In this research paper, we introduce a new Transform belonging to the class of Laplace Transforms, called the Yasser-Jassim Transform (YJ Transform). We explore its properties and compare it to the classical Laplace Transform. Furthermore, we provide proofs for the key properties associated with this Transform and demonstrate its application in solving differential and Integral equations. By employing this new Transform, we can reduce the original problem to an algebraic equation that can be solved directly, followed by applying the inverse Transform to obtain the solution to the original problem.

In this paper, we introduce  a single Integral Transform that  defines all known time scales generalized Integral Transforms in the family of Laplace Transform as the new general Integral Transform on time scales. As a result, a unified approach is developed for the use of Integral Transforms representing the family of Laplace Transform for solving problems on continuous and discrete cases dynamics. The convergence  conditions and some principal properties accompanying the convolution theorem are given. It is shown that all generalized Integral Transforms on time scales included in the family of the Laplace Transform are special cases of a new general Integral Transform. The applicability of this Transform is demonstrated by solving certain ordinary dynamic equations and Integral equations.

These days, Integral equations govern a wide range of physical processes, including those related to electricity, mechanics, fluid dynamics, ecology, soil moisture dynamics, shallow water wave propagation, and chemical science. As a result, creating new techniques and putting them into practice for solving Integral equations becomes more crucial. In this paper, the Upadhyaya Integral Transform is employed for determining the analytical solutions of the non-linear Volterra Integral equations (NVIEs) of the first kind. Four examples suggest the effectiveness of the Upadhyaya Integral Transform, particularly for NVIEs of the first kind. The calculation results suggest that the proposed method provides accurate solutions to the original problem and this method is a valuable tool for researchers and scientists working on the broader range of problems involving NVIEs of the first kind.

The proposed study is focused to introduce a novel Integral Transform op-erator, called Generalized Bivariate (GB) Transform. The proposed trans-form includes the features of the recently introduced Shehu Transform, ARA Transform, and Formable Transform. It expands the repertoire of existing Laplace-type bivariate Transforms. The primary focus of the present work is to elaborate fashionable properties and convolution theorems for the proposed Transform operator. The existence, inversion, and duality of the proposed Transform have been established with other existing Transforms. Implementation of the proposed Transform has been demonstrated by ap-plying it to different types of differential and Integral equations. It validates the potential and trustworthiness of the GB Transform as a mathematical tool. Furthermore, weighted norm inequalities for Integral convolutions have been constructed for the proposed Transform operator.

This paper deals with the one-dimensional discrete wavelet Transform (1D DWT) of four scaling coefficients are computed numerically by designing a convolutive operator. The near-zone contribution of the Integral is calculated through wavelet Transform and for the far-zone contribution the classic expansion of the spherical harmonics applied. Finally the geoidal heights are determined over a territory in Canada.