No abstract

This paper is devoted to the study of the initial value problem of nonlinear fractional differential equations involving a Caputo-type fractional derivative with respect to another function. Existence and uniqueness results for the problem are established by means of the some standard fixed point theorems. Next, we develop the Picard iteration method for solving numerically the problem and obtain results on the long-term behavior of solutions. Finally, we analyze a population growth model and a gross domestic product model with governing equations being fractional differential equations that we have introduced in this work.

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We introduce a fractional theory of the calculus of variations for multiple integrals. Our approach uses the recent notions of Riemann-Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler-Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.