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Extended cubic B-spline method for solving a linear system of second-order boundary value problems.

Ahmed Salem Heilat1, Nur Nadiah Abd Hamid1, Ahmad Izani Md Ismail1

  • 1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia.

Springerplus
|August 23, 2016
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Summary

This study introduces an extended cubic B-spline method for solving second-order boundary value problems. The optimized method demonstrates accurate and comparable results to existing techniques.

Keywords:
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Area of Science:

  • Numerical Analysis
  • Applied Mathematics

Background:

  • Second-order boundary value problems (BVPs) are fundamental in various scientific and engineering disciplines.
  • Existing numerical methods for BVPs often face challenges with accuracy and computational efficiency.

Purpose of the Study:

  • To propose and analyze a novel numerical method for solving linear systems of second-order BVPs.
  • To investigate the impact of parameter optimization on the accuracy of the proposed method.

Main Methods:

  • Development of a numerical technique utilizing extended cubic B-splines.
  • Optimization of two key free parameters within the B-spline formulation.
  • Calculation and analysis of the truncation error associated with the method.

Main Results:

  • The proposed extended cubic B-spline method was tested on three distinct examples.
  • Results indicate that the method achieves accuracy comparable to or exceeding that of the standard cubic B-spline and other existing numerical approaches.
  • Parameter optimization significantly influences the method's precision.

Conclusions:

  • The extended cubic B-spline method offers a robust and accurate approach for solving second-order BVPs.
  • The optimization of free parameters is crucial for enhancing the method's performance.
  • This method presents a viable alternative for researchers and practitioners dealing with such problems.