On the fractional abstract Schrodinger type evolution equations on the Hilbert space and its applications to the fractional dispersive equations

In this paper we prove the local and global well-posedness of the time fractional abstract Schr\"odinger type evolution equation on the Hilbert space and as an application, we prove the local and global well-posedness of the fractional dispersive equation with static potential under the only assumption that the symbol of P(D) behaves like a polynomial of highest degree m at infinity. In appendix, we also give the Holder regularities and the asymptotic behaviors of the mild solution to the linear time fractional abstract Schr\"odinger type equation. Because of the lack of the semigroup properties of the solution operators, we employ a strategy of proof based on the spectral theorem of the self-adjoint operators and the asymptotic behaviors of the Mittag-Leffler functions.Comment: 44 pages, 0 figure

Space-time fractional reaction-diffusion equations associated with a generalized Riemann-Liouville fractional derivative

This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann-Liouville fractional derivative defined in Hilfer et al. , and the space derivative of second order by the Riesz-Feller fractional derivative, and adding a function $\phi(x,t)$. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag-Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al., and the result very recently given by Tomovski et al.. At the end, extensions of the derived results, associated with a finite number of Riesz-Feller space fractional derivatives, are also investigated.Comment: 15 pages, LaTe

Application of the generalized Kudryashov method to the Eckhaus equation

In this paper, the generalized Kudryashov method is presented to seek exact solutions of the Eckhaus equation. From these solutions, we can derive solitary wave solutions as a special case. The proposed method is direct, effective and convenient and can be applied to many nonlinear evolution equations in mathematical physics

Coupled FCT-HP for Analytical Solutions of the Generalized Timefractional Newell-Whitehead-Segel Equation

This paper considers the generalized form of the time-fractional Newell-Whitehead-Segel model (TFNWSM) with regard to exact solutions via the application of Fractional Complex Transform (FCT) coupled with He’s polynomials method of solution. This is applied to two forms of the TFNWSM viz: nonlinear and linear forms of the time-fractional NWSM equation whose derivatives are based on Jumarie’s sense. The results guarantee the reliability and efficiency of the proposed method with less computation time while still maintaining high level of accuracy

Local Fractional Operator for the Solution of Quadratic Riccati Differential Equation with Constant Coefficients

In this paper, we consider approximate solutions of fractional Riccati differential equations via the application of local fractional operator in the sense of Caputo derivative. The proposed semi-analytical technique is built on the basis of the standard Differential Transform Method (DTM). Some illustrative examples are given to demonstrate the effectiveness and robustness of the proposed technique; the approximate solutions are provided in the form of convergent series. This shows that the solution technique is very efficient, and reliable; as it does not require much computational work, even without given up accuracy

On a Modified Iterative Method for the Solutions of Advection Model

Variational Iterative Method (VIM) has been reported in literature as a powerful semi-analytical method for solving linear and nonlinear differential equations; however, it has also been shown to have some weaknesses such as calculation of unneeded terms, and time-consumption regarding repeated calculations for series solution. In this work, a modified VIM is applied for approximate-analytical solution of homogeneous advection model. The result attest to the robustness and efficiency of the proposed method (MVIM)

On Existence and Uniqueness of Solutions for Semilinear Fractional Wave Equations

International audienceLet Ω be a C 2-bounded domain of R d , d = 2, 3, and fix Q = (0, T) × Ω with T ∈ (0, +∞]. In the present paper we consider a Dirichlet initial-boundary value problem associated to the semilinear fractional wave equation ∂ α t u + Au = f b (u) in Q where 1 1. We first provide a definition of local weak solutions of this problem by applying some properties of the associated linear equation ∂ α t u + Au = f (t, x) in Q. Then, we prove existence of local solutions of the semilinear fractional wave equation for some suitable values of b > 1. Moreover, we obtain an explicit dependence of the time of existence of solutions with respect to the initial data that allows longer time of existence for small initial data