New Methods to Solve Difference Equations Automatically

Difference Equations appear very often in discrete mathematics and computational science, as they describe how the future values of a system depend on their past values. Solving such equations is a very important task; Laplace transform has a long history and was used for solving difference equations. Although Laplace transform is useful in that regard, it suffers a serious drawback, which is the calculation of inverse Laplace transform. Such kind of inverse calculation is problematic or simply impossible in general. Sumudu transform is an integral transform which was introduced recently, with new and powerful features. In this work, we will explain that Sumudu transform can be used to solve difference equations in the same manner as Laplace transform. More importantly, new computational methods will be proposed to solve difference equations, differential-difference equations, and integro-differential-difference equations; these new methods combine both Laplace transform and Sumudu transform such that there is no calculation of the inverse transform. Demonstration code is provided to show that the methods can be implemented in computer algebra systems such as Maple to solve a wide range of equations automatically.