1)An elastic vibration of elliptic plate have been determined with the help of finite Mathiue transform, Laplace transform and finite Marchi-Fasulo transform techniques. The expressions is represented graphically. The results that is obtained can be useful to the design of structures or machines in engineering applications. ii) Choubey, N.K. ; Geeta Shrivastava and Mehta have studied the problem of heat conduction and investigated the unknown temperature of elliptical cylinder using finite Fourier sine transform technique. we have been made to determine the temperature distribution and unknown temperature gradient of elliptical cylinder with known radiation type boundary conditions, using Mathiue transform and finite Marchi-Fasulo transform techniques. iii)The determination of initial temperature distribution from a known physically realizable temperature distribution at any time and position is known as time reversal problem iv)The inverse problem is very important in view of its relevance's to various industrial machines subject to heating such as main shaft of lathe and turbine, roll of a rolling mill and measurement of aerodynamic heating. We attempt has been made to determine the temperature distribution and unknown temperature gradient, in which the solutions are expressed in Mathieu and modified Mathieu functions with known boundary conditions using finite Marchi-Fasulo transform and Mathieu transform techniques. v)Integral transform technique plays important role in solving problem of applied Mathematics