Grundwald-Letnikov Operator and Its Role in Solving Fractional Differential Equations

Abstract

Leibnitz in 1663 introduced the derivative notation for the order of natural numbers, and then the idea of fractional derivatives appeared. Only a century later, this idea began to be realized with the discovery of the concepts of fractional derivatives by several mathematicians, including Riemann (1832), Grundwal, Fourier, and Caputo in 1969. The concepts in the definitions of fractional derivatives by Riemann-Liouville and Caputo are more frequently used than other definitions, this paper will discuss the Grunwald-Letnikov (GL) operator, which has been discovered in 1867. This concept is less popular when compared to the Riemann-Liouville and Caputo concepts, however, this concept is quite interesting because the concept of derivation is developed from the definition of ordinary derivatives. In this paper will be shown that the formulas for the fractional derivative using the GL concept are the same as the results obtained using the Riemann-Liouville and Caputo concepts. As a complement, we will give an example of solving a fractional differential equation using Modified Homotopy Perturbation Methods.