Introduction to General Nonlocal

To reach this aim, third-order shear deformable model in Cartesian coordinate combined with a general nonlocal strain gradient theory is adopted.

The general nonlocal reduction of the coupled nonlocal NLS equation to the nonlocal NLS equation is discussed in detail.

The general nonlocal strain gradient theory including nonlocality and strain gradient characteristics of size-dependency in order is used to examine the small-scale effects.

Thus, in this effort, local and global sensitivity analysis and uncertainty quantification techniques are applied to determine the most influential parameters on the acoustic dispersion curves and the linear response of nanobeams which are modeled utilizing Euler-Bernoulli beam theory and the general nonlocal theory.

A higher-order refined shear deformation shell theory enriched by general nonlocal strain gradient elasticity accounting both nonlocality as well as strain gradient size-dependency are developed through assumptions of Hamiltonian principle, and afterward, a harmonic solution producer is handled with equations of motion is used to find the responses.

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time.

The paper develops some recent results in [1] on fractional differential equations to the case of more general nonlocal derivatives.

Furthermore, a general nonlocal strain gradient theory is employed in order to catch up with both phenomena of small-scale behaves.

We establish a new sub–supersolution method for general nonlocal elliptic equations and, consequently, we obtain the existence of positive solutions of a nonlocal logistic equation.

A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived.

Many difficulties are caused by general nonlocal operators, we develop new techniques to overcome them to construct the first nontrivial curve of Dancer–Fucik point spectrum by minimax methods, to show some qualitative properties of the curve, and to prove that the corresponding eigenfunctions are foliated Schwartz symmetric.

In this paper, we are concerned with the following general nonlocal problem\begin{equation*}\begin{cases}-\mathcal{L}_K u=\lambda_1u+f(x,u)& \text{in}\ \Omega,\\u=0& \text{in}\ \mathbb{R}^N\backslash\Omega,\end{cases}\end{equation*}where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary.

We apply this technique to a higher-order linear ODE involving general nonlocal conditions.

In the context of this model, the interatomic interactions are modeled based on the general nonlocal theory.