Vector-Valued Entire Functions of Several Variables: Some Local Properties

The present paper is devoted to the properties of entire vector-valued functions of bounded L-index in join variables, where L:Cn→R+n is a positive continuous function. For vector-valued functions from this class we prove some propositions describing their local properties. In particular, these functions possess the property that maximum of norm for some partial derivative at a skeleton of polydisc does not exceed norm of the derivative at the center of polydisc multiplied by some constant. The converse proposition is also true if the described inequality is satisfied for derivative in each variable.

Prediction of nonlinear dynamic coefficients of tilting-pad journal bearings with pad perturbation

In this paper, a modified partial derivative method is developed to predict the linear and nonlinear dynamic coefficients of tilting-pad journal bearings with journal and pad perturbation. To this end, Reynolds equation and its boundary conditions along with equilibrium equations of the pad are used. Finite difference, partial derivative method, and perturbation technique have been employed simultaneously for solving these equations. The accuracy of the results is investigated by comparing the linear dynamic coefficients of three types of tilting-pad journal bearings with those published the literature. It is shown that the nonlinear dynamic coefficients depend on Sommerfeld number, eccentricity ratio, and length to diameter ratio. Similar to the case of linear dynamic coefficients of TPJB, it is observed that the eccentricity ratio effects on nonlinear dynamic coefficients are more notable when the eccentricity ratio is higher than 0.8 or less than 0.2.

DERIVED TERMS WITHOUT DERIVATION A SHIFTED PERSPECTIVE ON THE DERIVED-TERM AUTOMATON

We present here a construction for the derived term automaton (aka partial derivative, or Antimirov, automaton) of a rational (or regular) expression based on a sole induction on the depth of the expression and without making reference to an operation of derivation of the expression. It is particularly well-suited to the case of weighted rational expressions.

Periodic solutions for nonresonant parabolic equations on $${\mathbb {R}}^N$$ with Kato–Rellich type potentials

A criterion for the existence of T-periodic solutions of nonautonomous parabolic equation $$u_t = \Delta u + V(x)u + f(t,x,u)$$ u t = Δ u + V ( x ) u + f ( t , x , u ) , $$x\in {\mathbb {R}}^N$$ x ∈ R N , $$t>0$$ t > 0 , where V is Kato–Rellich type potential and f diminishes at infinity, will be provided. It is proved that, under the nonresonance assumption, i.e. $${\mathrm {Ker}} (\Delta + V)=\{0\}$$ Ker ( Δ + V ) = { 0 } , the equation admits a T-periodic solution. Moreover, in case there is a trivial branch of solutions, i.e. $$f(t,x,0)=0$$ f ( t , x , 0 ) = 0 , there exists a nontrivial solution provided the total multiplicities of positive eigenvalues of $$\Delta +V$$ Δ + V and $$\Delta + V + f_0$$ Δ + V + f 0 , where $$f_0$$ f 0 is the partial derivative $$f_u(\cdot ,\cdot ,0)$$ f u ( · , · , 0 ) of f, are different mod 2.

Riemann–Hilbert Approach for Constructing Analytical Solutions and Conservation Laws of a Local Time-Fractional Nonlinear Schrödinger Type Equation

Fractal and fractional calculus have important theoretical and practical value. In this paper, analytical solutions, including the N-fractal-soliton solution with fractal characteristics in time and soliton characteristics in space as well as the long-time asymptotic solution of a local time-fractional nonlinear Schrödinger (NLS)-type equation, are obtained by extending the Riemann–Hilbert (RH) approach together with the symmetries of the associated spectral function, jump matrix, and solution of the related RH problem. In addition, infinitely many conservation laws determined by an expression, one end of which is the partial derivative of local fractional-order in time, and the other end is the partial derivative of integral order in space of the local time-fractional NLS-type equation are also obtained. Constraining the time variable to the Cantor set, the obtained one-fractal-soliton solution is simulated, which shows the solution possesses continuous and non-differentiable characteristics in the time direction but keeps the soliton continuous and differentiable in the space direction. The essence of the fractal-soliton feature is that the time and space variables are set into two different dimensions of 0.631 and 1, respectively. This is also a concrete example of the same object showing different geometric characteristics on two scales.

Laplace-SBA Method for Solving Nonlinear Coupled Burger's Equations

Burger’s equations, an extension of fluid dynamics equations, are typically solved by several numerical methods. In this article, the laplace-Somé Blaise Abbo method is used to solve nonlinear Burger equations. This method is based on the combination of the laplace transform and the SBA method. After reminders of the laplace transform, the basic principles of the SBA method are described. The process of calculating the Laplace-SBA algorithm for determining the exact solution of a linear or nonlinear partial derivative equation is shown. Thus, three examplesof PDE are solved by this method, which all lead to exact solutions. Our results suggest that this method can be extended to other more complex PDEs.

Global Regularity Criterion for the 3D Incompressible Navier–Stokes Equations Involving the Velocity Partial Derivative

In this paper, we study the regularity of the weak solutions for the incompressible 3D Navier–Stokes equations with the partial derivative of the velocity. By the embedded technology, we prove that the weak solution u is regular on (0, T] if ∂ 3 u ∈ L p 0 , T ; L q R 3 with 2 / p + 3 / q = 70 / 37 + 15 / 37 q , 15 / 4 ≤ q ≤ ∞ , or 2 / p + 3 / q = 34 / 19 + 9 / 19 q , 9 / 4 ≤ q ≤ ∞ .