Special transforms whose intervals are finite are unified and extended. A kernel is employed which may be determined to suit each particular type of problem. The Sturm-Liousville expansion is obtained for fx when fx is an integral function over a,b and a alxlb. The finite Sturm-Liouville transform is defined. Solutions are ontianed for some partial differential equations. Consideration is given to spherical harmonics Hermite and Tchebycheff polynomials and Bessel, Mathieu, and Wittaker functions. A heat conduction problem for which the solution was not known was successfully solved.