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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.

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Summary
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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.

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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.