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Linear B-spline finite element method for the generalized diffusion equation with delay.

Gemeda Tolessa Lubo1, Gemechis File Duressa2

  • 1Department of Mathematics, Wollega University, Nekemte, Ethiopia. gemedatolesa@gmail.com.

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|June 6, 2022
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Summary

A new linear B-spline finite element method was developed for generalized diffusion equations with delay. This numerical method provides a stable and convergent solution, verified through experiments.

Keywords:
Finite elementGeneralized diffusion equation with delayLinear B-spline

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Area of Science:

  • Numerical Analysis
  • Partial Differential Equations
  • Computational Mathematics

Background:

  • Generalized diffusion equations with delay present significant challenges in numerical computation.
  • Existing methods may lack stability or produce non-smooth solutions.

Purpose of the Study:

  • To develop and analyze a novel linear B-spline finite element method for solving generalized diffusion equations with delay.
  • To ensure the numerical method yields smooth, continuous solutions for accurate approximation.

Main Methods:

  • Spatial discretization using linear B-spline basis functions.
  • Temporal discretization employing the Crank-Nicolson scheme.
  • Derivation of conditions for asymptotic stability and convergence analysis.

Main Results:

  • The proposed method generates smooth, piecewise continuous numerical solutions.
  • Sufficient and necessary conditions for asymptotic stability were established.
  • Numerical experiments confirmed the method's applicability and accuracy.

Conclusions:

  • The developed linear B-spline finite element method is a stable and effective approach for generalized diffusion equations with delay.
  • The method's ability to provide smooth solutions enhances its utility for precise approximation.
  • Further numerical investigations support its practical implementation.