Root Operators and “Evolution” Equations
Abstract
:1. Introduction
2. Examples of Evolution Equation: Heat, Schrödinger and Paraxial Wave Equations
2.1. General Treatment: Hamiltonian and Evolution Operators
2.2. Evolution Operator as Poisson and Fresnel Transforms
2.3. Polynomial Solutions and Symmetry Transformations
3. Evolution Operator and Fractional Partial Differential Equations
4. Dirac-like Factorization to Disentangle Root OperatorFunctions
4.1. Square Root Function
4.2. Estension of the Procedure to Higher-Degree Root Operator Functions
4.2.1. Cube Root
4.2.2. Quartic Root
4.2.3. m-th Roots
5. Possible Applications: Practical and Conceptual Issues
5.1. Square Root of Differential Operators
5.1.1. Solving Relativistic-like Free Evolution Equations
5.1.2. Suggesting Alternative Formulations in Fractional Calculus
5.2. Cube Root of Differential Operators
6. Relativistic-like Evolution Equation: A “Direct” Solving Method
6.1. Evolution Operator
6.2. Evolution Operator as McDonald Transform
6.3. Symmetry Transformations and “Polynomial” Solutions
7. Conclusions
Author Contributions
Conflicts of Interest
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Dattoli, G.; Torre, A. Root Operators and “Evolution” Equations. Mathematics 2015, 3, 690-726. https://doi.org/10.3390/math3030690
Dattoli G, Torre A. Root Operators and “Evolution” Equations. Mathematics. 2015; 3(3):690-726. https://doi.org/10.3390/math3030690
Chicago/Turabian StyleDattoli, Giuseppe, and Amalia Torre. 2015. "Root Operators and “Evolution” Equations" Mathematics 3, no. 3: 690-726. https://doi.org/10.3390/math3030690