1,2M. Sandoval-Hernandez, 3G. J. Morales-Alarcon, 4U. A. Filobello-Nino, 4A. R. Escobar-Flores, 4H. Vazquez-Leal, 4J. E. Pretelin-Canela, 4J. E Perez-Jacome-Friscione, 4O. Alvarez-Gasca, 4A. D. Contreras-Hernandez, 4R. Castañeda-Sheissa, 4N. Carrillo-Ramon, 5S. Y. Campos-Dominguez, 5S. Ocaña-Pimentel, 4J.A. Domiguez-Chavez
1Centro de Bachillerato Tecnológico industrial y de servicios No. 190, Av. 15 Col. Venustiano Carranza 2da Sección, Boca del Río, 94297, Veracruz, México.
2Escuela de Ingeniería, Universidad de Xalapa, Carretera Xalapa-Veracruz Km 2 No. 341,91190 Xalapa,Veracruz, México
3Instituto de Psicología y Educación, Universidad Veracruzana, Agustín Melgar 2, col. 21 de Marzo, Xalapa, 91010 Veracruz, México.
4Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, 91090, Veracruz, México.
5Facultad de Arquitectura, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, 91090, Veracruz, México
DOI : https://doi.org/10.47191/ijmra/v7-i09-39Google Scholar Download Pdf
ABSTRACT:
The aim of this article is to assist novice students in understanding and applying numerical analysis techniques. This article presents a comparison between the numerical methods of least squares and Newton's interpolation for solving differential equations with boundary conditions. Two case studies are examined where these methodologies are applied, generating polynomial solutions of degree 20. Both methods demonstrate high accuracy, despite differences in the magnitude and signs of the coefficients for each power forming the polynomial. The solution obtained via interpolation exhibits better accuracy compared to LMS, with an improvement of up to 8 orders of magnitude near the right boundary.
KEYWORDS:Polynomial approximation, numerical analysis, ordinary differential equations, boundary value problems.
REFERENCES1) Edwards, C. H., and Penney, D. E. 2011. Ecuaciones diferenciales y problemas con valores de la frontera. Pearson Educación.
2) O'neil, P. V. 2004. Advanced Engineering Mathematics 5 ed. Tomson
3) Boelkins, M. R., Goldberg, J. L., and Potter, M. C. 2009. Differential equations with linear algebra. Oxford University Press.
4) Faires, J. D., and Burden, R. L. 2012. Numerical methods, 4th. Cengage Learning.
5) Chapra, S. C., Canale, 2011. Métodos numéricos para ingenieros New York, NY, USA: McGraw-Hill.
6) Maron, M. J., and López, R. J. 1995. Análisis numérico: un enfoque práctico. CECSA.
7) Sandoval-Hernandez, M., Vazquez-Leal, H., Filobello-Nino, U., De-Leo-Baquero, E., Bielma-Perez, A. C., Vichi-Mendoza, J. C., Alvarez-Gasca, O., Contreras-Hernandez, A. D., Bagatella-Flores, N., Palma-Grayeb, B. E., Sanchez-Orea, J. Cuellar-Hernandez, L., 2021b. The Quadratic Equation and its Numerical Roots. International Journal of Engineering Research and Technology, 10(6), 301-305. https://www.ijert.org/the-quadratic-equation-and-its-numerical-roots
8) Sandoval-Hernandez, M. A., Alvarez-Gasca, O., Contreras-Hernandez, A.D., Pretelin-Canela, J.E., Palma-Grayeb, B.E., Jimenez-Fernandez, V.M., Filobello-Nino, U., Pereyra-Diaz, D., Hernandez-Machuca, S.F., Sampieri-Gonzalez, C.E. and Gonzalez-Martinez, F.J., Castaneda-Sheissa, R., Hernandez-Mendez, S., Matias-Perez, J., Cuellar-Hernandez, L., Hoyos-Reyes, C., Cervantes-Perez, J., Varela-Lara, L.J., Vazquez-Aguirre, J.L., Gil-Adalid, L., Rocha-Fernandez, J.L., Bagatella-Flores, N. Vazquez-Leal, H. 2019. Exploring the classic perturbation method for obtaining single and multiple solutions of nonlinear algebraic problems with application to microelectronic circuits. International Journal of Engineering Research and Technology 8.9 (2019). https://www.ijert.org/exploring-the-classic-perturbation-method-for-obtaining-single-and-multiple-solutions-of-nonlinear-algebraic-problems-with-application-to-microelectronic-circuits
9) Filobello-Nino, U., Vazquez-Leal, H., Fariborzi Araghi, M. A., Huerta-Chua, J., Sandoval-Hernandez, M. A., Hernandez-Mendez, S., and Muñoz-Aguirre, E. 2020. A novel distribution and optimization procedure of boundary conditions to enhance the classical perturbation method applied to solve some relevant heat problems. Discrete Dynamics in Nature and Society, 2020(1). https://doi.org/10.1155/2020/1303701
10) Marinca, V., and Herisanu, N. 2012. Nonlinear dynamical systems in engineering: Some approximate approaches. Springer Science and Business Media.
