An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs.(English)Zbl 1328.65156

Summary: We developed multi-step iterative method for computing the numerical solution of nonlinear systems, associated with ordinary differential equations (ODEs) of the form $$L(x(t)) + f(x(t)) = g(t)$$: here $$L(\cdot)$$ is a linear differential operator and $$f(\cdot)$$ is a nonlinear smooth function. The proposed iterative scheme only requires one inversion of Jacobian which is computationally very efficient if either LU-decomposition or GMRES-type methods are employed. The higher-order Frechet derivatives of the nonlinear system stemming from the considered ODEs are diagonal matrices. We used the higher-order Frechet derivatives to enhance the convergence-order of the iterative schemes proposed in this note and indeed the use of a multi-step method dramatically increases the convergence-order. The second-order Frechet derivative is used in the first step of an iterative technique which produced third-order convergence. In a second step we constructed matrix polynomial to enhance the convergence-order by three. Finally, we freeze the product of a matrix polynomial by the Jacobian inverse to generate the multi-step method. Each additional step will increase the convergence-order by three, with minimal computational effort. The convergence-order (CO) obeys the formula $${\mathrm{CO}} = 3 m$$, where $$m$$ is the number of steps per full-cycle of the considered iterative scheme. Few numerical experiments and conclusive remarks end the paper.

MSC:

 65L05 Numerical methods for initial value problems involving ordinary differential equations 34A45 Theoretical approximation of solutions to ordinary differential equations
Full Text:

References:

 [1] Mandelzweig, V. B., Quasilinearization method and its verification on exactly solvable models in quantum mechanics, J. Math. Phys., 40, 12, 62666291, (1999) · Zbl 0969.81007 [2] Mandelzweig, V. B.; Tabakin, F., Quasilinearization approach to nonlinear problems in physics with application to nonlinear odes, Comput. Phys. Commun., 141, 2, 268281, (2001) · Zbl 0991.65065 [3] Mandelzweig, V. B., Quasilinearization method: nonperturbative approach to physical problems, Phys. At. Nucl., 68, 7, 12271258, (2005) [4] Bellman, R. E.; Kalaba, R. E., Quasilinearization and nonlinear boundary-value problems, (1965), Elsevier New York, NY, USA · Zbl 0139.10702 [5] Motsa, S. S.; Sibanda, P., Some modification of the quasilinearization method with higher-order convergence for solving nonlinear BVPs, Numer. Algorithms, 63, 3, 399417, (2013) [6] Alaidarous, E. S.; Ullah, M. Z.; Ahmad, F.; Al-Fhaid, A. S., An efficient higher-order quasilinearization method for solving nonlinear BVPs, J. Appl. Math., 03/2013, 11, (2013) · Zbl 1397.34046 [7] Ullah, M. Z.; Al-Fhaid, A. S.; Ahmad, F., Four-point optimal sixteenth-order iterative method for solving nonlinear equations, J. Appl. Math., 09/2013, (2013) · Zbl 1397.65074 [8] Ahmad, F.; Ullah, M. Z., Eighth-order derivative-free family of iterative methods for nonlinear equations, J. Mod. Methods Numer. Math., 02/2013, 4, 26-33, (2013) · Zbl 1413.65177 [9] Sargolzaei, P.; Soleymani, F., Accurate fourteenth-order methods for solving nonlinear equations, Numer. Algorithms, 58, 513-552, (2011) · Zbl 1242.65100 [10] Soleymani, F.; Sharifi, M.; Mousavi, B. S., An improvement of ostrowski’s and king’s techniques with optimal convergence order eight, J. Optim. Theory Appl., 153, 1, 225-236, (2011) · Zbl 1237.90229 [11] Soleymani, F.; Khattri, S. K.; Vanani, S. K., Two new classes of optimal jarratt-type fourth-order methods, Appl. Math. Lett., 25, 5, 847-853, (2012) · Zbl 1239.65030 [12] F. Soleymani, Regarding the accuracy of optimal eighth-order methods, Math. Comput. Modell. 53 (5) 1351-1357. · Zbl 1217.65089 [13] Soleymani, F.; Vanani, S. K.; Paghaleh, M. J., A class of three-step derivative-free root solvers with optimal convergence order, J. Appl. Math., (2012) · Zbl 1235.65047 [14] Soleymani, F.; Sharifi, M.; Shateyi, S.; Haghani, F. K., A class of Steffensen-type iterative methods for nonlinear systems, J. Appl. Math., 2014, (2014), Article ID 705375, 9p · Zbl 1474.65123 [15] Soleymani, F.; Lotfi, T.; Bakhtiari, P., A multi-step class of iterative methods for nonlinear systems, Optim. Lett., 8, 3, 1001-1015, (2014) · Zbl 1286.93068 [16] Montazeri, H.; Soleymani, F.; Shateyi, S.; Motsa, S. S., On a new method for computing the numerical solution of systems of nonlinear equations, J. Appl. Math., 2012, (2012), Article ID 751975, 15p · Zbl 1268.65075 [17] Sharma, J. R.; Arora, H., On efficient weighted-Newton methods for solving systems of nonlinear equations, Appl. Math. Comput., 222, 497-506, (2013) · Zbl 1329.65106 [18] Sharma, J. R.; Gupta, P., An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67, 591-601, (2014) · Zbl 1350.65048 [19] Sharma, J. R.; Sharma, R., Some third order methods for solving systems of nonlinear equations, World Acad. Sci. Eng. Technol., 60, (2011) · Zbl 1215.65096 [20] Sharma, J. R.; Arora, H., Efficient jarratt-like methods for solving systems of nonlinear equations, Calcolo, 51, 193-210, (2014) · Zbl 1311.65052 [21] J.R. Sharma, H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algorithms, http://dx.doi.org/10.1007/s11075-014-9832-1. [22] Sharma, J. R.; Arora, H., An efficient derivate free iterative method for solving systems of nonlinear equations, Appl. Anal. Discrete Math., 7, 390-403, (2013) · Zbl 1299.65102 [23] Candela, V.; Marquina, A., Recurrence relations for rational cubic methods. II: the Chebyshev method, Computing, 45, 355-367, (1990) · Zbl 0714.65061 [24] V. Kanwar, S. Kumar, R. Behl, Several new families of Jarratt’s method for solving systems of nonlinear equations, Appl. Appl. Math. ISSN: 1932-9466, . · Zbl 1283.65049 [25] D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations, J. Appl. Math. vol. 2012, Article ID 165452, 12p, http://dx.doi.org/10.1155/2012/165452. · Zbl 1268.65072 [26] Torregrosa, J. R.; Argyros, I. K.; Chun, C.; Cordero, A.; Soleymani, F., Iterative methods for nonlinear equations or systems and their applications, J. Appl. Math., 2013, (2013), Article ID 656953, 2p [27] Cordero, A.; Hueso, J. L.; Martínez, E.; Torregrosa, J. R., A modified Newton-jarratt’s composition, J. Numer. Algorithms, 55, 87-99, (2010) · Zbl 1251.65074 [28] Bratu, G., Sur LES equations integrales non-lineaires, Bull. Math. Soc. France, 42, 113-142, (1914) · JFM 45.1306.01 [29] Frank-Kamenetzkii, D. A., Diffusion and heat transfer in chemical kinetics, (1969), Plenum Press New York
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.