A new reconstruction of variational iteration method and its application to nonlinear Volterra integrodifferential equations. (English) Zbl 1474.65505
Summary: We reconstruct the variational iteration method that we call, parametric iteration method (PIM). The purposed method was applied for solving nonlinear Volterra integrodifferential equations (NVIDEs). The solution process is illustrated by some examples. Comparisons are made between PIM and Adomian decomposition method (ADM). Also exact solution of the 3rd example is obtained. The results show the simplicity and efficiency of PIM. Also, the convergence of this method is studied in this work.
References:
| [1] | Xu, L.; He, J. H.; Liu, Y., Electro spun nonporous spheres with Chinese drug, The International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 199-202 (2007) |
| [2] | Sun, F. Z.; Gao, M.; Lei, S. H.; Zhao, Y. B.; Wang, K.; Shi, Y. T.; Wang, N. H., The fractal dimension of the fractal model of dropwise condensation and its experimental study, International Journal of Nonlinear Sciences and Numerical Simulation, 8, 2, 211-222 (2007) |
| [3] | Jerri, A. J., , Introduction to Integral Equations with Applications (1985), New Yourk, NY, USA: Marcel Dekker, New Yourk, NY, USA · Zbl 0623.45001 |
| [4] | Saberi-Nadjafi, J.; Tamamgar, M., The variational iteration method: a highly promising method for solving the system of integro-differential equations, Computers & Mathematics with Applications, 56, 2, 346-351 (2008) · Zbl 1155.65399 · doi:10.1016/j.camwa.2007.12.014 |
| [5] | 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 & Computation, 136, 1, 151-159 (2003) · Zbl 1023.65149 · doi:10.1016/S0096-3003(02)00024-3 |
| [6] | Maleknejad, K.; Sohrabi, S.; Rostami, Y., Numerical solution of nonlinear Volterra integral equations of the second kind by using Chebyshev polynomials, Applied Mathematics & Computation, 188, 1, 123-128 (2007) · Zbl 1114.65370 · doi:10.1016/j.amc.2006.09.099 |
| [7] | Maleknejad, K.; Hashemizadeh, E.; Ezzati, R., A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation, Communications in Nonlinear Science and Numerical Simulation, 16, 2, 647-655 (2011) · Zbl 1221.65334 · doi:10.1016/j.cnsns.2010.05.006 |
| [8] | Liao, J. Sh., The proposed homotopy analysis technique for the solution of nonlinear problems [Ph.D. thesis] (1992), Shanghai Jiao Tong University |
| [9] | Abbasbandy, S.; Ashtiani, M.; Babolian, E., Analytic solution of the Sharma-Tasso-Olver equation by homotopy analysis method, Zeitschrift für Naturforschung Section A, 65, 4, 285-290 (2010) |
| [10] | Abbasbandy, S.; Shirzadi, A., Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems, Numerical Algorithms, 54, 4, 521-532 (2010) · Zbl 1197.65096 · doi:10.1007/s11075-009-9351-7 |
| [11] | He, J.-H., Variational iteration method-some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207, 1, 3-17 (2007) · Zbl 1119.65049 · doi:10.1016/j.cam.2006.07.009 |
| [12] | Ghorbani, A.; Gachpazan, M.; Saberi-Nadjafi, J., A modified parametric iteration method for solving nonlinear second order BVPs, Computational and Applied Mathematics, 30, 3, 499-515 (2011) · Zbl 1254.34023 · doi:10.1590/S1807-03022011000300002 |
| [13] | Liao, S., Beyond Perturbation: Introduction to the Homotopy Analysis Method. Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Series: Modern Mechanics and Mathematics, 2, xii+322 (2004), Boca Raton, Fla, USA: Chapman & Hall/CRC, Boca Raton, Fla, USA · Zbl 1051.76001 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
