Methods and Applications of Power Series
Jay A. Leavitt · Mathematics of Computation · 1966
Power series in the past played a minor role in the numerical solutions of ordinary and partial differential equations.There have been good reasons.It is often difficult to operate with power series.Finding the series expansion of d u _ " / du dk 'w\ dxk ~ V ' dx ' " ' ' (¿rFy can be arduous.Furthermore if the series can be found, often it will converge in too small a region.There are, however, certain advantages which make their use desirable.A truncated series forms a closed approximation of the solution which can be evaluated at any point in the region where the series converges.Instability, which causes difficulties for finite difference solutions, does not affect the power series solutions.The series solution, with its great accuracy, permits study of the analytic properties of the solution to an extent which is unachievable with a finite difference solution; and the series solution can be used as an intermediate result which can be integrated and differentiated easily.If a finite difference solution is only an intermediate step in the solution of a problem, computer storage problems can be a major concern.Derivatives and interpolated values of the difference solution can be very unreliable.In this paper we intend to show how many of the disadvantages of power series can be overcome by automatic coding procedures and to indicate some of their useful properties and results.In this paper we use the convention that a sum is zero if the upper limit is less than the lower.Let P(I) denote the coefficient of x/_1 in the polynomial P(x) = X^=i P(I)x'~1 and let Q(I, J) denote the coefficient of x'_1?/_I in the polynomial Q(x, y) = TUYsUaa^w-y-1.The following formulas for integration and differentiation are well known: If Q(x, y) = ¡x P(x, y) dx or R(x, y) = j" P(x, y) dy then Q(I,J) =P(/-1, J)/(I-1) and R(I, J) = P(I, J-l)/(J -1).If Q(x, y) = dP(x, y)/dx or R(x, y) = dP(x, y)/dy then Q(I,J)=IP(I + 1,J) and R(I, J) = J P(I, J + 1).The formulas for multiplication and division are as follows:If R(x, y) = P(x, y)-Q(x, y) then R(I, J) = 12 12P(L, M)Q(I -L + 1, J -M + 1).£=1 U-l If R(x, y) = P(x, y)/Q(x, y) then RU' '/) -ñ7T-ñ (P(/> J) -12 12 R(K, L)Q(I -K+l,J-L+l) V(l, 1) \ JC=1 Z.-1 -¿ R(I,L)Q(l,J -L+1)Y L-l / They have been used in machine calculations by R. D. Richtmyer [1].