SOME APPLICATIONS OF GENERALIZED CALCULUS TO DIFFERENTIAL AND INTEGRO DIFFERENTIAL EQUATIONS

Abstract

This chapter discusses some applications of generalized calculus to differential and integrodifferential equations. A study is made about the use of fractional (generalized) calculus methods in solving some classes of differential and integrodifferential equations. This may be accomplished through representations of equations by their equivalent operator equations or, as they are called in some other studies, transform equations. Then the operator equations may be solved by using some operational properties of the integrodifferential operators of fractional (generalized) orders. The chapter discusses equivalence relations and properties and describes some methods of solving the operator equations that yield solutions to their equivalent differential or integrodifferential or integral equations. In particular, the method is applied to classes of differential or integrodifferential equations whose solutions represent some generalized special functions such as hypergeometric functions, Laguerre functions, Legendre functions, Bessel functions, etc.