Utilization of Haar wavelet collocation technique for fractal-fractional order problem

Kamal Shah; Rohul Amin; Thabet Abdeljawad

doi:10.1016/j.heliyon.2023.e17123

Utilization of Haar wavelet collocation technique for fractal-fractional order problem

Heliyon. 2023 Jun 13;9(6):e17123. doi: 10.1016/j.heliyon.2023.e17123. eCollection 2023 Jun.

Authors

Kamal Shah^{1

2}, Rohul Amin³, Thabet Abdeljawad^{1

4}

Affiliations

¹ Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.
² Department of Mathematics, University of Malakand, Chakdara, Dir(L) 18000, KPK, Pakistan.
³ Department of Mathematics, University of Peshawar, KPK, Pakistan.
⁴ Department of Medical Research, China Medical University, Taichung 40402, Taiwan.

Abstract

This work is devoted for establishing adequate results for the qualitative theory as well as approximate solution of "fractal-fractional order differential equations" (F-FDEs). For the required numerical results, we use Haar wavelet collocation (H-W-C) method which has very rarely utilized for F-FDEs. We establish the general algorithm for F-FDEs to compute numerical solution for the considered class. Also, we establish a result devoted to the qualitative theory via Banach fixed point result. A results devoted to Ulam-Hyers (U-H) stability are also included. Two pertinent examples are given along with the comparison and different norms of errors displayed in figures as well as tables.

Keywords: F-FDEs; Fractal-fractional Caputo derivative; H-W-C; Numerical results.