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Application of the perturbation iteration method to boundary layer type problems.

Mehmet Pakdemirli1

  • 1Applied Mathematics and Computation Center, Celal Bayar University, 45140 Muradiye, Manisa, Turkey.

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Summary

The novel perturbation iteration algorithm effectively solves singular boundary layer problems. This new technique combines inner and outer solutions for accurate domain-wide results in fluid dynamics and related fields.

Keywords:
Boundary layer problemsOrdinary differential equationsPerturbation methodsPerturbation–iteration algorithmSingular perturbation problems

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Area of Science:

  • Applied Mathematics
  • Numerical Analysis
  • Fluid Dynamics

Background:

  • Singular boundary layer problems present significant challenges in various scientific and engineering disciplines.
  • Traditional numerical and analytical methods often struggle with the complexities of these problems.

Purpose of the Study:

  • To introduce and evaluate the perturbation iteration algorithm for the first time on boundary layer type singular problems.
  • To demonstrate the efficacy of the simplest perturbation-iteration algorithm (PIA(1,1)) in handling both linear and nonlinear singular problems.

Main Methods:

  • Application of the perturbation iteration algorithm, specifically the PIA(1,1) variant.
  • Determination of inner and outer solutions using the iterative algorithm.
  • Matching of inner and outer solutions to construct a composite expansion.
  • Validation against exact or existing numerical solutions.

Main Results:

  • The perturbation iteration algorithm successfully determined inner and outer solutions for linear and nonlinear singular boundary layer problems.
  • A composite expansion was constructed by matching the inner and outer solutions, providing a domain-wide valid solution.
  • The results obtained using the perturbation iteration algorithm showed good agreement with available exact or numerical solutions.

Conclusions:

  • The perturbation iteration algorithm is a powerful and effective new technique for solving boundary layer type singular problems.
  • The PIA(1,1) algorithm provides a viable approach for tackling complex fluid dynamics and related singular problems.