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4. Series-Expansion Methods
expansion be written as
N
f/>(x , t) = I:>k(t)qJk(X) ,
k=l
(4.2)
where qJI , • •• , qJN are predetennined expansion functions satisfying the required
boundary conditions. Then the task of solving (4.1) is transfonned into a problem
of calculating the unknown coefficients al (r) , ... , aN(t) in a way that minimizes
the error in the approximate solution. One might hope to obtain solvable expressions for the ak(t) by substituting (4.2) into the goveming equation (4.1). For
example, if Fis a linear function of a n1/Jlaxn with constant coefficients and the
expansion functions are Fourier series, direct substitution will yield a system of
ordinary differential equations for the evolution of the ak (r). Unfortunately, direct
substitution yields a solvable system of equations for the expansion coefficients
only when the qJk are eigenfunctions of the differential operator F -direct substitution works in precisely those special cases for which analytic solutions are
available. This, of course, is a highly restrictive limitation.
In the general case where the qJk are not eigenfunctions of F, it is impossible to
specify al (r), ... , aN(t) such that an expression of the form (4.2) exactly satisfies (4.1). As an example, suppose F(1/J) = 1/Ja1/Jlax and the expansion functions
are the Fourier components qJk = e ikx , - N
k
N. If this Fourier series
is substituted into (4.1), the nonlinear product in F (1/J) introduces spatial variations at wave numbers that were not present in the initial truncated series, e.g.,
F(e i N x ) = i N e i 2N x • A total of 4N + I equations are obtained after substituting the expansion functions into (3.1) and requiring that the coefficients of each
Fourier mode sum to zero. It is not possible to choose the 2N + I Fourier coefficients in the original expansion to satisfy these 4N + I equations simultaneously.
The best one can do is to select the expansion coefficients to minimize the error.
Since the actual error in the approximate solution 1I1/J - f/>II cannot be determined, the most practical way to try to minimize the error is to minimize the
residual,
R(f/» = af/>
at
+ F(f/»,
(4.3)
which is the amount by which the approximate solution fails to satisfy the goveming equation. Three different strategies are available for constraining the size
of the residual. Each strategy leads to a system of N coupled ordinary differential equations for the time-dependent coefficients al (r), . .. , aN (t). This transformation of the partial differential equation into a system of ordinary differential
equations is similar to that which occurs in grid-point methods when the spatial
derivatives are replaced with finite differences.
One strategy for constraining the size of the residual is to pick the ak (t) to
minimize the square of the 12-nonn of the residual :
4. Series-Expansion Methods
expansion be written as
N
f/>(x , t) = I:>k(t)qJk(X) ,
k=l
(4.2)
where qJI , • •• , qJN are predetennined expansion functions satisfying the required
boundary conditions. Then the task of solving (4.1) is transfonned into a problem
of calculating the unknown coefficients al (r) , ... , aN(t) in a way that minimizes
the error in the approximate solution. One might hope to obtain solvable expressions for the ak(t) by substituting (4.2) into the goveming equation (4.1). For
example, if Fis a linear function of a n1/Jlaxn with constant coefficients and the
expansion functions are Fourier series, direct substitution will yield a system of
ordinary differential equations for the evolution of the ak (r). Unfortunately, direct
substitution yields a solvable system of equations for the expansion coefficients
only when the qJk are eigenfunctions of the differential operator F -direct substitution works in precisely those special cases for which analytic solutions are
available. This, of course, is a highly restrictive limitation.
In the general case where the qJk are not eigenfunctions of F, it is impossible to
specify al (r), ... , aN(t) such that an expression of the form (4.2) exactly satisfies (4.1). As an example, suppose F(1/J) = 1/Ja1/Jlax and the expansion functions
are the Fourier components qJk = e ikx , - N
k
N. If this Fourier series
is substituted into (4.1), the nonlinear product in F (1/J) introduces spatial variations at wave numbers that were not present in the initial truncated series, e.g.,
F(e i N x ) = i N e i 2N x • A total of 4N + I equations are obtained after substituting the expansion functions into (3.1) and requiring that the coefficients of each
Fourier mode sum to zero. It is not possible to choose the 2N + I Fourier coefficients in the original expansion to satisfy these 4N + I equations simultaneously.
The best one can do is to select the expansion coefficients to minimize the error.
Since the actual error in the approximate solution 1I1/J - f/>II cannot be determined, the most practical way to try to minimize the error is to minimize the
residual,
R(f/» = af/>
at
+ F(f/»,
(4.3)
which is the amount by which the approximate solution fails to satisfy the goveming equation. Three different strategies are available for constraining the size
of the residual. Each strategy leads to a system of N coupled ordinary differential equations for the time-dependent coefficients al (r), . .. , aN (t). This transformation of the partial differential equation into a system of ordinary differential
equations is similar to that which occurs in grid-point methods when the spatial
derivatives are replaced with finite differences.
One strategy for constraining the size of the residual is to pick the ak (t) to
minimize the square of the 12-nonn of the residual :