Preface
In this section, we discuss how to solve nonhomogeneous boundary value problems. Most of our attention is directed toward problems in which the differential equations of the second order are nonhomogeneous while the boundary conditions remain homogeneous. We assume that the solution can be expanded in a series of eigenfunctions of a related homogeneous problem, and then we determine the coefficients in this series.
Let us consider a nonhomogeneous differential equation for Sturm--Liouville self-adoint operator
\begin{equation} \label{EqInhomo.1}
L\left[ x, \texttt{D} \right] y = q(x)\,y - \frac{\text d}{{\text d}x} \left( p(x)\,\frac{{\text d}y}{{\text d}x} \right) = \mu\,w(x)\,y(x) + f(x) , \qquad 0 < x < \ell ;
\end{equation}
where μ is a iven constant,
p(
x) ∈
C¹(0, ℓ),
q(
x), and
f(
x) are given continuous function on the open interval (0, ℓ); moreover,
p(
x) is assumed to be continuously differentiable. WE consider the differential equation under the two-point homogeneous boundary conditions
\begin{equation} \label{EqInhomo.2}
\alpha_0 y(0) - \alpha_1 y' (0) =0, \qquad \beta_0 y(\ell ) + \beta_2 y' (\ell ) = 0.
\end{equation}
Upon considering the corresponding homogeneous Sturm--Liouville problem
\begin{equation} \label{EqInhomo.3}
L\left[ x, \texttt{D} \right] y = w(x)\,\lambda\,y(x)
\end{equation}
and corresponding homogeneous boundary conditions \eqref{EqInhomo.2}, we conclude that this problem has infinite many eigenvalues λ₁ < λ₂ < λ₃ < ··· < &lamda;
n < ··· <. Let ϕ₁, ϕ₂,
ϕ₃, … be the corresponding eigenfunctions. Now we assume that the solution
y = ϕ(
x) of the nonhomogeneous problem \eqref{EqInhomo.1}, \eqref{EqInhomo.2} can be expanded with respect to the eigenfunctions:
\begin{equation} \label{EqInhomo.4}
y = \phi (x) = \sum_{n\ge 1} c_n \phi_n (x).
\end{equation}
Since the eigenfunctiosn are orthogonal, we find
\begin{equation} \label{EqInhomo.5}
c_n = \frac{\langle f\,, \, \phi_n \rangle}{\| \phi_n \|^2} = \dfrac{\int_0^{\ell} f(x)^{\ast} \phi_n (x)\,{\text d}x}{\int_0^{\ell} \left\vert \phi_n (x) \right\vert^2 {\text d} x} , \qquad n=1,2,3,\ldots .
\end{equation}
However, since we do not know ϕ(
x), we cannot use Eq.\eqref{EqInhomo.5} to calculate
cn. Instead, we will try to determine
cn so that the problem \eqref{EqInhomo.1}, \eqref{EqInhomo.2}
is satisfied and then use Eq.\eqref{EqInhomo.4} to determine
y = ϕ(
x). Observe that the series \eqref{EqInhomo.4} automatically satisfies the homogeneous boundary conditions \eqref{EqInhomo.2} because every term does.
Substituting series \eqref{EqInhomo.4} into the differential equation \eqref{EqInhomo.1}, we get
\[
L \left[ x, \texttt{D} \right] y = L \left[ \sum_{n\ge 1} c_n \phi_n (x) \right] = \sum_{n\ge 1} c_n \lambda_n w(x)\,\phi_n (x) ,
\]
where we have assumed that we can interchange the operations of summation and differentiation.
Since the weight function w(x) appear at the right-hand side as a multiple, we expand the ratio
\[
\frac{f(x)}{w(x)} = \sum_{n\ge 1} b_n \phi_n (x) ,
\]
where
\begin{equation} \label{EqInhomo.6}
b_n = \frac{1}{\| \phi_n \|^2} \left\langle \frac{f}{w} . \phi_n \right\rangle = \frac{1}{\| \phi_n \|^2} \int_0^{\ell} w(x) \,\frac{f(x)}{w(x)} \,\phi_n (x)\,{\text d} x = \frac{1}{\| \phi_n \|^2} \int_0^{\ell} f(x)\,\phi_n (x)\,{\text d}x, \qquad n=1,2,3,\ldots .
