Matrix Boundary Method

A finite-dimensional method for integrals of the form e^(bx) f(x) over [0, a], where f lies in a function space closed under differentiation. Differentiation becomes a constant matrix, and the integral reduces to one linear system and a boundary evaluation, with extensions to products, matrix exponentials, the Laplace transform, and first-order linear ODEs.

Differentiation as a matrix

Chapter 5 begins with linear spaces of functions and focuses on finite-dimensional spaces closed under differentiation: polynomials of bounded degree, the pair $\{\cos\omega x,\sin\omega x\}$, exponential–trigonometric pairs, and products of these. If the vector $\Phi(x)$ collects a basis, closure means there is a constant matrix $\Omega$ with

so differentiation inside the space is multiplication by $\Omega$, and each function in the space is represented by its coefficient vector $u$.

From an integral to a linear system

To integrate $e^{bx}f(x)$, the book proposes the candidate antiderivative $F(x)=e^{bx}\Phi(x)^{\mathsf T}z$. Differentiating, matching with the integrand, and using the linear independence of the basis leaves one linear system, $(bI+\Omega^{\mathsf T})z=u$. When the matrix is invertible, the truncated Laplace-type integral becomes a boundary evaluation:

The book calls this the Matrix Boundary Method (MBM). In practice you choose a basis, build $\Omega$, solve once, and evaluate at the two endpoints. For the basis $(\cos\omega x,\sin\omega x)$ the resolvent is $\frac{1}{b^2+\omega^2}\begin{pmatrix}b&-\omega\\ \omega&b\end{pmatrix}$, and the book lists ready matrices of this kind for polynomial, trigonometric, exponential–trigonometric, and exponential–hyperbolic bases.

The method replaces repeated integration by parts with a computation whose size is fixed by the basis. Integrals such as $\int_0^a e^{bx}x^2\,dx$ or $\int_0^a e^{bx}\cos\omega x\,dx$ become one small linear solve, and the same matrix serves every integrand in the space. The book's worked examples apply the same idea to first-order ODEs: for $y'-3y=\cos 2x$ with $y(0)=0$ it obtains $y=\tfrac{3}{13}e^{3x}-\tfrac{3}{13}\cos 2x+\tfrac{2}{13}\sin 2x$.

Resolvent and exponential views

On the closed space, $(bI+\Omega^{\mathsf T})^{-1}$ represents the inverse of $D+b$, which is why the same boundary principle solves first-order linear ODEs whose forcing lies in that space. Writing $\Phi(x)=e^{x\Omega}\Phi(0)$ gives an exponential-matrix version,

and letting $a\to\infty$, when the matrix exponential decays, yields the Laplace transform in resolvent form, $u^{\mathsf T}(sI-\Omega)^{-1}\Phi(0)$. Products $f(x)G(x)$ are handled by a Kronecker-product state $\Phi_g\otimes\Phi_f$, whose differentiation matrix is the Kronecker sum of the two factors' matrices.

How it connects to the other areas

The MBM is the algebraic counterpart of the earlier chapters. Truncated integrals like those of higher-order derivatives become boundary terms, the Laplace transform of the transform framework reappears as a resolvent, and the trigonometric and hyperbolic ready matrices mirror the circular and hyperbolic regimes of mathematical geometry.