Efficiency of Newton’s Iteration of Order 3, in the Numerical Solution by the Shooting Method
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Alex Youn Aro Huanacuni
Oscar Santander Mamani
Abstract
The shooting method is an iterative technique that uses shooting speed to obtain an optimal solution. The purpose of the research is to verify the effectiveness of Newton iterations of order 3 adapted and coupled to RK4, in the shooting method. Numerical experiments were performed with examples using the shooting method in nonlinear ordinary differential equations of order 2, and shooting velocities and errors were obtained in each iteration. The results obtained on final velocities were satisfactory with the iterations NW, NWO3-1, NWO3-2 and NWO3-3 applied to the shooting method and of which, NWO3-2 and NWO3-3, have high accuracy in the experiments performed.
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Ahsan, M., & Farrukh, S. (2013). A new type of shoo- ting method for nonlinear boundary value problems. Alexandria Engineering Journal, 52(4), 801–805. https://doi.org/10.1016/j.aej.2013.07.001
Attili, B., & Syam, M. (2008). Efficient shooting method for solving two point boundary value problems. Chaos, Solitons and Fractals, 35(5), 895–903. https://doi.org/10.1016/j.chaos.2006.05.094
Bailey, P., & Shampine, L. (1968). On shooting methods for two-point boundary value problems. Journal of Mathematical Analysis and Applications, 23(2), 235–249. https://doi.org/10.1016/0022-247X(68)90064-4
Burden, R., Faires, J., & Burden, A. (2017). Análisis Numérico (10th ed.). Cengage Learning.
Butcher, J. (1996). History of Runge-Kutta methods. Appllied Numerical Mathematics, 20(3), 247–260. https://doi.org/10.1016/0168-9274(95)00108-5
Darvishi, M., & Barati, A. (2007). A third-order Newton- type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2), 630–635. https://doi.org/10.1016/j.amc.2006.08.080
Filipov, S., Gospodinov, I., & Faragó, I. (2017). Shooting- projection method for two-point boundary value problems. Applied Mathematics Letters, 72, 10–15. https://doi.org/10.1016/j.aml.2017.04.002
Granas, A., Guenther, R., & Lee, J. (1979). The Shooting Method for the Numerical Solution of a Class of Nonlinear Boundary Value Problems. SIAM Journal on Numerical Analysis, 16(5), 828–836. https://doi.org/10.1137/0716062
Ha, S. N. (2001). A nonlinear shooting method for two-point boundary value problems. Computers and Mathematics with Applications, 42(10–11), 1411–1420. https://doi.org/10.1016/S0898-1221(01)00250-4
Keller, H. B. (2018). Numerical Methods for Two-Point Boundary- Value Problems. Dover Publications.
Kutta, W. (1901). Beitrag zur näherungsweisen Integration otaler Differentialgleichungen. Zeit. Math. Phys., 46, 435–453.
Liu, C. (2006). The Lie-group shooting method for non- linear two-point boundary value problems exhibiting multiple solutions. CMES - Computer Modeling in Engineering and Sciences, 13(2), 149–163. https://doi.org/10.3970/cmes.2006.013.149
Magreñán Ruiz, Á., & Argyros, I. (2014). Two-step Newton methods. Journal of Complexity, 30(4), 533–553. https://doi.org/10.1016/j.jco.2013.10.002
Mataušek, M. (1974). Direct shooting method, linearization, and nonlinear algebraic equations. Journal of Optimization Theory and Applications, 14(2), 199–212. https://doi.org/10.1007/BF00932940
NumPy. (2022). NumPy User Guide v1.3. https://numpy.org/doc/1.23/numpy-ref.pdf
Osborne, M. (1969). On shooting methods for boundary value problems. Journal of Mathematical Analysis and Applications, 27(2), 417–433. https://doi.org/10.1016/0022-247X(69)90059-6
Schrader, K. (1969). Existence theorems for second order boundary value problems. Journal of Differential Equations, 5(3), 572–584. https://doi.org/10.1016/0022-0396(69)90094-1
SymPy. (2022). SymPy Documentation. https://github.com/sympy/sympy/releases
Weerakoon, S., & Fernando, T. (2000). A variant of Newton’s method with accelerated third-order convergence. Applied Mathematics Letters, 13(8), 87–93. https://doi.org/10.1016/S0893-9659(00)00100-2
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