BY FINITE DIFFERENCES OF PHYSICAL PROBLEMS, ETC. 



313 



writing algebraic symbols for the unknown values of 8"^ and 8" *< and finding them 

 from S S"" 1 ^ = S"<, and from the given difference equation simultaneously. On the 

 other hand if any of the first (n 1) differential coefficients are unknown at x u the 

 summation from column to column across the table cannot be carried out, and so the 

 step method is impossible. The conditions for stepwise integration of partial equations 

 are not so simple and require investigation. 



2 '2. As an example of stepwise integration, let us take the equation for diffusion 

 in a tube, o-3 2 </3, 2 = 3</3. In the first place, if <f> = F (x, t) is a solution of 

 3 2 (/3.x 2 = 3(/>/3<, then <f> = F (ax, aV<) is a solution of the given equation. So we 

 need only concern ourselves with 3 2 </3. a = d<f)/dt, and the results will apply to bodies 

 of any linear dimensions and any uniform diffusivity. Let us suppose the boundary 

 conditions are : (f> = when x = |- for all values of t ; <j> = 1 for all values of x when 

 t = ; in fact the familiar case of a uniformly heated slab, the faces of which are 

 suddenly cooled. The method of this example is so simple that it can hardly be 

 novel. It is introduced to show how easy it sometimes is to obtain approximate 

 integrals by arithmetic of equations usually treated by complex analysis. We draw 

 up a table with a row for each O'l of x and a column at every O'OOl of t. (The reason 

 for making the time step small will appear later, 3'2'1.) The given boundary values 

 are next inserted. 



TABLE I. 



In satisfying the equation we must be careful to equate values of S^/ftr 2 and 

 ; 8<f>l ; 8t, which are centered at the same point. This causes a little difficulty at starting. 

 When t = O'OOl let the values of < be a, b, c, d, e, as indicated in Table I. Then if 

 the difference equation be satisfied at t = O'OOOS, it takes the form of 5 simultaneous 

 equations involving a, b, c, d, e. Solving these equations, we find the numbers given 

 in the column = O'OOl. Having got over this rather troublesome first step, we can 

 find the rest much more simply by centering all differences on the columns t O'OOl, 

 0'002, 0'003, &c., and deducing each number from the two preceding columns. The 

 errors resulting from the above process may be found by comparison with the Fourier 



m 1 -I 



solution <f> = 2 ( 1) 2 - e~ mvt cos (mirx]. This series has been computed when 



m odd 



t = 0'005, and the numbers so found are given in the table. It is seen that the step 



VOL. CCX. A. 2 S 



