论文标题
托马斯 - 费尔米边界价值问题的新方法
A new approach to the Thomas-Fermi boundary-value problem
论文作者
论文摘要
给定Thomas-Fermi方程SQRT（x）phi'= phi*（3/2），本文首先通过定义y（x）= sqrt（x phi（x））更改因变量。边界条件要求y（x）必须以sqrt（x）的形式消失，而它的行为与独立变量x的幂（1/2）（1-chi）的无穷大（1/2）（1-chi）的行为降低，chi为正数。这样的边界条件导致以SQRT（X）形式的1参数家族（X）乘以整数的有限线性组合和X的半odd幂的比例。如果将CHI设置为等于3，为了完全同意Sommerfeld的渐近解决方案，则可以为所有X的所有值获得近似解的明确形式。他们完全同意小X处的Majoraana解决方案，并且对于所有x的所有值都非常接近数值解决方案。值得注意的是，在不使用串联的情况下，我们的近似解决方案实现了从小X到大X行为的平稳过渡。最终，研究了包括相对论，非扩展和热效应的广义托马斯 - 弗米方程，在该方程中的物理参数的小或有限值中找到了在大小x处的近似溶液。
Given the Thomas-Fermi equation sqrt(x)phi''=phi*(3/2), this paper changes first the dependent variable by defining y(x)=sqrt(x phi(x)). The boundary conditions require that y(x) must vanish at the origin as sqrt(x), whereas it has a fall-off behaviour at infinity proportional to the power (1/2)(1-chi) of the independent variable x, chi being a positive number. Such boundary conditions lead to a 1-parameter family of approximate solutions in the form sqrt(x) times a ratio of finite linear combinations of integer and half-odd powers of x. If chi is set equal to 3, in order to agree exactly with the asymptotic solution of Sommerfeld, explicit forms of the approximate solution are obtained for all values of x. They agree exactly with the Majorana solution at small x, and remain very close to the numerical solution for all values of x. Remarkably, without making any use of series, our approximate solutions achieve a smooth transition from small-x to large-x behaviour. Eventually, the generalized Thomas-Fermi equation that includes relativistic, non-extensive and thermal effects is studied, finding approximate solutions at small and large x for small or finite values of the physical parameters in this equation.
