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Approximation to Hadamard Derivative via the Finite Part Integral.

Chuntao Yin1, Changpin Li2, Qinsheng Bi2

  • 1Department of Mathematics, Shanghai University, Shanghai 200444, China.

Entropy (Basel, Switzerland)
|December 3, 2020
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Summary
This summary is machine-generated.

This study introduces finite part integral methods to approximate Hadamard derivatives, offering practical solutions for differential equations in elasticity and wave propagation problems.

Keywords:
Hadamard derivativefinite part integral

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Area of Science:

  • Mathematical Physics
  • Numerical Analysis
  • Continuum Mechanics

Background:

  • Hadamard's 1923 work on integrals with strong singularities.
  • The practical utility of finite part integrals in elasticity and wave equations.
  • Challenges in solving differential equations involving singular integrals.

Purpose of the Study:

  • To develop novel numerical methods for approximating the Hadamard derivative.
  • To apply these methods to solve differential equations with Hadamard derivatives.
  • To demonstrate the effectiveness of the finite part integral approach.

Main Methods:

  • Development of rectangular and trapezoidal formulas for Hadamard derivative approximation.
  • Integration of the finite part integral concept.
  • Application to differential equations with Hadamard derivatives.

Main Results:

  • Successful approximation of Hadamard derivatives using finite part integrals.
  • Demonstration of the numerical methods' effectiveness on relevant differential equations.
  • Validation of the proposed technique through numerical examples.

Conclusions:

  • The finite part integral method provides a viable approach for handling Hadamard derivatives.
  • The developed rectangular and trapezoidal formulas are effective for numerical solutions.
  • This technique offers practical benefits for modeling crack problems and wave phenomena.