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A practical proposal to obtain solutions of certain variational problems avoiding Euler formalism.

U Filobello-Nino1, H Vazquez-Leal1,2, J Huerta-Chua3

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Summary

This study presents two novel methods, Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and Moving Boundary Condition Employing Transversality Conditions (METC), for solving variational problems with moving boundaries without traditional Euler formalism.

Keywords:
Euler equationsMathematicsOrdinary differential equationsVariable end point conditionsVariational problems

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Area of Science:

  • Variational calculus
  • Mathematical physics
  • Optimization problems

Background:

  • Traditional methods for variational problems with moving boundaries often rely on Euler formalism and transversality conditions.
  • These established approaches can be complex and computationally intensive.

Purpose of the Study:

  • To introduce novel methodologies for solving variational problems with moving boundaries.
  • To provide exact and analytical approximate solutions.
  • To circumvent the necessity of Euler formalism and complex transversality conditions.

Main Methods:

  • Development of the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) method.
  • Development of the Moving Boundary Condition Employing Transversality Conditions (METC) method.
  • Utilization of the Generalized Bernoulli Method (GBM) as a foundational tool.

Main Results:

  • The MWTC method successfully avoids the use of transversality conditions.
  • The METC method incorporates transversality conditions through a simplified, systematic procedure.
  • Both methods yield exact and analytical approximate solutions for the addressed problems.

Conclusions:

  • The proposed MWTC and METC methods offer efficient alternatives for solving variational problems with moving boundaries.
  • The Generalized Bernoulli Method is a key component for achieving these solutions.
  • These new approaches simplify the solution process without compromising accuracy.