Logarithmic Bernstein functions for fractional Rosenau–Hyman equation with the Caputo–Hadamard derivative

• Published in: Results in physics 2024-12, Vol.67, p.108055, Article 108055
• Main Authors: Heydari, M.H., Heydari, F., Bavi, O., Bayram, M.
• Format: Article
• Language: English
• Subjects: Caputo–Hadamard fractional derivative; Hadamard fractional integral; Logarithmic Bernstein functions; Rosenau–Hyman equation
• ISSN: 2211-3797; 2211-3797
• DOI: 10.1016/j.rinp.2024.108055

In this study, the Caputo–Hadamard derivative is fittingly used to define a fractional form of the Rosenau–Hyman equation. To solve this equation, the orthonormal logarithmic Bernstein functions (BFs) are created as a suitable basis for handling this type of derivative. The primary benefit of these functions lies in the ease of computing their Hadamard fractional integral and derivative. These logarithmic functions, combined with the orthonormal Bernstein polynomials (BPs), are simultaneously employed to develop a hybrid strategy for solving the aforementioned equation. More precisely, the orthonormal logarithmic BFs are utilized to approximate the solution in the temporal domain and the orthonormal BPs are employed in the spatial domain. In addition, a matrix is extracted for the Hadamard integral of the orthonormal logarithmic BFs due to the implementation of the presented method. The effectiveness of the established scheme in finding accurate numerical solutions is evaluated through the resolution of three examples. •A fractional version of the Rosenau–Hyman equation is defined using the Caputo–Hadamard derivative.•The orthonormal logarithmic Bernstein functions, as a new family of basic functions, are introduced.•An formula for computing the Hadamard fractional integral of these logarithmic functions is obtained.•The logarithmic Bernstein functions and orthonormal Bernstein polynomials are employed mutally for this equation.•The applicability and validity of the developed method are checked in some examples.