Quadratic‐phase wavelet transform with applications to generalized differential equations

摘要

In this article, we introduce a novel integral transform coined as quadratic‐phase wavelet transform by employing the convolution structure of quadratic‐phase Fourier transforms. Firstly, we explore some mathematical properties of the quadratic‐phase wavelet transform, including the orthogonality relation, inversion formula, range theorem and some notable inequalities. Secondly, we derive an extension of the Heisenberg's uncertainty principle and formulate an analogue of the Pitt's inequality for the quadratic‐phase Fourier and wavelet transforms. Finally, we broaden the scope of the proposed work by studying the differentiation properties of the quadratic‐phase Fourier transform and then addressing the analytical solutions of certain generalized partial differential equations, such as the generalized Laplace, wave, and the Schrödinger equations by utilizing the quadratic‐phase wavelet transform.