Tuesday, February 2, 2010

Galerkin's Method

One of the most important weighted residual methods was invented by the Russian mathematician Boris Grigoryevich Galerkin (February 20, 1871 - July 12, 1945). Galerkin's method selects the weight function functions in a special way: they are chosen from the basis functions, i.e. [Graphics:Images/GalerkinMod_gr_59.gif]. It is required that the following [Graphics:Images/GalerkinMod_gr_60.gif] equations hold true

(6) [Graphics:Images/GalerkinMod_gr_61.gif] for [Graphics:Images/GalerkinMod_gr_62.gif].

To apply the method, all we need to do is solve these [Graphics:Images/GalerkinMod_gr_63.gif] equations for the coefficients [Graphics:Images/GalerkinMod_gr_64.gif].

Galerkin's Method for solving an I. V. P.

Suppose we wish to solve the initial value problem

(i) [Graphics:Images/GalerkinMod_gr_65.gif],
with
[Graphics:Images/GalerkinMod_gr_66.gif] over the interval [Graphics:Images/GalerkinMod_gr_67.gif].

We use the trial function

(ii) [Graphics:Images/GalerkinMod_gr_68.gif].

There are [Graphics:Images/GalerkinMod_gr_69.gif] equations to solve [Graphics:Images/GalerkinMod_gr_70.gif] for [Graphics:Images/GalerkinMod_gr_71.gif], i.e.

(iii) [Graphics:Images/GalerkinMod_gr_72.gif] for [Graphics:Images/GalerkinMod_gr_73.gif].

Remark: For the solution of an I. V. P. we choose [Graphics:Images/GalerkinMod_gr_74.gif].


Galerkin's Method for solving an a B. V. P.

Suppose we wish to solve a boundary value problem over the interval [Graphics:Images/GalerkinMod_gr_142.gif],

(I) [Graphics:Images/GalerkinMod_gr_143.gif],
with
[Graphics:Images/GalerkinMod_gr_144.gif]

We define [Graphics:Images/GalerkinMod_gr_145.gif] and use the trial function

(II) [Graphics:Images/GalerkinMod_gr_146.gif].

There are [Graphics:Images/GalerkinMod_gr_147.gif] equations to solve [Graphics:Images/GalerkinMod_gr_148.gif] for [Graphics:Images/GalerkinMod_gr_149.gif], i.e.

(III) [Graphics:Images/GalerkinMod_gr_150.gif] for [Graphics:Images/GalerkinMod_gr_151.gif].

Remark: The functions [Graphics:Images/GalerkinMod_gr_152.gif] must all be chosen with the boundary properties

[Graphics:Images/GalerkinMod_gr_153.gif] and [Graphics:Images/GalerkinMod_gr_154.gif] for [Graphics:Images/GalerkinMod_gr_155.gif].


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Introductory Methods of Numerical Analysis