An exact solution of a molecular field equation in the theory of ferromagnetism

The theory of complex variables is used to develop an exact closed-form solution of a transcendental equation basic to the molecular field theory of ferromagnetism. The analysis yields analytical expressions, in terms of elementary quadratures, for the reduced magnetization as it depends on the temperature and magnetic field.

---

Zusammenfassung Im Rahmen der Theorie komplexer Variabler wird für eine, der Molekularfeldtheorie des Ferromagnetismus' zugrunde liegenden transzendenten Gleichung, eine exakte Lösung in geschlossener Form entwickelt. Die Rechnung liefert analytische Ausdrücke, in Form elementarer Quadraturen, welche die reduzierte Magnetisierung als Funktion der Temperatur und des Magnetfeldes beschreiben.

     

Approximation and solution of a general symmetric functional equation

In this paper, we investigate the Hyers–Ulam–Rassias stability of a general equation f(_φ1(x,y))=_φ2(f(x),f(y))$f\left({\phi }_1(x,y)\right)={\phi }_2(f\left(x\right),f\left(y\right))$ in metric spaces. As a consequence, we obtain some stability results in the sense of Hyers–Ulam–Rassias.     

On the numerical solution of the spherically symmetric diffusion equation

Summary A new stability proof for a difference approximation to the spherically symmetric diffusion equation is given.     

The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation

In this paper we establish a blow up rate of the large positive solutions of the singular boundary value problem -Δu=λu-b(x)u^p,u_|∂Ω=+∞$-\mathrm{\Delta }u=\lambda u-b(x)u^p,u{|}_{\mathrm{\partial }\Omega }=+\infty$ with a ball domain and radially function b(x)$b(x)$. All previous results in the literature assumed the decay rate of b(x)$b(x)$ to be approximated by a distance function near the boundary ∂Ω$\mathrm{\partial }\Omega$. Obtaining the accurate blow up rate of solutions for general b(x)$b(x)$ requires more subtle mathematical analysis of the problem.     

Barenblatt solution and spherically symmetric solutions of the porous media equation

We consider the Cauchy problem of the porous media equation. We show that it is spherically symmetric solution has the same property as Barenblatt solution, with respct to some regularity property.     

分子束外延方程的时间周期解

利用Galerkin方法和Leray-Schauder不动点定理,研究了一类描述分子束表面增长模型的广义时间周期解,以及时间周期古典解的存在唯一性.     

A non-radially symmetric solution to a class of elliptic equation with Kirchhoff term

We consider the following equation with Kirchhoff term $-(a+b\int_{\mathbb{R}^3} {|\nabla u|^2} dx)$ $\Delta u + u =|u|^{p-2}u$, $u \in H^1 (\mathbb{R}^3)$, where $a, b$ are positive constants and $2 < p < 6$. By deducing a variant variational identity and a constraint set, we are able to prove the existence of a non-radially symmetric solution $u(x_1, x_2, x_3)$ for the full range of $p\in (2,6)$. Moreover this solution $u(x_1, x_2, x_3)$ is radially symmetric with respect to $(x_1,x_2)$ and odd with respect to $x_3$.     

A self-similar solution to the equation of gas filtration in a spherically symmetric porous medium

We consider the Cauchy problem for the Boussinesq equation which describes filtration of a gas in a spherically symmetric porous medium. For the self-similar solution to this problem we construct a formal in the neighborhood of the point r → ∞ expansion and a convergent near r = 0 one.     

Existence of solution for mean field equation for the equilibrium turbulence

This paper is concerned with the mean field equation of the equilibrium turbulence on a closed Riemannian surface. The existences of solution are shown in the critical and supercritical case.     

Asymptotic behavior of the solution to the non-isothermal phase field equation

We consider the Penrose–Fife phase field model [Thermodynamically consistent models of phase-field type for the kinetics of phase transitions, Physica D 43 (1990) 44–62] with homogeneous Neumann boundary condition to the nonlinear heat flux q=∇(1/θ)$\mathbf{q}=\nabla (1/\theta )$, i.e., q=0$\mathbf{q}=0$ on the boundary, where θ>0$\theta >0$ is the temperature. There is a unique H¹$H^1$ solution globally in time with the non-empty, connected, compact ω$\omega$-limit set composed of stationary solutions, and the linearized stable stationary solution is dynamically stable.     

An iterative method for the least squares symmetric solution of matrix equation AXB = C

This paper an iterative method is presented to solve the minimum Frobenius norm residual problem: with unknown symmetric matrix . By the iterative method, for any initial symmetric matrix , a solution can be obtained within finite iteration steps in the absence of roundoff errors, and the solution with least Frobenius norm can be obtained by choosing a special kind of initial symmetric matrix. In addition, in the solution set of the minimum Frobenius norm residual problem, the unique optimal approximation solution to a given matrix in Frobenius norm can be expressed as , where is the least norm symmetric solution of the new minimum residual problem: with . Given numerical examples are show that the iterative method is quite efficient.Research supported by Scientific Research Fund of Hunan Provincial Education Department of China (05C797), by China Postdoctoral Science Foundation (2004035645) and by National Natural Science Foundation of China (10571047).     

Central symmetric solution to the Neumann problem for a time-fractional diffusion-wave equation in a sphere

In this paper, a time-fractional central symmetric diffusion-wave equation is investigated in a sphere. Two types of Neumann boundary condition are considered: the mathematical condition with the prescribed boundary value of the normal derivative and the physical condition with the prescribed boundary value of the matter flux. Several examples of problems are solved using the Laplace integral transform with respect to time and the finite sin-Fourier transform of the special type with respect to the spatial coordinate. Numerical results are illustrated graphically.     

Spherically symmetric solutions of a reaction-diffusion equation

Two theorems are proved for the spherically symmetric solutions of the “bistable” reaction-diffusion equation $\text{u}{}_{\mathrm{t}}\text{= Δ}{}_{\mathrm{x}}\text{u + ?(u)}$, where ? is cubic-like and x ∈ Rⁿ. The first theorem says that, for a suitable class of initial data, there are only two types of asymptotic behavior, u(x, t) tends to an equilibrium solution as t → + ∞ or u(x, t) → 1 uniformly on compact sets. The second theorem says that in the latter case, if the solution is followed out along any ray, it approaches, in shape, the one-dimensional travelling wave.     

Support theorem for the fundamental solution to the Schrödinger equation on certain compact symmetric spaces

In this paper we construct the fundamental solution to the Schödinger equation on a compact symmetric space with even root multiplicities using shift operators of Heckman and Opdam. Next, we prove that the support of the fundamental solution becomes a lower dimensional subset at a rational time whereas its support and its singular support coincide with the whole symmetric space at an irrational time. Moreover, we also show that generalized Gauss sums appear in the expression of the fundamental solution.     

Global stability of a rational symmetric difference equation

Motivated by ideas from papers: C. Cinar, I. Yalçinkaya, S. Stević, A note on global asymptotic stability of a family of rational equations, Rostock. Math. Kolloq. 59 (2004), 41–49, and S. Stević, Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl. 316 (2006), 60–68, here we confirm a conjecture on a rational symmetric difference equation from the paper: K. Berenhaut, J. Foley, S. Stević, The global attractivity of the rational difference equation , Appl. Math. Lett. 20 (2007), 54–58.