We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian with . Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for the corresponding...
This paper is mainly concerned with existence of mild solutions and optimal controls for nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces. We do not assume that the relevant strongly continuous semigroup is compact.
In this paper, we discuss some generalized stability of solutions to a class of nonlinear impulsive evolution equations in the certain piecewise essentially bounded functions space. Firstly, stabilization of solutions to nonlinear impulsive evolution equations are studied by means of fixed point methods at an appropriate decay rate. Secondly, stable manifolds for the associated singular perturbation problems with impulses are compared with each other. Finally, an example on initial boundary value...
Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional...
Protein function and dynamics are closely related to its sequence and structure.However, prediction of protein function and dynamics from its sequence and structure is still a fundamental challenge in molecular biology. Protein classification, which is typically done through measuring the similarity between proteins based on protein sequence or physical information, serves as a crucial step toward the understanding of protein function and dynamics. Persistent homology is a new branch of algebraic...
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