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Frozen Jacobian multistep iterative method for solving nonlinear IVPs and BVPs. (English) Zbl 1367.65077
Summary: In this paper, we present and illustrate a frozen Jacobian multistep iterative method to solve systems of nonlinear equations associated with Initial Value Problems (IVPs) and Boundary Value Problems (BVPs). We have used Jacobi-Gauss-Lobatto collocation (J-GL-C) methods to discretize the IVPs and BVPs. Frozen Jacobian multistep iterative methods are computationally very efficient. They require only one inversion of the Jacobian in the form of LU-factorization. The LU factors can then be used repeatedly in the multistep part to solve other linear systems. The convergence order of the proposed iterative method is $$5 m - 11$$, where $$m$$ is the number of steps. The validity, accuracy, and efficiency of our proposed frozen Jacobian multistep iterative method is illustrated by solving fifteen IVPs and BVPs. It has been observed that, in all the test problems, with one exception in this paper, a single application of the proposed method is enough to obtain highly accurate numerical solutions. In addition, we present a comprehensive comparison of J-GL-C methods on a collection of test problems.
##### MSC:
 65H10 Numerical computation of solutions to systems of equations 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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##### References:
 [1] Doha, E. H.; Bhrawy, A. H.; Abdelkawy, M. A.; Van Gorder, R. A., Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrödinger equations, Journal of Computational Physics, 261, 244-255, (2014) · Zbl 1349.65511 [2] Bhrawy, A. H.; Doha, E. H.; Abdelkawy, M. A.; Van Gorder, R. A., Jacobi-Gauss-Lobatto collocation method for solving nonlinear reaction-diffusion equations subject to Dirichlet boundary conditions, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, 40, 3, 1703-1716, (2016) [3] Bhrawy, A. H., An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Applied Mathematics and Computation, 247, 30-46, (2014) · Zbl 1339.65188 [4] Dehghan, M.; Fakhar-Izadi, F., The spectral collocation method with three different bases for solving a nonlinear partial differential equation arising in modeling of nonlinear waves, Mathematical and Computer Modelling, 53, 9-10, 1865-1877, (2011) · Zbl 1219.65106 [5] Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien, S. S., Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations, Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, 35, 12, 5662-5672, (2011) · Zbl 1228.65126 [6] Doha, E. H.; Bhrawy, A. H.; Hafez, R. M., On shifted Jacobi spectral method for high-order multi-point boundary value problems, Communications in Nonlinear Science and Numerical Simulation, 17, 10, 3802-3810, (2012) · Zbl 1251.65112 [7] Tohidi, E.; Lotfi Noghabi, S., An efficient legendre pseudospectral method for solving nonlinear quasi bang-bang optimal control problems, Journal of Applied Mathematics, Statistics and Informatics, 8, 2, 73-85, (2012) · Zbl 1277.65104 [8] Szego, G., Orthogonal Polynomials, Colloquium Publications XXIII, (1939), American Mathematical Society · JFM 65.0278.03 [9] Shen, J.; Tang, T.; Wang, L.