THE NEW SOLUTION OF FRACTIONAL LAPLACE TYPE EQUATION IN CONFORMABLE SENSE

Abstract

The fundamental characteristics of conformable integrals and derivatives have garnered significant attention in recent research [1, 3, 4, 5]. In this work, we delve into the advanced field of fractional calculus, focusing on the development and application of a novel framework. We introduce and elaborate on various definitions and types of fractional derivatives, with a special emphasis on the ”new conformable derivative,” a concept introduced in [4]. This derivative is defined as: ( D α G ) ( z ) = lim x −→ 0 G ( z + x e ( α − 1 ) z ) − G ( z ) x , where α represents the order of the derivative. The new conformable derivative offers substantial advancements in the study of fractional differential equations, particularly through its application to the fractional Laplace equation (FLE). We demonstrate how this derivative enhances the analysis and solution of (FLE) by employing fractional Fourier series (FFS). Our study involves a comprehensive comparison of results derived from this approach with those obtained from fractional Laplace-type equations in the conformable sense [2], as well as with non-conformable fractional Laplace equations for varying values of α see [7]. This work not only expands the theoretical framework but also offers practical tools for tackling complex differential equations using conformable fractional calculus.