Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 47

A special case of an initial value problem is when the number of dimensions n in an initial value problem is equal to 1. In this case we speak of a scalar problem, and it is useful to study these problems if one wishes to get some insights into how finite difference methods work. In this section we discuss some numerical properties of one-step finite difference schemes for the linear scalar problem:

where

The reader can check that the one-step methods (Equations (2.10), (2.11) and (2.12) can all be cast as the general form recurrence relation:

where Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows:

with:

for a mesh function . A special case is when the coefficients and are constant , that is:

Then the general solution is given by:

where we note that power of constant and .

In order to prove this, we need the formula for the sum of a series:

For a readable introduction to difference schemes, we refer the reader to Goldberg (1986).