A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

摘要

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=xa, x'(a)=x'a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies ek=0(h2). If e(t) is the error function on a⩽t⩽b; it is also shown (in a variety of norms) that e(t)⩽Ch2 and e'(t)⩽C1h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that eRk=0(h4). Thus our results are comparable to Runge-Kutta with half the function evaluations.