An Improvement of Takahasi's Method finding the Graphical Solution of Heat Conduction

Abstract

Recently Y. Takahasi has given a very convenient method solving graphically the problems of heat conduction. If u(x, t) is the temperature distribution at time t, then the temperature at t+τ is given by if we assume that τ is very small and neglect τ_??_. This can be easily verified by the aid of Taylor expansion and of the differential equation of heat conduction, i.e. Takabasi's method consists in evaluating the integral in (1) by the use of a planimeter. It is desirable, if possible, to replace the use of a planimeter by some simple constructions. It
can be realized as shown in the following.
We start, in place of (1), from which can be verified in similar way. Taking k=√6 and applying the Simpson's 1/3-rule, we have The operation on the right hand is easily parformable on the graph. Namely, the point of symmetry, of the point representing u(x, t) on the xu-plane, with respect to the middle point of u(x-b, t) and u(x+b, t) gives u(x, t+τ) (Fig. 1).