In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solved. The compatible pairs of nonlocal Poisson brackets of hydrodynamic type ha...

In the theory of differential equations, the basic concepts have been formulated studying the problems of classical mathematical physics. However, the modern problems motivate to formulate and investigate the new ones, for example, a class of nonlocal problems. Nonlocal conditions arise when we cannot measure data directly at the boundary. In this case, the problem is formulated, where the valu...

Free vibration characteristics of polymer composite plates reinforced by graphene nanosheets employing the Eringen nonlocal elasticity theory were investigated. Theoretical formulations are derived based on Hamilton’s principle implementing linear orthotropic constitutive equations of lamina while the behavior of nanostructure points affected by all other nonlocal points is also taken into acco...

The nonlocal nonlinear buckling of a double layer graphene sheet (DLGS) covered by zinc oxide (ZnO) piezoelectric layers is investigated in this study. The surrounding circumstances of the system are considered as a Pasternak foundation including spring constants and a shear layer. Graphene sheets are subjected to longitudinal magnetic field and biaxial forces. On the other hand, the ZnO piezoe...

In this article, the nonlocal biaxial buckling load and bending analysis of polymeric piezoelectric nanoplate reinforced by carbon nanotube (CNT) considering the surface stress effect is presented. This plate is subjected to electro-magneto-mechanical loadings. Eshelby-Mori-Tanaka approach is used for defining the piezoelectric nanoplate material properties. Navier’s type solution is employed t...

The damping vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam resting on viscoelastic foundation based on nonlocal strain gradient elasticity theory (NSGT) is studied in this article. For this purpose, by considering the effects of Winkler-Pasternak, the viscoelastic medium consists of linear and viscous layers. with respect to the displacement field in accordance with th...

We apply nonlocal symmetry-like operators to systems of two first and two second-order ordinary differential equations to seek reduction to quadratures. The reduction of order of such systems is carried out with the help of analytic continuation of scalar equations in the complex plane. Examples are taken from the literature. Precisely it is shown how the reduction to quadratures of a system of...

Complete dynamical PDE systems of one-dimensional nonlinear elasticity satisfying the principle of material frame indifference are derived in Eulerian and Lagrangian formulations. These systems are considered within the framework of equivalent nonlocally related PDE systems. Consequently, a direct relation between the Euler and Lagrange systems is obtained. Moreover, other equivalent PDE system...

We establish sharp boundary regularity estimates in C and C domains for nonlocal problems of the form Lu = f in Ω, u = 0 in Ω. Here, L is a nonlocal elliptic operator of order 2s, with s ∈ (0, 1). First, in C domains we show that all solutions u are C up to the boundary and that u/d ∈ C(Ω), where d is the distance to ∂Ω. In C domains, solutions are in general not comparable to d, and we prove a...