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

Area of Science:

• Mathematics
• Applied Mathematics
• Numerical Analysis

Background:

• Fractal-fractional differential equations (F-FDEs) are increasingly important in modeling complex phenomena.
• Existing methods for solving F-FDEs have limitations in terms of applicability and efficiency.
• The qualitative theory and numerical solutions of F-FDEs require further investigation.

Purpose of the Study:

• To establish adequate results for the qualitative theory of F-FDEs.
• To develop an approximate numerical solution for F-FDEs using a novel method.
• To analyze the Ulam-Hyers stability of the proposed solutions.

Main Methods:

• The Haar wavelet collocation (H-W-C) method is employed for numerical approximation.
• A general algorithm is established for computing numerical solutions of F-FDEs.
• Banach fixed point theorems are utilized to establish qualitative theory results.

Main Results:

• The Haar wavelet collocation method is successfully applied to F-FDEs, a rarely used approach.
• A general algorithm for the numerical solution of F-FDEs is presented.
• Theoretical results concerning the qualitative aspects and Ulam-Hyers stability are established.

Conclusions:

• The Haar wavelet collocation method offers an effective approach for solving F-FDEs.
• The study contributes to the qualitative theory and numerical analysis of F-FDEs.
• The findings are validated through numerical examples and error analysis.