= Solution
{c}
Subtract the particular <plane Poiseuille flow> profile from the full <rectangular duct flow>. The remainder is a <harmonic function>:
$$
\begin{gathered}
w_2=w-w_1,\qquad w_{2,xx}+w_{2,yy}=0,\\
w_2(x,\pm b)=0,\qquad
w_2(\pm a,y)=-\frac G{2\mu}(b^2-y^2).
\end{gathered}
$$
The data are even in $x$ and $y$. The <boundary value problem> is unique: if two solutions differed by $v$, <Green's first identity> and the zero <Dirichlet boundary condition> would give
$$
\int_{-a}^{a}\int_{-b}^{b}|\nabla v|^2\,dy\,dx
=\int_{\partial\Omega}v\partial_n v\,ds-\int_\Omega v\nabla^2v\,dA=0.
$$
Thus $v$ is constant, and its zero boundary values force $v=0$. Reflection of a solution in either coordinate axis therefore gives the same solution. We may seek even separated <eigenfunctions>.
Use <separation of variables> with
$$
k_n=\frac{(2n+1)\pi}{2b},\qquad
w_2(x,y)=\sum_{n=0}^{\infty}A_n
\frac{\cosh(k_nx)}{\cosh(k_na)}\cos(k_ny).
$$
Each summand solves the <Laplace equation>, vanishes at $y=\pm b$, and has the required evenness. The <hyperbolic cosine> ratio equals one at either side wall. The functions $\cos(k_ny)$ have <orthogonality> relation
$$
\int_{-b}^{b}\cos(k_ny)\cos(k_my)\,dy
=\begin{cases}b,&m=n,\\0,&m\ne n.\end{cases}
$$
They are the even <Dirichlet eigenfunctions> on $[-b,b]$, so their <Fourier series> represents the even boundary data. Projecting that data onto the $n$th <eigenfunction> yields the <Fourier coefficient>
$$
A_n=-\frac{G}{2\mu b}\int_{-b}^{b}(b^2-y^2)\cos(k_ny)\,dy
=-\frac{16Gb^2}{\mu\pi^3}\frac{(-1)^n}{(2n+1)^3}.
$$
For completeness, two <integrations by parts> give the <quadratic arch expansion in half-integer cosine modes>: the displayed <integral> is $4(-1)^n/k_n^3$, agreeing with the identity supplied in the paper.
Adding the particular solution gives
$$
\boxed{\begin{aligned}
w(x,y)&=\frac{G}{2\mu}(b^2-y^2)\\
&\quad-\frac{16Gb^2}{\mu\pi^3}\sum_{n=0}^{\infty}
\frac{(-1)^n}{(2n+1)^3}
\frac{\cosh(k_nx)}{\cosh(k_na)}\cos(k_ny),\\
k_n&=\frac{(2n+1)\pi}{2b}.
\end{aligned}}
$$
The coefficients are absolutely summable and the <hyperbolic cosine> ratio is at most one throughout the closed rectangle, giving <uniform convergence> and the prescribed boundary values. On compact subsets of the interior, the <hyperbolic cosine> ratio supplies exponential decay in $n$, justifying differentiation of the <series> and the <Laplace equation>. The side-wall correction slows the <plane Poiseuille flow> enough to satisfy all four <no-slip boundary conditions>.
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