In this paper, we give several upper and lower bounds for the numerical radius of 2 ×2 block matrices. Several special cases of our results are given.

In this paper, we present several singular value inequalities for special types of functions of products and sums of matrices. Some of special cases of our results give a generalization of some recent inequalities

This study introduces a novel integral transform derived by integrating the ARA and Sawi transforms. The paper explores the foundational properties and establishes the existence of this new transform. It presents advanced results for partial differential equations in higher dimensions and extends the double convolution theorem to two dimensions. These developments are applied to solve specific types of differential equations, demonstrating practical applications in physics and related scientific fields.