NUMERICAL METHODS
These can then be treated in the way indicated in the previous paragraph. The
extension to more than one dependent variable is straightforward.
In practical problems it often happens that boundary conditions applicableto a higher-order equation consist not of the values of the function and all its
derivatives at one particular point but of, say, the values of the function at two
separate end-points. In these cases a solution cannot be found using an explicit
step-by-step ‘marching’ scheme, in which the solutions at successive values of the
independent variable are calculated using solution values previously found. Other
methods have to be tried.
One obvious method is to treat the problem as a ‘marching one’, but to use anumber of (intelligently guessed) initial values for the derivatives at the starting
point. The aim is then to find, by interpolation or some other form of iteration,
those starting values for the derivatives that will produce the given value of the
function at the finishing point.
In some cases the problem can be reduced by a differencing scheme to a matrixequation. Such a case is that of a second-order equation fory(x) with constant
coefficients and given values ofyat the two end-points. Consider the second-order
equation
y′′+2ky′+μy=f(x), (27.85)with the boundary conditions
y(0) =A, y(1) =B.If (27.85) is replaced by a central difference equation,
yi+1− 2 yi+yi− 1
h^2+2kyi+1−yi− 1
2 h+μyi=f(xi),we obtain from it the recurrence relation
(1 +kh)yi+1+(μh^2 −2)yi+(1−kh)yi− 1 =h^2 f(xi).Forh=1/(N−1) this is in exactly the form of theN×Ntridiagonal matrix
equation (27.30), with
b 1 =bN=1,c 1 =aN=0,ai=1−kh, bi=μh^2 − 2 ,ci=1+kh, i=2, 3 ,...,N− 1 ,andy 1 replaced byA,yNbyBandyibyh^2 f(xi)fori=2, 3 ,...,N−1. The
solutions can be obtained as in (27.31) and (27.32).
27.8 Partial differential equationsThe extension of previous methods to partial differential equations, thus involving
two or more independent variables, proceeds in a more or less obvious way. Rather