講義概要

現実の問題では，微分方程式の厳密解を求めることが困難な場合が多い。一方，純粋な数値計算では答えが個別的であり，全体の傾向をつかみにくい。そこで，本講義では，まず「近似的な解析解」を求める手法として重み付き残差法の一つであるGalerkin法を中心に講述する。また，Fourier級数を始めとする直交関数展開，統計解析として教えられている最少自乗法や主成分分析，数値計算手法の一つである有限要素法の数式構造が基本的に同じであることを示し，応用数学の理解を深める。なお，本講義は若干の演習も含んでいる。

We must solve many kinds of differential equations to understand and predict natural phenomena in the environment. Under the condition of practical problems in the environment, however, it is often difficult to obtain a strict solution of differential equations in an explicit form. On the other hand, purely numerical approach, which always gives individual solution under a specified condition, is not convenient to survey the total picture of the system’s response. In this lecture, the weighted residual method (WRM) and its derivatives are introduced to give an idea “how to obtain an approximate analytical solution” of given differential equation under practical conditions. It is also described that WRM is a general form of mathematics to understand connectedly a variety of mathematical techniques through the idea of orthogonality.