Tuesday, February 2, 2010

Finite Difference Method for ODE's

Methods involving difference quotient approximations for derivatives can be used for solving certain second-order boundary value problems. Consider the linear equation

(1) [Graphics:Images/FiniteDifferenceMod_gr_26.gif]

over [a,b] with [Graphics:Images/FiniteDifferenceMod_gr_27.gif]. Form a partition of [a, b] using the points [Graphics:Images/FiniteDifferenceMod_gr_28.gif], where [Graphics:Images/FiniteDifferenceMod_gr_29.gif] and [Graphics:Images/FiniteDifferenceMod_gr_30.gif] for [Graphics:Images/FiniteDifferenceMod_gr_31.gif]. The central-difference formulas discussed in Chapter 6 are used to approximate the derivatives

(2) [Graphics:Images/FiniteDifferenceMod_gr_32.gif]

and

(3) [Graphics:Images/FiniteDifferenceMod_gr_33.gif]

Use the notation [Graphics:Images/FiniteDifferenceMod_gr_34.gif] for the terms [Graphics:Images/FiniteDifferenceMod_gr_35.gif] on the right side of (2) and (3) and drop the two terms [Graphics:Images/FiniteDifferenceMod_gr_36.gif]. Also, use the notations [Graphics:Images/FiniteDifferenceMod_gr_37.gif], [Graphics:Images/FiniteDifferenceMod_gr_38.gif], and [Graphics:Images/FiniteDifferenceMod_gr_39.gif] this produces the difference equation

[Graphics:Images/FiniteDifferenceMod_gr_40.gif]

which is used to compute numerical approximations to the differential equation (1). This is carried out by multiplying each side by [Graphics:Images/FiniteDifferenceMod_gr_41.gif] and then collecting terms involving [Graphics:Images/FiniteDifferenceMod_gr_42.gif] and arranging them in a system of linear equations:

[Graphics:Images/FiniteDifferenceMod_gr_43.gif]

for [Graphics:Images/FiniteDifferenceMod_gr_44.gif] , where [Graphics:Images/FiniteDifferenceMod_gr_45.gif] and [Graphics:Images/FiniteDifferenceMod_gr_46.gif] . This system has the familiar tridiagonal form.

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Introductory Methods of Numerical Analysis