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Lie symmetry analysis for fractional evolution equation with $ψ$-Riemann-Liouville derivative

Junior C. A. Soares, Felix S. Costa, J. Vanterler C. Sousa, Maria V. S. Sousa, Amália R. E. Pereira

Abstract

We present the applycation of theory of Lie group analysis with $ψ$-Riemann-Liouville fractional derivative detailing the construction of infinitesimal prolongation to obtain Lie symmetries. In additional, is addressed the invariance condition without the need to impose that the lower limit of fractional integral is fixed. We find an expression that expands the knowledge regarding the study of exact solutions for fractional differential equations. We use of the framework developed in \cite{zaky2022note} to present our understanding of the extension of $ψ$-Riemann-Liouville fractional derivative. It is demonstrate the Leibniz type rule for the derivative operator in question for built the prolongation. At last, we calculate the Lie symmetries of the generalized Burgers equation and fractional porous medium equation.

Lie symmetry analysis for fractional evolution equation with $ψ$-Riemann-Liouville derivative

Abstract

We present the applycation of theory of Lie group analysis with -Riemann-Liouville fractional derivative detailing the construction of infinitesimal prolongation to obtain Lie symmetries. In additional, is addressed the invariance condition without the need to impose that the lower limit of fractional integral is fixed. We find an expression that expands the knowledge regarding the study of exact solutions for fractional differential equations. We use of the framework developed in \cite{zaky2022note} to present our understanding of the extension of -Riemann-Liouville fractional derivative. It is demonstrate the Leibniz type rule for the derivative operator in question for built the prolongation. At last, we calculate the Lie symmetries of the generalized Burgers equation and fractional porous medium equation.
Paper Structure (8 sections, 17 theorems, 124 equations, 1 table)

This paper contains 8 sections, 17 theorems, 124 equations, 1 table.

Table of Contents

  1. Introduction
  2. Mathematical background and main results
  3. Lie symmetry for Riemann-Liouville fractional differential equation
  4. Lie point symmetry for time-fractional evolution equation
  5. Gazizov's method for GFBE
  6. Prolongation for $\psi$-Riemann-Liouville derivative
  7. Some results
  8. Considerations

Key Result

Lemma 2.2

sousa2019leibniz Admitting the sets defined above and the conditions for the functions $f$ and $\psi(t)$ in Definition def1 we can rewrite the Eq.(ipsi) as: where $t>a$.

Theorems & Definitions (33)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 2.4: Leibniz rule
  • Theorem 4.1: Z.Y. Zhang's Theorem zhang2020symmetry
  • Corollary 4.2: zhang2020symmetry
  • Theorem 4.3: zhang2020symmetry
  • Theorem 4.4: soares2023note
  • proof
  • Theorem 4.5
  • ...and 23 more