The existence of the uniform global attractor for a second order non-autonomous lattice dynamical system (LDS) with almost periodic symbols has been carefully studied. Considering the nonlinear operators f 1i. u j | j ∈ I iq 1 i∈Z n and (f 2i (u j | j ∈ I iq 2)) i∈Z n of this LDS, up to our knowledge it is the first time to investigate the existence of uniform global attractors for such second order LDSs. In fact there are some previous studies for first order autonomous and non-autonomous LDSs with similar nonlinear parts, cf. [3, 24]. Moreover, the LDS under consideration covers a wide range of second order LDSs. In fact, for specific choices of the nonlinear functions f 1i and f 2i we get the autonomous and non-autonomous second order systems given by [1, 25, 26].

In a weighted Hilbert space, the existence of a uniform global attractor for a family of first order non-autonomous deterministic lattice dynamical systems with a nonlinear part of the form [Formula: see text] is carefully investigated, where [Formula: see text] is an element of the hull of an almost periodic function in a suitable Banach space.

We propose an asymmetric model position dependent mass and study its quantum mechanical behaviour on different potentials such as harmonic oscillator potential, double well potential, Gaussian single well potential and triangular single well model potential. It is observed from our study that the model asymmetric mass works well for weak coupling preserving the symmetric phase portrait. However, the dominance of asymmetric feature of the mass in the system clearly visible for higher values of the constant associated with the mass. Though, both position dependent mass and potential have significant role in controlling the spectral feature of the system, one may dominate over other for certain cases.

We propose nonlinear model T-symmetry operators having quartic, sextic, octic anharmonicity and inverse quadratics under real spectra. In fact, the model operator is non-Hermitian but real in nature. A comparison with the corresponding hermitian counterpart shows higher energy levels (ET ≫ Ehermitian).

<abstract><p>Heydari-Hosseininia (HH) fractional derivative is a newly introduced concept of fractional calculus which conquers the restrictions of non-singular fractional derivatives in the Caputo–Fabrizio (CF) and Atangana–Baleanu senses. For instance, it is not easy to get the closed-form of the fractional derivative of functions using CF because of the construction of its kernel function. In this paper, we present a powerful numerical scheme based on energy boundary functions to get the approximate solutions of the time-fractional inverse Burger equation containing HH-derivative: $ ^{HH}\mathcal{D}_{\tau}^{\alpha}h(z, \tau)-h(z, \tau)h_z(z, \tau) = h_{zz}(z, \tau)+H(z, \tau) $, which $ ^{HH}\mathcal{D}_{\tau}^{\alpha} $ is the HH-derivative with regard to $ {\alpha} $-order. This problem has never been investigated earlier so, this is our motivation to work on this important problem. Some numerical examples are presented to verify the efficiency of the presented technique. Graphs of the exact and numerical solutions along with the plot of absolute error are provided for each example. Tables are given to see and compare the results point by point for each example.</p></abstract>