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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
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.
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.
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