Many significant equations in mathematical physics and engineering, such as the heat equation, wave equation, Boltzmann’s equation, and the Navier-Stokes equation, can be expressed in the form of evolution equations. Mathematicians and physicists now create evolution equations in fractional-order forms. Many researchers believe that fractional-order differential equations can represent real-world situations better and more accurately than integer-order differential equations. Since the Caputo type generalized fractional derivative is well-known for being the generalization of Caputo fractional derivatives, this article’s studies contribute to the solving of a variety of fractional differential equations in the sense of Caputo type generalized fractional derivative and Caputo fractional derivative. Moreover, the fractional Green’s functions for those fractional differential equations are obtained. The generalized Laplace transform and generalized Mellin transform are used to effectively and successfully achieve the desired results. Importantly, the generalized Mellin transform is firstly proposed here.
Reference: Wannika Sawangtong and Panumart Sawangtong
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