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High Accuracy Spline Explicit Group (SEG) Approximation for Two Dimensional Elliptic Boundary Value Problems.

Joan Goh1, Norhashidah Hj M Ali1

  • 1School of Mathematical Sciences, Universiti Sains Malaysia, Pulau Pinang, Malaysia.

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Summary
This summary is machine-generated.

This study introduces an efficient numerical method using cubic splines and finite differences to solve elliptic partial differential equations. The approach demonstrates high accuracy and computational efficiency for physical problems.

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Area of Science:

  • Numerical analysis
  • Computational mathematics
  • Applied physics

Background:

  • Cubic splines are established for accurate differential equation approximation.
  • Numerical solutions for partial differential equations are crucial in science and engineering.
  • Existing methods may face challenges in computational efficiency or accuracy.

Purpose of the Study:

  • To present a novel numerical method for solving two-dimensional elliptic partial differential equations.
  • To combine cubic spline approximation in one direction with finite differences in another.
  • To apply an accelerated iterative scheme for enhanced computational performance.

Main Methods:

  • Utilizing a cubic spline approximation in the x-direction.
  • Employing the finite difference method in the y-direction.
  • Applying a four-point explicit group (EG) iterative scheme with an acceleration tool.

Main Results:

  • The proposed method provides accurate numerical solutions for the targeted partial differential equations.
  • The computational complexity of the method is analyzed.
  • Comparative results highlight the efficiency and accuracy of the developed technique.

Conclusions:

  • The combined cubic spline and finite difference method offers an efficient approach for solving elliptic partial differential equations.
  • The explicit group iterative scheme with acceleration enhances the performance for physical problem applications.
  • The method is validated through detailed formulation, implementation, and comparative analysis.