Modified Fourier Sine and Cosine transforms for the Hadamard fractional calculus

Nowadays, the study of Hadamard fractional calculus is a hot topic, where the Hadamard fractional calculus is more suitable for describing the very slow process. Due to the logarithmic integral kernel of Hadamard calculus, it brings great difficulties to the corresponding theoretical analysis and numerical calculation. In this research, we introduce a novel modified Fourier Sine transform and a Fourier Cosine transform and then study the corresponding convolution theorem for the Fourier Sine and Cosine transforms. Finally, we provide the transformation results of the Hadamard fractional integral and derivative separately, which successfully overcome the difficulties caused by logarithmic singular kernels.