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This study introduces a new numerical method for the advection-diffusion equation using subdivision schemes and collocation. The technique accurately solves physical science and engineering problems, transforming complex equations into simpler algebraic ones.

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Area of Science:

  • Numerical Analysis
  • Computational Physics
  • Engineering Mathematics

Background:

  • The advection-diffusion equation is fundamental in modeling transport phenomena.
  • Existing numerical methods may face challenges with accuracy and efficiency.
  • Novel approaches are needed for complex physical and engineering problems.

Purpose of the Study:

  • To present a novel numerical technique for the one-dimensional advection-diffusion equation.
  • To explore the application of subdivision schemes in physical sciences and engineering.
  • To validate the accuracy and correctness of the proposed method.

Main Methods:

  • A subdivision scheme-based collocation method for spatial interpolation.
  • The finite difference method for temporal discretization.
  • Transformation of the differential equation into a system of algebraic equations.

Main Results:

  • The proposed technique was tested on various problems.
  • Quantitative results were presented in tables and figures.
  • Comparative analysis confirmed the method's accuracy against existing approaches.

Conclusions:

  • The novel technique provides an accurate and efficient solution for the advection-diffusion equation.
  • Subdivision schemes show significant potential for applications in physical sciences and engineering.
  • The method's ability to simplify problems into algebraic equations is a key advantage.