-L., Spectral Methods: Algorithms, Analysis and Applications, 41, (2011), Springer · Zbl 1227.65117 [10] Traub, J. F., Iterative methods for the solution of equations. Iterative methods for the solution of equations, Prentice-Hall Series in Automatic Computation, (1964), Englewood Cliffs, NJ, USA: Prentice-Hall, Englewood Cliffs, NJ, USA · Zbl 0121.11204 [11] Ortega, J. M.; Rheinboldt, W. C., Iterative solution of nonlinear equations in several variables, (1970), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0241.65046 [12] Cordero, A.; Kansal, M.; Kanwar, V.; Torregrosa, J. R., A stable class of improved second-derivative free Chebyshev-Halley type methods with optimal eighth order convergence, Numerical Algorithms, 72, 4, 937-958, (2016) · Zbl 1347.65097 [13] Arroyo, V.; Cordero, A.; Torregrosa, J. R., Approximation of artificial satellites’ preliminary orbits: the efficiency challenge, Mathematical and Computer Modelling, 54, 7-8, 1802-1807, (2011) · Zbl 1235.70032 [14] Budzko, D. A.; Cordero, A.; Torregrosa, J. R., New family of iterative methods based on the Ermakov-Kalitkin scheme for solving nonlinear systems of equations, Computational Mathematics and Mathematical Physics, 55, 12, 1947-1959, (2015) · Zbl 1336.65088 [15] Qasim, S.; Ali, Z.; Ahmad, F.; Serra-Capizzano, S.; Ullah, M. Z.; Mahmood, A., Solving systems of nonlinear equations when the nonlinearity is expensive, Computers & Mathematics with Applications, 71, 7, 1464-1478, (2016) [16] Qasim, U.; Ali, Z.; Ahmad, F.; Serra-Capizzano, S.; Ullah, M. Z.; Asma, M., Constructing frozen Jacobian iterative methods for solving systems of nonlinear equations, associated with ODEs and PDEs using the homotopy method, Algorithms (Basel), 9, 1, (2016) · Zbl 07042327 [17] Ahmad, F.; Tohidi, E.; Carrasco, J. A., A parameterized multi-step Newton method for solving systems of nonlinear equations, Numerical Algorithms, 71, 3, 631-653, (2016) · Zbl 1350.65046 [18] Ullah, M. Z.; Serra-Capizzano, S.; Ahmad, F., An efficient multi-step iterative method for computing the numerical solution of systems of nonlinear equations associated with ODEs, Applied Mathematics and Computation, 250, 249-259, (2015) · Zbl 1328.65156 [19] Ahmad, F.; Tohidi, E.; Ullah, M. Z.; Carrasco, J. A., Higher order multi-step Jarratt-like method for solving systems of nonlinear equations: application to PDEs and ODEs, Computers & Mathematics with Applications. An International Journal, 70, 4, 624-636, (2015) [20] Alaidarous, E. S.; Ullah, M. Z.; Ahmad, F.; Al-Fhaid, A. S., An efficient higher-order quasilinearization method for solving nonlinear BVPs, Journal of Applied Mathematics, 2013, (2013) · Zbl 1397.34046 [21] Ullah, M. Z.; Soleymani, F.; Al-Fhaid, A. S., Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs, Numerical Algorithms, 67, 1, 223-242, (2014) · Zbl 1316.65053 [22] Montazeri, H.; Soleymani, F.; Shateyi, S.; Motsa, S. S., On a new method for computing the numerical solution of systems of nonlinear equations, Journal of Applied Mathematics, 2012, (2012) · Zbl 1268.65075 [23] Soleymani, F.; Lotfi, T.; Bakhtiari, P., A multi-step class of iterative methods for nonlinear systems, Optimization Letters, 8, 3, 1001-1015, (2014) · Zbl 1286.93068 [24] Abbasbandy, S.; Babolian, E.; Ashtiani, M., Numerical solution of the generalized Zakharov equation by homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14, 12, 4114-4121, (2009) · Zbl 1221.65269 [25] Chang, Q. S.; Jiang, H., A conservative difference scheme for the Zakharov equations, Journal of Computational Physics, 113, 2, 309-319, (1994) · Zbl 0807.76050 [26] Chang, Q. S.; Guo, B. L.; Jiang, H., Finite difference method for generalized Zakharov equations, Mathematics of Computation, 64, 210, 537-553, (1995) · Zbl 0827.65138 [27] Javidi, M.; Golbabai, A., Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos, Solitons and Fractals, 36, 2, 309-313, (2008) · Zbl 1350.35182 [28] Bao, W.; Sun, F.; Wei, G. W., Numerical methods for the generalized Zakharov system, Journal of Computational Physics, 190, 1, 201-228, (2003) · Zbl 1236.76043 [29] Bao, W.; Sun, F., Efficient and stable numerical methods for the generalized and vector Zakharov system, SIAM Journal on Scientific Computing, 26, 3, 1057-1088, (2005) · Zbl 1076.35114
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