Matrix Boundary Method: Truncated Laplace Integrals and Differential Equations

We present a unified treatment of the Matrix Boundary Method (MBM). Part I covers truncated Laplace-type integrals when f lies in a finite-dimensional space closed under differentiation with constant coefficients. Part II adapts MBM to solve linear ODEs 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) are included. This paper studies the evaluation of integrals using a matrix-based approach, where the integrand is assumed to be closed under differentiation. Within this framework, the Laplace transform appears as a special case when the limits of integration extend from zero to infinity.