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Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems.

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Title: Employing CNPS and CPS approaches to calculate numerical roots of ninth-order linear and nonlinear boundary value problems.
Authors: Khalid, Aasma1 (AUTHOR) aasmakhalid@gcwuf.edu.pk, Haq, Inamul1 (AUTHOR) inamulhaq@gcwuf.edu.pk, Rehan, Akmal2 (AUTHOR) akmalrehan@uaf.edu.pk, Alshehri, Hashim M.3 (AUTHOR) hmalshehri@kau.edu.sa, Osman, M. S.4 (AUTHOR) mofatzi@cu.edu.eg
Source: International Journal of Modern Physics C: Computational Physics & Physical Computation. Oct2024, Vol. 35 Issue 10, p1-28. 28p.
Subjects: Nonlinear boundary value problems, Numerical roots, Boundary value problems, Splines, Finite differences
Abstract: The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of h = 1 / 1 0 and h = 1 / 5. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
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Abstract:The extremely accurate findings of ninth-order linear and nonlinear BVPs are described in this paper. CNPS and CPS are used to find out the numerical roots of linear and nonlinear, ninth-order BVPs. The proposed methods transform the ninth-order BVPs into a system of linear equations. The algorithms developed in this study not only provide approximate solutions to the BVPs using CNPS and CPS, but they also estimate the derivatives from the first to the ninth-order of the analytical solution simultaneously. These approaches are implemented across five diverse problems, showcasing the effectiveness of these techniques by utilizing step sizes of h = 1 / 1 0 and h = 1 / 5. In order to gauge the accuracy of the methods, a comparison is drawn between the outcomes and the exact solutions, which are then presented in tabular and graphical format. The precision of these techniques is demonstrated through a detailed investigation and is found to be superior to the CBS, the Petrov–Galerkin Method (PGM) using Splines functions as basis functions and Septic B Splines functions as weight functions, the collocation method for ninth-order BVPs by Quintic B Splines and Sextic B Splines as evidenced by the comparison of AEs of CPS and CNPS with these alternative methods. [ABSTRACT FROM AUTHOR]
ISSN:01291831
DOI:10.1142/S0129183124501213