Matrix Boundary Method: Truncated Laplace Integrals and Differential Equations

We present a unified treatment of the Matrix Boundary Method (MBM). Part I coverstruncated Laplace-type integrals when f lies in a finite-dimensional spaceclosed under differentiation with constant coefficients. Part II adapts MBM to solve linearODEs with constant coefficients, replacing convolution integrals by boundary evaluations.Throughout, we reuse the resolvent M(b) = (bI+ΩT)−1 with bases closed under differentiation(trigonometric, exp–trig, polynomial). Worked examples and exercises (with solutions) areincluded.