Implicit Function Method for Solving Difference Equations

An alternative new method to solve difference equations is outlined in this work. The method is named as implicit function method. The method relies on implicit functional forms that enable transformation of the difference equation into a more solvable simple form. First order variable coefficient difference equations are treated first. A second order variable coefficient difference equation is solved next. Perturbed first and second order difference equations are solved by the combination of the method and perturbation techniques. Some nonlinear equations are treated finally. Exact and approximate analytical solutions are constructed using the new method. The advantage of the method is that it can be applicable to any type of difference equation. The equations treated may be of arbitrary order, linear, nonlinear, constant coefficient or variable coefficient. The disadvantage of the method is that the transformation of the equation to a simpler form is not straightforward and requires some experience.