The Cauchy Problem for the Heat Equation
Part I is devoted to the consideration of a Cauchy-like problem for the heat equation. Let $u(x,t)$ satisfy the heat equation in $D = \{ (x,t):0 < x < s(t),0 < t \leqq T\} $ and let $u(x,0) = \varphi (x),u(s(t),t) = f(t)$, and $u_x (s(t),t) = g(t)$. An a ...
Spline Function Approximations for Solutions of Ordinary Differential Equations
A procedure for obtaining spline function approximations for solutions of the initial value problem in ordinary differential equations is presented. The proposed method with quadratic and cubic splines is shown to be related to the well-known trapezoidal ...
Laguere’s Method and a Circle which Contains at Least One Zero of a Polynomial
Given an Nth degree polynomial $P(z)$ one may use Laguerre’s method to generate a sequence of complex numbers $x_0 ,x_1 ,x_2 , \ldots $ which usually converges to a, zero of $P(z)$. This note shows that each circle $\left| {z - x_n } \right| \leqq \sqrt N ...
