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