Abstract
We introduce a class of orthogonal functions associated with integral and fractional differential equations with a logarithmic kernel. These functions are generated by applying a log transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced logarithmic Jacobi functions, we develop an efficient spectral logarithmic Jacobi collocation method for the integrated form of the Caputo–Hadamard fractional nonlinear differential equations. To demonstrate the proposed approach's spectral accuracy, an error estimate is derived, which is then confirmed by numerical results.
| Original language | English |
|---|---|
| Pages (from-to) | 326-346 |
| Number of pages | 21 |
| Journal | Applied Numerical Mathematics |
| Volume | 181 |
| DOIs | |
| Publication status | Published - Nov 2022 |
Funding
The authors are grateful to the editors and the anonymous referees for their constructive feedback and helpful suggestions, which highly improved the paper. M. A. Zaky was supported by the Nazarbayev University Program 091019CRP2120 . This research was partially funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan Grant OR11466188 (Grant “Dynamical Analysis and Synchronization of Complex Neural Networks with Its Applications”). A. S. Hendy wishes to acknowledge the support of the RSF grant, project 22-21-00075 .
Keywords
- Caputo–Hadamard derivative
- Convergence analysis
- Logarithmic Jacobi function
- Spectral collocation method
ASJC Scopus subject areas
- Numerical Analysis
- Computational Mathematics
- Applied Mathematics