeme, Atifa “Approximated Methods for First Order Volterra Integro-Differential Equations, M.Sc. Thesis, University Technology” 2003
 H. Brunner “Implicit Runge-Kutta Methods of Optimal Order for Volterra Integro-Differential Equations”, Mathematics of Computation, 1984, 42: (165) 95-109
 p. Brunner “The collocation Methods for Ordinary Differential Equations”, An Introduction, Cambridge University Press, 1996
 M. Dadkhah, M. T. Kajani and S. Mahdavi “Numerical Solution of Non-linear Fredholm-Volterra Integro-Differential Equations Using Legendre Wavelets”, 2010, ICMA
 J. T. Day (1967). “Note on the Numerical Solution of Integro-Differential Equations”. Journal of Computation, 1967, 9: pp.394-395
 G. Ebadi, M. Y. Rahimi-Ardabili and S. Shahmorad “Numerical Solution of the System of Nonlinear Volterra Integro-Differential Equations”, Southeast Asian Bulletin of Mathematics, 2009, 33: 835-846
 A. Feldstein and J. R. Sopka “Numerical Methods for Non-linear Volterra Integro-Differential Equations”, SIAM Journal Numerical Analysis, 1974, 11:826-846
 H. D. Gherjalar and H. Mohammadikia “Numerical Solution of Functional Integral and Integro-Differential Equations by Using B-Splines”, Applied Mathematics. 2012, 3:1940-1944
 R. Behrouz “Numerical Solutions of the Linear Volterra Integro-differential Equations by Homotopy Perturbation Method and Finite Difference Method”, World Applied Sciences Journal 9 (Special Issue of Applied Math): 2010. 07-12
 S. Abbasbandy and A. Taati “Numerical Solution of the System of Nonlinear Volterra Integro-Differential Equations with Nonlinear Differential Part by the Operational Tau Method and Error Estimation”, Journal of Computational and Applied Mathematics, 2009, 23:106-113
 A. Al-Jubory “Some Approximation Methods for Solving Volterra-Fredholm Integral and Integro- Differential Equations”, Ph.D. Thesis, Department of Mathematics, University of Technology Nest Lafayette, 2013, 195pp
 N. M. Kamoh, T. Aboiyar, and E. S. Onah “On One Investigating Some Quadrature Rules For The Solution Of Second Order Volterra Integro-Differential Equations”, IOSR Journal of Mathematics (IOSR-JM), 2017, 13(5):pp 45-50