Analytic and Numerical Solutions of Time-Fractional Linear Schrödinger Equation

Fractional Schrödinger equation is a basic equation in fractional quantum mechanics. In this paper, we consider both analytic and numerical solutions of time-fractional linear Schrödinger Equations. This is done via a proposed semi-analytical method upon the modification of the classical Differential Transformation Method (DTM). Some illustrative examples are used; the results obtained converge faster to their exact forms. This shows that this modified version is very efficient, and reliable; as less computational work is involved, even without given up accuracy. Therefore, it is strongly recommended for both linear and nonlinear time-fractional partial differential equations (PDEs) with applications in other areas of applied sciences, management, and finance

On Some Rigidity Properties in PDEs

This thesis is dedicated to the study of three rigidity properties arising in different partial differential equations: (1) the backward uniqueness property of the heat equation in two-dimensional conical domains, (2) the weak and strong unique continuation principles for fractional Schrödinger equations with rough potentials and (3) the rigidity and non-rigidity of exactly stress-free configurations of a differential inclusion describing the cubic-to-orthorhombic phase transition in the geometrically linearized theory of elasticity

Some new properties and applications of a fractional Fourier transform

In this paper, we deal with the fractional Fourier transform in the form introduced a little while ago by the first named author and his coauthors. This transform is closely connected with the Fractional Calculus operators and has been already employed for solving of both the fractional diffusion equation and the fractional Schrödinger equation. In this paper, we continue the investigation of the fractional Fourier transform, and in particular prove some new operational relations for a linear combination of the left- and righthand sided fractional derivatives. As an application of the obtained results, we provide a schema for solving the fractional differential equations with both leftand right-hand sided fractional derivatives without and with delays and give some examples of realization of our method for several fractional differential equations

A conservative exponential integrators method for fractional conservative differential equations

The paper constructs a conservative Fourier pseudo-spectral scheme for some conservative fractional partial differential equations. The scheme is obtained by using the exponential time difference averaged vector field method to approximate the time direction and applying the Fourier pseudo-spectral method to discretize the fractional Laplacian operator so that the FFT technique can be used to reduce the computational complexity in long-time simulations. In addition, the developed scheme can be applied to solve fractional Hamiltonian differential equations because the scheme constructed is built upon the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated by solving the fractional Schrödinger equation

Nonlinear fractional magnetic Schr\"odinger equation: existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schr\"odinger equation with magnetic field \begin{equation*} \varepsilon^{2s}(-\Delta)_{A/\varepsilon}^{s}u+V(x)u=f(|u|^{2})u \quad \mbox{ in } \mathbb{R}^{N}, \end{equation*} where $\varepsilon>0$ is a parameter, $s\in (0, 1)$, $N\geq 3$, $(-\Delta)^{s}_{A}$ is the fractional magnetic Laplacian, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ and $A:\mathbb{R}^{N}\rightarrow \mathbb{R}^N$ are continuous potentials and $f:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick-Schnirelmann theory, we prove existence and multiplicity of solutions for $\varepsilon$ small.Comment: 23 page

Concentrating solutions for a fractional Kirchhoff equation with critical growth

In this paper we consider the following class of fractional Kirchhoff equations with critical growth: \begin{equation*} \left\{ \begin{array}{ll} \left(\varepsilon^{2s}a+\varepsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u \quad &\mbox{ in } \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \quad u>0 &\mbox{ in } \mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b>0$ are constants, $s\in (\frac{3}{4}, 1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the fractional critical exponent, $(-\Delta)^{s}$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{\varepsilon}$ which concentrates around a local minimum of $V$ as $\varepsilon\rightarrow 0$.Comment: arXiv admin note: text overlap with arXiv:1810.0456

Existence and concentration results for some fractional Schr\"odinger equations in RN\mathbb{R}^{N} with magnetic fields

We consider some nonlinear fractional Schr\"odinger equations with magnetic field and involving continuous nonlinearities having subcritical, critical or supercritical growth. Under a local condition on the potential, we use minimax methods to investigate the existence and concentration of nontrivial weak solutions.Comment: arXiv admin note: text overlap with arXiv:1807.0744