2 Young’s integrals and stochastic differential equations driven by fractional Brownian motions 4 2.1 Young’s integral and basic estimates . . . . . . . . . . . . . . . . . . 4 2.2 Stochastic differential equations driven by a Hölder path . . . . . . . 7 2.3 Multidimensional extension . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Fractional calculus . . . . . . . . . . . . . . . . . . . ...

Fractional calculus, which has almost the same history as classic calculus, did not attract enough attention for a long time. However, in recent decades, fractional calculus and fractional differential equations become more and more popular because of its powerful potential applications. A large number of new differential equations (models) that involve fractional calculus are developed. These ...

We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained.This approach can be suit...

Regular and singular perturbations of fractional ordinary differential equations (fODEs) are considered. This is likely the first attempt to describe these problems. Similarities and differences between these cases and the analogous ones for classical (integer-order) differential equations are pointed out. Examples, including the celebrated Bagley-Torvik equations are discussed. Asymptotic-nume...

In this paper, a new fractional sub-equation method is proposed for finding exact solutions of fractional partial differential equations (FPDEs) in the sense of modified Riemann-Liouville derivative. With the aid of symbolic computation, we choose the space-time fractional Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation in mathematical physics with a source to illustrate the validity a...

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differential equations of arbitrary positive real order by using the fractional B-Splines wavelets and theMittagLeffler function. The differential operators are taken in the Riemann-Liouville sense and the initial values are zeros. The scheme of solving the fractional differential equations and the explicit ex...