Explicit and implicit methods for second order ordinary differential equations

A family of explicit formulas is developed for solving a system of second order linear ordinary differential equations with constant coefficients and with initial conditions specified. A family of implicit formulas for solving the same system with specified boundary conditions is also developed. Both families are based on Padé approximants to the exponential function and for each formula developed the order of the formula is seen to be one higher than the order of the Padé approximant used. In the case of the family of implicit formulas it is seen that the order of the formula is made arbitrarily high by using an appropriate Padé approximant. It is shown that the families are readily applicable to the numerical solution of second order hyperbolic partial differential equations with constant coefficients. The formulas developed are tested on four problems

Extrapolation methods for first order ordinary differential equations

Given a system of fist order differential equations, whose coefficient matrix has constant elements, with initial conditions specified, a family of extrapolating algorithms based on Pade approximants to the exponential function is developped. An important application of such methods is seen to be the numerical solution of the diffusion equation

Extrapolation techniques for first order hyperbolic partial differential equations

A uniform grid of step size h is superimposed on the space variable x in the first order hyperbolic partial differential equation ∂u/∂t + a ∂u/∂x = 0 (a > 0, x > 0, t > 0). The space derivative is approximated by its backward difference and central difference replacements and the resulting linear systems of first order ordinary differential equations are solved employing Padé approximants to the exponential function. A number of difference schemes for solving the hyperbolic equation are thus developed and each is extrapolated to give higher order accuracy. The schemes, and their extrapolated forms, are applied to two problems, one of which has a discontinuity in the solution across a characteristic

A note on methods with 0(H4) and 0(H6) phase lags for periodic

Two families of computational methods are discussed for the solution of second order periodic initial value problems. The first is a family with 0(H4) phase-lag which contains the recently published "Numerov made explicit" method of Chawla [2]. The second is a family with 0(H6) phase-lag and periodicity interval given by H2 Є (0,12)

Global extrapolation procedures for linear partial differential equations

Global extrapolation procedures, in space and time are considered for the numerical Solution of linear partial differential equations. Global extrapolation procedures in time only are reviewed. The procedures are tested on three problems from the literature, one of which has a nonlinear source term

A two - grid, fourth order method for nonlinear fourth order boundary value problems

A fourth order convergent finite difference method is developed for the numerical solution of the nonlinear fourth order boundary value problem y(iv)(x) = f(x,y), a<x <b, y(a) = A0 , y"(a) = B0 , y(b) = A1, y" (b) = B1 . The method is based on a second order convergent method which is used on two grids, fourth order convergence being obtained by considering a linear combination of the individual results relating to the two grids. Special formulas are developed for application to grid points adjacent to the boundaries x = a and x = b , the principal parts of the local truncation errors of these formulas being the same as that of the second order method used at other points of each grid. Modifications to these special formulas are noted for problems with boundary conditions of the form y (a) = Ao , y'(a) = Co , y(b) = A1, y'(b) =c1

Partial differential equations in medical biophysics

A number of examples of collaborative research are outlined which show how mathematicians and medical biophysicists have contributed to a wider understanding of some problems in applied physiology