Modified backward differentiation methods of the Adams-type based on exponential interpolation (Q806976)

Adams-Bashforth and Adams-Moulton finite difference schemes of the usual form \(y_{p+1}=y_ p+h\cdot \sum^{m+q}_{j=m}\beta_ jf_{p+1-j},\) \(m=0\) or 1, \(h>0\), are derived from the integral formulation of the Cauchy problem for the ordinary differential equation \(y'=f(x,y):\) \[ y(x_{p+1})=y(x_ p)+h\cdot \int^{x_{p+1}}_{x_ p}f(t,y(t))dt \] where f(t,y(t)) is replaced by its Lagrange interpolatory polynomial with nodes \(x_{p+1-j}=x_ 0+(p+1-j)h\), \(j=m,...,q\), \(m=0\) or 1, \(h>0.\) The authors propose to apply here functions of the form \(e^{kx}P_ q(x)\), where \(P_ q\) is a polynomial of degree \(\leq q\), and k is a real free parameter, instead of the usual polynomials. If calculations were performed without round-off errors, finite difference schemes of this type would fit the exact solution, provided that this last one is of the form \(y(x)=e^{kx}\sum^{q}_{j=0}a_ jx^ j.\) In this paper, coefficients \(\beta_ j\) for such schemes are derived for \(q=1,2\) and 3. In contrast to ``traditional'' multistep methods, here the coefficients \(\beta_ j\) depend on h and k. The right choice of the free parameter k is suggested by the form of the remainder term of interpolation. This term, usually expressed with an application of the ``midpoint-value theorem'', involves certain unknown point inside the interval of interpolation. The value of the remainder is expressed as the \((q+1)\)-st derivative of certain known function depending on y and k. The authors propose to make vanishing this derivative in some point of the interval. This suggests an equation from which k may be eventually found. The paper contains few numerical tests showing a real improvement of results with respect to traditional algorithms, when the proposed method is applied to Cauchy problems having solutions of exponential character.