The new integral transform, DPR Transform and its Application

Abstract

In this paper, a new integral DPR transform is developed and introduced. It is a superior transform, exhibiting dualities with more than eighty other integral transforms. Many researchers have been introduced different integral transform with various kernels. This transform is generalization of all such more than eighty integral transforms. Instead of studying eighty different transforms this single transform can solve the various problem solved by these different integral transform. A connection is established between this recent transformation and the previously recognized advantageous transforms, thereby extending the theoretical framework.We further discuss fundamental properties of the DPR transform which include linearity property, scaling property, domain shifting property, and its operation on derivatives are rigorously demonstrated. The transform is applied to a variety of equations,which include ordinary differential equations, partial differential equations, integro-differential equations, fractional differential equations, difference equations, and differential–difference equations. In addition, the practical utility of the DPR integral transform is illustrated through applications in forensic science, where it provides outstanding results without requiring extensive computational effort.