Mathematics serves as a fundamental intelligent theoretic basis for computation, and mathematical analysis is very useful to develop computational methods to solve various problems in science and engineering. Integral transforms such as Laplace Transform have been playing an important role in computational methods. In this paper, we will introduce Sumudu Transform in a new computational approach, in which effective computational methods will be developed and implemented. Such computational methods are straightforward to understand, but powerful to incorporate into computational science to solve different problems automatically. We will provide computational analysis and essentiality by surveying and summarizing some related recent works, with additional automatic proof details by applying system built-in functions. Applications include the computation of coefficients of Taylor's expansions, calculation of generating functions, mathematical identity proofs, solving differential equations and integral equations. For demonstration purposes, some of the methods were implemented in Maple with demonstrational results matching the expected values.