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This paper discusses how to derive multidimensional conformable diffusion equation from continuous time random walks process by employing fractional Laplace transform and the conformable fractional derivative defined by Khalil et al in [10]. The results show that there exists intimate relationship between the stretched exponential function and conformable diffusion equation involving the conformable fractional derivative. The relationship is analogous to the relationship between the exponential function and usual diffusion equation involving usual derivative and the relationship between the Mittag-Leffler function and subdiffusion or slow diffusion equation involving Caputo fractional derivative. Multidimensional conformable FokkerPlanck equation is also derived here. The equation describes diffusion process influenced by external force fields. Conformable semigroup associated with the solution to the conformable diffusion equation has similar properties to the properties of the semigroup associated with the solution to usual diffusion except the semigroup property. For sufficiently large time, the graphs of the solution to the conformable diffusion equation and the subdiffusion equation involving Caputo fractional derivative are sufficiently close. Especially, for, the graphs are sufficiently close for . In presence of nonzero constant external force fields, the particle involved in the diffusion process moves faster toward the directions of the external force fields. [ABSTRACT FROM AUTHOR]

We derive nonlocal reaction-diffusion system with subdiffusive kinetics from random walks using a probability measure on a n-multidimensional unit ball S n−1 . The system describes two particles moving with subdiffusive kinetics and then undergoing a chemical reaction which occurs if and only if a pair of the particles are in a distance less than R. We also study the existence and uniqueness of a mild solution to a fractional nonlinear Cauchy problem associated with the system by applying Banach’s Fixed Point Theorem. The result shows that the mild solution to the problem exists uniquely under some Lipschitz continuity on the nonlinear part of the problem. Consequently, the system also has a unique mild solution since the reaction term of the system satisfies the Lipschitz continuity. [ABSTRACT FROM AUTHOR]

In this article, we study the existence and uniqueness of a local mild solution for a class of semilinear differential equations involving the Caputo fractional time derivative of order α (0 < α < 1) and, in the linear part, a sectorial linear operator A. We put some conditions on a nonlinear term ƒ and an initial data u0 in terms of the fractional power of A. By applying Banach's Fixed Point Theorem, we obtain a unique local mild solution with smoothing effects, estimates, and a behavior at t close to 0. An example as an application of our results is also given. [ABSTRACT FROM AUTHOR]