11) Vazquez-Leal, H., Sandoval-Hernandez, M., Castaneda-Sheissa, R., Filobello-Nino, U., and Sarmiento-Reyes, A. 2015, Modified Taylor solution of equation of oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics. International Journal of Applied Mathematical Research, 4(2), 253-258. https://doi.org/10.14419/ijamr.v4i2.4273
12) Sandoval-Hernandez, M. A., Filobello-Nino, U. A., Vazquez-Leal, H., Huerta-Chua, J., Castaneda-Sheissa, R., Diaz-Arango, G. U., Callejas-Molina R.A, Torreblanca-Bouchan, S.E, Chong-Duran, J., and Pablo-Lopez, B., 2021, Taylor Series Method Coupled with Shooting Technique for Solving Boundary Value Problems, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH AND TECHNOLOGY (IJERT) Volume 10, Issue 10 (October 2021). https://www.ijert.org/taylor-series-method-coupled-with-shooting-technique-for-solving-boundary-value-problems
13) Sandoval-Hernandez, M. A., Vazquez-Leal, H., Filobello-Nino, U., and Hernandez-Martinez, L., 2019. New handy and accurate approximation for the Gaussian integrals with applications to science and engineering. Open Mathematics, 17(1), 1774-1793. https://doi.org/10.1515/math-2019-013
14) Vazquez-Leal, H., Sandoval-Hernandez, M. A., Garcia-Gervacio, J. L., Herrera-May, A. L., and Filobello-Nino, U. A. 2019. PSEM approximations for both branches of Lambert W function with applications. Discrete Dynamics in Nature and Society, 2019(1). https://doi.org/10.1155/2019/8267951
15) Sandoval-Hernandez, M. A., Vazquez-Leal, H., Sarmiento-Reyes, A., Filobello-Nino, U., Castro-Gonzalez, F., Herrera-May, A. L, Callejas-Molina, R. A. and Ruiz-Gomez, R. 2018. Optimized direct Padé and HPM for solving equation of oxygen diffusion in a spherical cell. Discrete Dynamics in Nature and Society, 2018(1). https://doi.org/10.1155/2018/9142124.
16) Sandoval-Hernandez, M., Shukla, A. K., Vazquez-Leal, H., Ramamohan, T. R., Castaneda-Sheissa, R., Filobello-Nino, U., Sampieri-Gonzalez, C.E., Hernandez-Martinez, L. 2018. Exploring a novel approach for homotopy solutions for nonlinear problems with two-point neumann boundary conditions. Nonlinear Science Letters A, 9, 360-377.
17) Vazquez-Leal, H., Sandoval-Hernandez, M. A., Filobello-Nino, U., and Huerta-Chua, J. 2020. The novel Leal-polynomials for the multi-expansive approximation of nonlinear differential equations. Heliyon, 6(4).
18) Villadsen, J., and Michelsen, M. L. 1978. Solution of differential equation models by polynomial approximation, prentice-Hall.
19) Finlayson, B. A. 1980. Nonlinear analysis in chemical engineering, McGraw-Hill
20) Carey, G. F., and Finlayson, B. A. 1975. Orthogonal collocation on finite elements. Chemical Engineering Science, 30(5-6), 587-596.
21) Sandoval-Hernandez, M., Vazquez-Leal, H., Hernandez-Martinez, L., Filobello-Nino, U. A., Jimenez-Fernandez, V. M., Herrera-May, A. L., Castaneda-Sheissa, R., Ambrosio-Lazaro, R.C., and Diaz-Arango, G. 2018. Approximation of Fresnel integrals with applications to diffraction problems. Mathematical Problems in Engineering, 2018(1), 4031793. https://doi.org/10.1155/2018/4031793
22) Douglas M. Bates, and Donald G. Watts. 1988. Nonlinear regression: iterative estimation and linear approximations, in: Nonlinear Regression Analysis and Its Applications, pp. 32–66.
23) Ola Ragb, L.F. Seddek, M.S. Matbuly, Iterative differential quadrature solutions for Bratu problem, Comput. Math. Appl. 74 (2) (2017) 249–257.
24) Yinghong Xu, Xiangjie Li, Lipu Zhang, The particle swarm shooting method for solving the Bratu’s problem, J. Algorithms Comput. Technol. 9 (3) (2014) 291–302.
25) Sandoval-Hernandez, M. A., Filobello-Nino, U., Vazquez-Leal, H., Torreblanca-Bouchan, S., Alvarez-Gasca, O., Contreras-Hernandez, A. D., Palma-Grayeb, B. E., De-Leo-Baquero, E., Bielma-Pérez, A. C., Vichi-Mendoza, J.C. Sanchez-Orea, J. Callejas-Molina, R. A., Hoyos-Reyes, C. 2021. An Analytical Approximate Solution for the Bratu Problem by using Nonlinearities Distribution Homotopy Perturbation Method. International Journal of Engineering Research and Technology, 10, 823-826. https://www.ijert.org/an-analytical-approximate-solution-for-the-bratu-problem-by-using-nonlinearities-distribution-homotopy-perturbation-method
26) Sandoval-Hernandez, M. A., Morales-Alarcon, G. J., Filobello-Nino, U. A., Vazquez-Leal, H., Alday-Garcia, A., Sampieri-Gonzalez, C. E., Pretelin-Canela, J. E., Escobar-Flores, A. R., Gasca- Herrera, A. E. Alvarez-Gasca, O., Contreras-Hernandez, A. D., Perez-Jacome-Friscione, J. E., Torreblanca-Bouchan, S.E. 2024. Solving Ordinary Differential Equations with Boundary Conditions Numerically. International Journal of Multidisciplinary Research and analysis. 7(02), 522-528. https://doi.org/10.47191/ijmra/v7-i02-13
Volume 07 Issue 09 September 2024
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