\end{equation}
Upon substituting every series into the differential equation
\[
L \left[ x, \texttt{D} \right] y = \mu\,w(x) \,y + f(x) ,
\]
we obtain
\[
\sum_{n\ge 1} c_n \lambda_n w(x)\,\phi_n (x) = \mu\,w(x) \sum_{n\ge 1} c_n \phi_n (x) + w(x) \sum_{n\ge 1} b_n \phi_n (x) .
\]
After collecting terms and canceling the common nonzero factor
w(
x), we have
\[
\sum_{n\ge 1} \left[ \left( \lambda_n - \mu \right) c_n - b_n \right] \phi_n (x) = 0 .
\]
This equation is to hold for each
x in the interval 0 <
x < ℓ, the coefficient of ϕ(
x) must be zero for each
n. Therefore, we have
\begin{equation} \label{EqInhomo.7}
\left( \lambda_n - \mu \right) c_n - b_n = 0, \qquad n=1,2,3,\ldots .
\end{equation}
If μ ≠ λ
n for
n = 1, 2, 3, … ,
then
\[
c_n = \frac{b_n}{\lambda_n - \mu} , \qquad n=1,2,3,\ldots ,
\]
and
\begin{equation} \label{EqInhomo.8}
y = \phi (x) = \sum_{n\ge 1} \frac{b_n}{\lambda_n - \mu} \,\phi_n (x) .
\end{equation}
Equation \eqref{EqInhomo.8} with
bn given by Eq.\eqref{EqInhomo.6}, provides a formal solution of the nonhomogeneous boundary value problem
\eqref{EqInhomo.1}, \eqref{EqInhomo.2}. It is reasonable to expect that the series \eqref{EqInhomo.8} converges and then it provides the solution.
Now suppose that μ is equal to one of the eigenvalue of the corresponding homogeneous problem, say, μ = λm. Then the situation becomes quite different. In this event, for n = m, Eq.\eqref{EqInhomo.7} hs the form 0 · cm −bm = 0, which leads to bm = 0. There are two options.
If μ = λm and bm ≠ 0, then there is no solution.
If μ = λm and bm = 0, then Eq.\eqref{EqInhomo.6} is satisfied regardless of the value of cm; in other words, cm remains arbitrary. In this case, the two point inhomogeneous boundary value problem \eqref{EqInhomo.1}, \eqref{EqInhomo.2} does have a solution, but it is not unique since it contains an arbitrary multiple of the eigenfunction ϕm(x).
The condition bm = 0 means that
\begin{equation} \label{EqInhomo.9}
\int_0^{\ell} f(x)\,\phi_m (x)\,{\text d} x = 0.
\end{equation}
Hence, if μ = λ
m, the nonhomogeneous boundary value problem \eqref{EqInhomo.1}, \eqref{EqInhomo.2} can be solved only when
f is orthogonal to the eigenfunction corresponding to the eigenvalue
λ
m.
Theorem 7: Fredholm alternative.
Suppose that we have a regular Sturm–Liouville problem \eqref{EqSturm.5}. Then either the homogeneous boundary value problem
\[
\frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}y}{{\text d}x} \right) - q(x)\,y(x) + \lambda\,\rho (x)\,y(x) = 0 ,
\]
\[
\alpha_0 y(a) - \alpha_1 y' (a) = 0 , \qquad \beta_0 y(b) + \beta_1 y' (b) = 0
\]
has a nonzero solution (when λ is an eigenvalue), or the nonhomogeneous boundary value problem
\[
\frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}y}{{\text d}x} \right) - q(x)\,y(x) + \lambda\,\rho (x)\,y(x) = f(x) ,
\]
\[
\alpha_0 y(a) - \alpha_1 y' (a) = 0 , \qquad \beta_0 y(b) + \beta_1 y' (b) = 0
\]
has a unique solution for any continuous function
f(
x) on the interval [𝑎,
b].
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