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.
[1] 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.
[2] P. Linz “Linear Multistep methods for Volterra Integro-Differential Equations”, Journal of the Association for Computing Machinery, 1969, 16: (2) 295-301
[3] 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in Ordinary Differential Equations”, John Wiley, 1973, New York B. Smith, “An approach to graphs of linear forms (Unpublished work style),” unpublished.
[4] 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
[5] 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
[6] V. Volterra “Theory of Functional and of Integral and Integro-Differential Equations”, Dover publications; Ing New York, 1959, 304p
[7] A. Wazwaz “Linear and Nonlinear Integral Equations Methods and Applications” Higher Education Press, Beijing and Springer-Verlag Berlin Heideberg, 2011
[8] 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
[9] Al-Timeme, Atifa “Approximated Methods for First Order Volterra Integro-Differential Equations, M.Sc. Thesis, University Technology” 2003
[10] H. Brunner “Implicit Runge-Kutta Methods of Optimal Order for Volterra Integro-Differential Equations”, Mathematics of Computation, 1984, 42: (165) 95-109
[11] p. Brunner “The collocation Methods for Ordinary Differential Equations”, An Introduction, Cambridge University Press, 1996
[12] M. Dadkhah, M. T. Kajani and S. Mahdavi “Numerical Solution of Non-linear Fredholm-Volterra Integro-Differential Equations Using Legendre Wavelets”, 2010, ICMA
[13] J. T. Day (1967). “Note on the Numerical Solution of Integro-Differential Equations”. Journal of Computation, 1967, 9: pp.394-395
[14] 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
[15] A. Feldstein and J. R. Sopka “Numerical Methods for Non-linear Volterra Integro-Differential Equations”, SIAM Journal Numerical Analysis, 1974, 11:826-846
[16] 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
[17] 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
[18] 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
[19] 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
[20] 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
[1] 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.
[2] P. Linz “Linear Multistep methods for Volterra Integro-Differential Equations”, Journal of the Association for Computing Machinery, 1969, 16: (2) 295-301
[3] 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.
[4] 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
[5] 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
[6] V. Volterra “Theory of Functional and of Integral and Integro-Differential Equations”, Dover publications; Ing New York, 1959, 304p
[7] A. Wazwaz “Linear and Nonlinear Integral Equations Methods and Applications” Higher Education Press, Beijing and Springer-Verlag Berlin Heideberg, 2011
[8] 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
[9] Al-Timeme, Atifa “Approximated Methods for First Order Volterra Integro-Differential Equations, M.Sc. Thesis, University Technology” 2003
[10] H. Brunner “Implicit Runge-Kutta Methods of Optimal Order for Volterra Integro-Differential Equations”, Mathematics of Computation, 1984, 42: (165) 95-109
[11] p. Brunner “The collocation Methods for Ordinary Differential Equations”, An Introduction, Cambridge University Press, 1996
[12] M. Dadkhah, M. T. Kajani and S. Mahdavi “Numerical Solution of Non-linear Fredholm-Volterra Integro-Differential Equations Using Legendre Wavelets”, 2010, ICMA
[13] J. T. Day (1967). “Note on the Numerical Solution of Integro-Differential Equations”. Journal of Computation, 1967, 9: pp.394-395
[14] 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
[15] A. Feldstein and J. R. Sopka “Numerical Methods for Non-linear Volterra Integro-Differential Equations”, SIAM Journal Numerical Analysis, 1974, 11:826-846
[16] 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
[17] 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
[18] 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
[19] 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
[20] 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
@article{"International Journal of Engineering, Mathematical and Physical Sciences:78776", author = "N. M. Kamoh and D. G. Gyemang and M. C. Soomiyol", title = "Comparing the Efficiency of Simpson’s 1/3 and 3/8 Rules for the Numerical Solution of First Order Volterra Integro-Differential Equations", abstract = "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.
", keywords = "Collocation shifted Legendre polynomials, Simpson’s rule and Volterra integro-differential equations. ", volume = "13", number = "5", pages = "136-5", }
