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Fractional calculus with exponential memory
Hui Fu1, Guo-Cheng Wu1, Guang Yang1
1Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, People's Republic of China.
Chaos (Woodbury, N.Y.)
|April 3, 2021
Summary
A novel fractional integral with an exponential kernel is introduced, offering efficient solutions for fractional nonlinear differential equations. This new method simplifies complex calculations and demonstrates practical applicability in scientific research.
Area of Science:
- Fractional Calculus
- Mathematical Analysis
- Numerical Methods
Background:
- The standard Riemann-Liouville integral definition has limitations.
- Fractional calculus requires robust integral definitions for advanced applications.
Purpose of the Study:
- To propose a new fractional integral with an exponential kernel.
- To analyze its properties and applications in solving fractional differential equations.
Main Methods:
- Revisiting the standard Riemann-Liouville integral definition.
- Developing a new fractional integral with an exponential kernel.
- Deriving properties like composition and Leibniz integral law.
- Obtaining exact solutions for fractional homogeneous and non-homogeneous equations.
- Proposing a finite difference scheme for nonlinear fractional differential equations.
Main Results:
- A new fractional integral with an exponential kernel is successfully defined.
- Key properties of the new integral, including composition and Leibniz rule, are established.
- Exact solutions for fractional differential equations are derived.
- A finite difference scheme is presented for solving fractional nonlinear differential equations with exponential memory.
Conclusions:
- The new fractional integral with an exponential kernel is efficient and convenient.
- The proposed methods offer effective solutions for fractional nonlinear differential equations.
- This work advances the field of fractional calculus and its applications.
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