×
Daliri Birjandi, M. H.; Saberi-Nadjafi, J.; Ghorbani, A.

An efficient numerical method for a class of nonlinear Volterra integro-differential equations. (English) Zbl 1437.65235

J. Appl. Math. 2018, Article ID 7461058, 7 p. (2018).
Summary: We investigate an efficient numerical method for solving a class of nonlinear Volterra integro-differential equations, which is a combination of the parametric iteration method and the spectral collocation method. The implementation of the modified method is demonstrated by solving several nonlinear Volterra integro-differential equations. The results reveal that the developed method is easy to implement and avoids the additional computational work. Furthermore, the method is a promising approximate tool to solve this class of nonlinear equations and provides us with a convenient way to control and modify the convergence rate of the solution.

Cited in 1 Document

MSC:

65R20 Numerical methods for integral equations
45J05 Integro-ordinary differential equations
45D05 Volterra integral equations
PDF BibTeX XML Cite
\textit{M. H. Daliri Birjandi} et al., J. Appl. Math. 2018, Article ID 7461058, 7 p. (2018; Zbl 1437.65235)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Abdul, I., Introduction to Integral Equations with Application, (1999), NewYork, NY, USA: Wiley, NewYork, NY, USA
[2] Shaw, S.; Whiteman, J. R., Adaptive space-time finite element solution for Volterra equations arising in viscoelasticity problems, Journal of Computational and Applied Mathematics, 125, 1-2, 337-345, (2000) · Zbl 0990.74071
[3] Volterra, V., Theory of Functionals And of Integral And Integro-Differential Equations, (1959), New York, NY, USA: Dover Publications, Inc., New York, NY, USA
[4] Jerri, A. J., Introduction to Integral Equations with Applications, (1999), New Yourk, NY, USA: Marcel Dekker, New Yourk, NY, USA · Zbl 0938.45001
[5] Dehghan, M.; Salehi, R., The numerical solution of the non-linear integro-differential equations based on the meshless method, Journal of Computational and Applied Mathematics, 236, 9, 2367-2377, (2012) · Zbl 1243.65154
[6] Maleknejad, K.; Mahmoudi, Y., Taylor polynomial solution of high-order nonlinear Volterra-Fredholm integro-differential equations, Applied Mathematics and Computation, 145, 2-3, 641-653, (2003) · Zbl 1032.65144
[7] Maleknejad, K.; Mirzaee, F., Using rationalized Haar wavelet for solving linear integral equations, Applied Mathematics and Computation, 160, 2, 579-587, (2005) · Zbl 1299.76043
[8] El-Sayed, S. M.; Abdel-Aziz, M. R., A comparison of Adomian’s decomposition method and wavelet-Galerkin method for solving integro-differential equations, Applied Mathematics and Computation, 136, 1, 151-159, (2003) · Zbl 1023.65149
[9] He, J.-H., The homotopy perturbation method nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151, 1, 287-292, (2004) · Zbl 1039.65052
[10] Behzadi, S.; Abbasbandy, S.; Allahviranloo, S. T.; Yildirim, A., Application of homotopy analysis method for solving a class of nonlinear Volterra-Fredholm integro-differential equations, Journal of Applied Analysis and Computation, 1-14, (2012) · Zbl 1304.34110
[11] Zarebnia, M., Sinc numerical solution for the Volterra integro-differential equation, Communications in Nonlinear Science and Numerical Simulation, 15, 3, 700-706, (2010) · Zbl 1221.65346
[12] Tohidi, E.; Samadi, O. R. N., Optimal control of nonlinear Volterra integral equations via Legendre polynomials, IMA Journal of Mathematical Control and Information, 30, 1, 67-83, (2013) · Zbl 1275.49056
[13] Karamete, A.; Sezer, M., A Taylor collocation method for the solution of linear integro-differential equations, International Journal of Computer Mathematics, 79, 9, 987-1000, (2002) · Zbl 1006.65144
[14] Brunner, H., Collocation Method for Volterra Integral and Related Functional Equations, (2004), Cambridge, London, UK: Cambridge University Press, Cambridge, London, UK · Zbl 1059.65122
[15] Daliri Birjandi, M. H.; Saberi-Nadjafi, J.; Ghorbani, A., A novel method for solving nonlinear volterra integro-differential equation systems, Abstract and Applied Analysis, 2018, (2018) · Zbl 1470.65211
[16] Ghorbani, A.; Saberi-Nadjafi, J., A piecewise-spectral parametric iteration method for solving the nonlinear chaotic Genesio system, Mathematical and Computer Modelling, 54, 1-2, 131-139, (2011) · Zbl 1225.65120
[17] Linz, P., Analytical and Numerical Methods for Volterra Equations. Analytical and Numerical Methods for Volterra Equations, SIAM Studies in Applied Mathematics, 7, (1985), Philadelphia, Pa, USA: Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA · Zbl 0566.65094
[18] Weideman, J. A. C.; Reddy, S. C., A MATLAB diferentiation matrix suite, ACM Transactions on Mathematical Software, 26, 4, 465-519, (2000)
[19] Damian, T., Matrix Based Operatorial Approach to Differential and Integral Problems, (2011), Romania: Babes-Bolyai University of Cluj-Napoca Romania, Romania
[20] Small, R. D., Population Growth in a Closed System, SIAM Review, 25, 1, 93-95, (1983) · Zbl 0502.92012
[21] Maleknejad, K.; Tamamgar, M., A New Reconstruction of Variational Iteration Method and Its Application to Nonlinear Volterra Integrodifferential Equations, Abstract and Applied Analysis, 2014, 1-6, (2014)
[22] Kim, K.; Jang, B., A novel semi-analytical approach for solving nonlinear Volterra integro-differential equations, Applied Mathematics and Computation, 263, 25-35, (2015) · Zbl 1410.65496
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.