Theory of Functional Connections Extended to Fractional Operators
Abstract
:1. Introduction
2. Background on Fractional Calculus
2.1. The Gamma Function
- ;
- , for ;
- is logarithmically convex (or superconvex).
2.2. Riemann–Liouville Fractional Integral
2.3. Riemann–Liouville Fractional Derivative
2.4. Caputo Fractional Derivative
2.5. Grünwald–Letnikov Definitions
3. Background on the Theory of Functional Connections
A Simple Explanatory Example
4. Shifted Chebyshev Polynomials
Example
5. Numerical Examples
5.1. Single Fractional Constraint
5.2. Three Mixed Constraints
5.3. Two Linear Combinations of Fractional Constraints
6. Discussion
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
TFC | Theory of functional connections |
SCP | Shifted Chebyshev polynomials |
Appendix A. Some Fractional Integrals and Derivatives with Closed-Form Expressions
Appendix B. Non-Locality of Fractional Operators (Memory Effect)
Appendix C. Least-Squares Approaches
- The common solution: ;
- The QR decomposition: , then , where and R an upper-triangular matrix;
- The SVD decompositions: , then where and and where is the pseudo-inverse of , which is formed by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix;
- The Cholesky decomposition: , then , where U is an upper-triangular matrix.
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Mortari, D.; Garrappa, R.; Nicolò, L. Theory of Functional Connections Extended to Fractional Operators. Mathematics 2023, 11, 1721. https://doi.org/10.3390/math11071721
Mortari D, Garrappa R, Nicolò L. Theory of Functional Connections Extended to Fractional Operators. Mathematics. 2023; 11(7):1721. https://doi.org/10.3390/math11071721
Chicago/Turabian StyleMortari, Daniele, Roberto Garrappa, and Luigi Nicolò. 2023. "Theory of Functional Connections Extended to Fractional Operators" Mathematics 11, no. 7: 1721. https://doi.org/10.3390/math11071721