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Closed solutions to a differential-difference equation and an associated plate solidification problem.

Olawanle P Layeni1, Adegbola P Akinola1, Jesse V Johnson2

  • 1Department of Mathematics, Obafemi Awolowo University, Ile-Ife, 220005 Nigeria.

Springerplus
|August 20, 2016
PubMed
Summary

Novel methods were developed to find exact solutions for differential-difference equations in plate solidification. This research provides traveling wave and similarity solutions for specific temperature conditions, advancing materials science understanding.

Keywords:
Clarkson–Kruskal’s methodDifferential-difference equationStefan problem

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Area of Science:

  • Materials Science
  • Mathematical Physics
  • Solidification Phenomena

Background:

  • Plate solidification involves complex differential-difference equations with variable coefficients.
  • Finding exact closed-form solutions is crucial for accurate modeling and prediction.
  • Existing methods may not fully address the complexities of variable coefficients in this context.

Purpose of the Study:

  • To introduce two novel formalisms for deriving exact closed solutions.
  • To obtain exact closed traveling wave and similarity solutions for a plate solidification problem.
  • To analyze solutions under specific conditions of time-varying surface temperature.

Main Methods:

  • Development of two distinct mathematical formalisms.
  • Application of these formalisms to derive exact solutions.
  • Analysis of special cases involving time-varying plate surface temperature.

Main Results:

  • Successful derivation of exact closed solutions for a class of variable-coefficient differential-difference equations.
  • Obtained exact closed traveling wave solutions for the plate solidification problem.
  • Obtained exact closed similarity solutions for specific time-varying temperature scenarios.

Conclusions:

  • The introduced formalisms are effective for solving complex differential-difference equations in solidification problems.
  • The obtained solutions provide valuable insights into the dynamics of plate solidification.
  • This work contributes to the theoretical understanding and predictive capabilities in materials processing.