this paper proposed analytical solutions for the buckling analysis of rectangular single-layered graphene sheets under in-plane loading on all edges simply is supported. the characteristic equations of the graphene sheets are derived and the analysis formula is based on the nonlocal mindlin plate. this theory is considering both the small length scale effects and transverse shear deformation ef...

Torsional dynamic analysis of carbon nanotubes under the effect of longitudinal magnetic field is carried out in the present study. Torque effect of an axial magnetic field on a carbon nanotube has been defined using Maxwell’s relation. Nonlocal governing equation and boundary conditions for carbon nanotubes are obtained by using Hamilton’s minimum energy principle. Eringen’s nonlocal stress gr...

Weighted inequality theory for fractional integrals is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Basic weighted inequalities are often associated Hardy, Littlewood and Sobolev [6, 11], Caffarelli, Kohn Nirenberg [4], respectively Stein Weiss [12]. A key attempt in the present paper prove Stein–Weiss with lack symmetr...

which was discovered in a symmetry classification of nonlocal PDEs with quadratic or cubic nonlinearity. By using the perturbation symmetry approach [7], Novikov found the first few symmetries and a scalar Lax pair for Eq. (1), then proved that it is integrable [9]. Hone and Wang [5] gave a matrix Lax pair for the Novikov equation and found its infinitely many conserved quantities, as well as a...

We show non-existence of solutions of the Cauchy problem in R for the nonlinear parabolic equation involving fractional diffusion ∂tu + (-∆) s φ(u) = 0, with 0 < s < 1 and very singular nonlinearities φ . More precisely, we prove that when φ(u) = −1/u with n > 0, or φ(u) = log u, and we take nonnegative L initial data, there is no (nonnegative) solution of the problem in any dimension N ≥ 2. We...

Localized wrinkles of thin premixed flames subject to hydrodynamic instability and geometrical stretch of uniform intensity (S) are studied. A stretch-affected nonlinear and nonlocal equation, derived from an inhomogeneous Michelson-Sivashinsky equation, is used as a starting point, and pole decompositions are used as a tool. Analytical and numerical descriptions of isolated (centered or multic...

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation ut+(−∂ x) u+uux = 0 with α ∈ (0, 1], supplemented with an initial datum approaching the constant states u± (u− < u+) as x → ±∞, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536–1549) that, for α ∈ (1, 2), the large time asymptotics of solutions is described by...

We discuss the role of nonlocal interactions in determining the mechanisms of pattern formation and selection in a nonlinear system formed by a Liquid Crystal Light Valve inserted in an optical feedback loop. We present experimental results, and compare them with the predictions of a linear stability analysis of the equations governing the system. Nonlocality is introduced in the system via a l...