These are, properly speaking, the coefficients of partial differences of the first order of uy But on the assumption that Ax and Ay are each equal to unity, an assumption which we can always legitimate, Chap. I. Art. 2, the above are the partial differences of the first order of u2y• On the same assumption the general form of a partial difference of u is When the form of u is given, this expression is to be interpreted by performing the successive operations indicated, each elementary operation being of the kind indicated in (1). in which Ax is an absolute constant, it follows that (3), A A to substitute for (A)TM (A)TM Ax' Ay their symbolical expressions, to effect their symbolical expansions by the binomial theorem, and then to perform the final operations on the subject function Uz.y Though in what follows each increment of an independent variable will be supposed equal to unity, it will still be A A necessary to retain the notation for the sake of disAx' Ay tinction, or to substitute some notation equivalent by definition, e. g. A, Ay. These things premised, we may define a partial differenceequation as an equation expressing an algebraic relation between any partial differences of a function ux, y, z...) the function itself, and the independent variables x, y, z... Or instead of the partial differences of the dependent function, its successive values corresponding to successive states of increment of the independent variables may be involved. are, on the hypothesis of Ax and Ay being each equal to unity, different but equivalent forms of the same partial difference-equation. Mixed difference-equations are those in which the subject function is presented as modified both by operations of the A A Ax' Ay' form and by operations of the form is a mixed difference-equation. Upon the obvious subordinate distinction of ordinary mixed difference-equations and partial mixed difference-equations it is unnecessary to enter. 2. Partial Difference-equations. When there are two independent variables x and y, while the coefficients are constant and the second member is 0, the proposed equation may be presented, according to convenience, in any of the forms F (A2, ▲,) u = 0, F(A,, E) u = 0, F (Ex, E,) u = 0, F(E2, ▲ ̧) u = 0. Now the symbol of operation relating to x, viz. A, or E,, combines with that relating to y, viz. A, or E,, as a constant with a constant. Hence a symbolical solution will be obtained by replacing one of the symbols by a constant quantity a, integrating the ordinary difference-equation which results, replacing a by the symbol in whose place it stands, and the arbitrary constant by an arbitrary function of the independent variable to which that symbol has reference. This arbitrary function must follow the expression which contains the symbol corresponding to a. The condition last mentioned is founded upon the interpretation of (E-a)X, upon which the solution of ordinary difference-equations with constant coefficients is ultimately dependent. For (Chap. XI. Art. 11) the constants following the factor involving a. The difficulty of the solution is thus reduced to the difficulty of interpreting the symbolical result. +1 Ex. 1. Thus the solution of the equation u1au = 0, of which the symbolical form is the solution of the equation u,y,+= 0, of which the symbolic form is will be Eu2- Eu = 0, u2,, = (E,)* (y). = To interpret this we observe that since E," we have This equation, on putting u for u,,,, may be presented in the form Now replacing E, by a, the solution of the equation ..(1). where (y) is an arbitrary function of y. Now, developing the binomial, and applying the theorem developing the binomial in ascending powers of E,, and interpreting, we have Or, treating the given equation as an ordinary differenceequation in which y is the independent variable, we find as the solution Any of these three forms may be used according to the requirements of the problem. Thus if it were required that when x = 0, u should assume the form em, it would be best to employ (3) or to revert to (2) which gives (y) = em, whence 3. There is another method of integrating this class of equations with constant coefficients which deserves attention. We shall illustrate it by, the last example. |