The Solvability of Mixed Value Problem for the First and Second Approximations of One-Dimensional Nonlinear System of Moment Equations with Microscopic Boundary Conditions

The paper gives a derivation of a new one-dimensional non-stationary nonlinear system of moment equations, that depend on the flight velocity and the surface temperature of an aircraft. Maxwell microscopic condition is approximated for the distribution function on moving boundary, when one fraction of molecules reflected from the surface specular and another fraction diffusely with Maxwell distribution. Moreover, macroscopic boundary conditions for the moment system of equations depend on evenness or oddness of approximation $${f}_{k}(t,x,c)$$ f k ( t , x , c ) , where $${f}_{k}(t,x,c)$$ f k ( t , x , c ) is partial expansion sum of the molecules distribution function over eigenfunctions of linearized collision operator around local Maxwell distribution. The formulation of initial and boundary value problem for the system of moment equations in the first and second approximations is described. Existence and uniqueness of the solution for the above-mentioned problem using macroscopic boundary conditions in the space of functions $$C\left(\left[0,T\right];{L}^{2}\left[-a,a\right]\right)$$ C 0 , T ; L 2 - a , a are proved.

Equivalence Conditions and Invariants for the General Form of Burgers’ Equations

Equivalence of differential equations is one of the most important concepts in the theory of differential equations. In this paper, the moving coframe method is applied to solve the local equivalence problem for the general form of Burgers’ equation, which has two independent variables under action of a pseudo-group of contact transformations. Using this method, we found the structure equations and invariants of these equations, as a result some conditions for equivalence of them will be given.

A Different View on Dynamics of Space Curves Geometry

In this study, we define the X-torque curves, $$X-$$ X - equilibrium curves, X-moment conservative curves, $$X-$$ X - gyroscopic curves as new curves derived from a regular space curve by using the Frenet vectors of a space curve and its position vector, where $$X\in \left\{ T\left( s\right) , N\left( s\right) , B\left( s\right) \right\} $$ X ∈ T s , N s , B s and we examine these curves and we give their properties.

A Note on Function Space and Boundedness of the General Fractional Integral in Continuous Time Random Walk

The general fractional calculus becomes popular in continuous time random walk recently. However, the boundedness condition of the general fractional integral is one of the fundamental problems. It wasn’t given yet. In this short communication, the classical norm space is used, and a general boundedness theorem is presented. Finally, various long–tailed waiting time probability density functions are suggested in continuous time random walk since the general fractional integral is well defined.

Almost Cosymplectic $$(k,\mu )$$-metrics as $$\eta$$-Ricci Solitons

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k<0$$ k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k<0$$ k < 0 . In addition, a concrete example is constructed to illustrate the above result.

Global Existence and Blow-Up for the Pseudo-parabolic p(x)-Laplacian Equation with Logarithmic Nonlinearity

In this paper, we study the initial boundary value problem of the pseudo-parabolic p(x)-Laplacian equation with logarithmic nonlinearity. The existence of the global solution is obtained by using the potential well method and the logarithmic inequality. In addition, the sufficient conditions of the blow-up are obtained by concavity method.

A New Application of the $${\bar{\partial }}$$-Method

By constructing a new calculating rule of Lie bracket, we construct a new nonlinear Schrödinger hierarchy and its reduction equations via using the $${\bar{\partial }}$$ ∂ ¯ -method. Furthermore, some soliton solutions of such the equation are obtained by making use of Dirac function.

KdV Equation Model in Open Cylindrical Channel under Precession

A new model for Korteweg and de-Vries equation (KdV) is derived. The system under study is an open channel consisting of two concentric cylinders, rotating about their vertical axis, which is tilted by slope $$\tau$$ τ from the inertial vertical $$z$$ z , in uniform rate $${\Omega }_{1}=\tau \Omega$$ Ω 1 = τ Ω , and the whole tank is elevated over other table rotating at rate $$\Omega$$ Ω . Under these conditions, a set of Kelvin waves is formed on the free surface depending on the angle of tilt, characterized by the slope $$\tau$$ τ , volume of water, and rotation rate. The resonant mode in the system appears in the form of a single Kelvin solitary wave, whose amplitude satisfies the Korteweg-de Vries equation with forced term. The equation was derived following classical perturbation methods, the additional term made the equation a non-integrable one, that cannot be solved without the help of numerical methods. Invoking the simple finite difference scheme method, it was found that the numerical results are in a good agreement with the experiment.

Dirac–Witten Operators and the Kastler–Kalau–Walze Type Theorem for Manifolds with Boundary

In this paper, we obtain two Lichnerowicz type formulas for the Dirac–Witten operators. And we give the proof of Kastler–Kalau–Walze type theorems for the Dirac–Witten operators on 4-dimensional and 6-dimensional compact manifolds with (resp. without) boundary.