To recapitulate, the problem of solving the harmonic and biharmonic equations has been
reduced to solving a set of one-dimensional periodic functions. They can be solved efficiently
by using highly accurate spectral (Fourier series) method. The solution to the set of the complex
periodic functions 0(x)={ Gl'(x), GP(x), Rc'(x), RP(x), QP(x), and Q (x) can be sought in
the form of Fourier series as
(x) = 0 m exp(-imx)
(4-45)
where Om ,={gm,g,r r', ,qm,,q,} are the respective Fourier coefficients. In the absence of
singularity of the field variables, the series solution converges exponentially. Hence only 10-40
terms of the Fourier series are typically needed, depending on the values of (K, a, h), to ensure
that the error in the solutions for the electric and velocity fields do not exceed a predefined
threshold value (10-8 in this study).
For the infinite interval of integration defined in Equations 4-20, 4-37 and 4-38, the
interval is split into three parts: (-oo,-Lint), [-Lint, Lint], and (Lint, oo). The middle interval, [-Lint,
Lint], is bounded so that it can be accurately and efficiently evaluated using the built-in adapative
quadrature routine based on Simpson's rule in the MATLAB. In the present study, Lint = 1,0247t
and a tolerance of 10-8 are used. For the unbound intervals, the details of the evaluation of the
velocity are illustrated below. From Equation 4-35 u(x, y) can rewritten as
u(x, y)= 2By K y2 + 1 intu()d
47r 2z i (4-46)
where the integrand is
int u(=) = z (-) P )( '(;) + 2 P
int< Z)2 Z2--o z z2z
+(( )-z)2 () (()2 2 '() =(y+ix)2