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Fundamental solutions for semidiscrete evolution equations via Banach algebras
Jorge González-Camus1, Carlos Lizama1, Pedro J Miana2
1Departamento de Matemáticas y Ciencias de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile.
Advances in Difference Equations
|January 13, 2021
Summary
This study provides novel representations for solutions to time-fractional differential equations using discrete Fourier transforms and explores the properties of finite difference operators. Results offer insights into fractional calculus and operator theory on sequence spaces.
Area of Science:
- Mathematics
- Numerical Analysis
- Functional Analysis
Background:
- Time-fractional differential equations (TFDEs) are crucial in modeling complex phenomena.
- Understanding operators on sequence spaces is fundamental in applied mathematics.
- Discrete convolutions and Fourier transforms offer powerful analytical tools.
Purpose of the Study:
- To develop representations for solutions of TFDEs.
- To analyze finite difference operators and their fractional powers.
- To investigate the algebraic and spectral properties of these operators.
Main Methods:
- Utilizing the discrete Fourier transform for discrete convolutions.
- Examining finite difference operators as generators of semigroups and cosine functions.
- Applying the subordination principle to fractional powers of operators.
Main Results:
- Obtained representations for TFDE solutions on Lebesgue spaces of sequences.
- Characterized linear and algebraic structures, norms, and spectra of operators.
- Identified fractional powers of finite difference operators.
Conclusions:
- The findings contribute to the theoretical understanding of TFDEs.
- The methods provide a framework for analyzing fractional calculus problems.
- Results have potential applications in areas utilizing TFDEs.
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