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Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system.

Muhammad Abbas1, Ahmad Abd Majid2, Ahmad Izani Md Ismail2

  • 1Department of Mathematics, University of Sargodha, Sargodha, Pakistan ; School of Mathematical Sciences, Universiti Sains Malaysia, Penang, Malaysia.

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Summary
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A new numerical method using cubic B-splines accurately solves reaction-diffusion systems. This stable, efficient approach shows excellent agreement with exact solutions for complex problems.

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Partial Differential Equations

Background:

  • Reaction-diffusion systems are crucial in modeling phenomena across various scientific fields.
  • Solving these strongly coupled systems numerically presents significant challenges.
  • Existing methods often struggle with stability and accuracy for complex dynamics.

Purpose of the Study:

  • To present a novel numerical method for solving strongly coupled reaction-diffusion systems.
  • To evaluate the stability and accuracy of the proposed cubic B-spline collocation scheme.
  • To demonstrate the method's effectiveness on a relevant test problem.

Main Methods:

  • A cubic B-spline collocation scheme is employed on a uniform grid.
  • The time derivative is discretized using a standard finite difference scheme.
  • Cubic B-splines serve as interpolation functions for the spatial dimension.
  • The von Neumann method is used to rigorously assess the scheme's unconditional stability.

Main Results:

  • The proposed numerical scheme is proven to be unconditionally stable.
  • Accuracy is demonstrated through application to a test problem, yielding good agreement with exact solutions.
  • Performance is quantified using L∞ and L2 error norms at various time levels.

Conclusions:

  • The cubic B-spline collocation method offers a stable and accurate solution for strongly coupled reaction-diffusion systems.
  • The numerical results validate the scheme's reliability and efficiency.
  • This method provides a valuable tool for simulating complex reaction-diffusion processes.