Matrix-based solution methods for deformable derivative systems: applications to growth-decay and mortgage models

Abstract

In this work, we study linear systems with deformable derivatives and provide a matrix-based approach to their solution. The given system is first transformed into a comparable classical matrix differential equation using a deformation-adjusted system matrix. This allows us to use popular techniques like the Putzer algorithm & the Cayley-Hamilton theorem to generate explicit solutions. One advantage of this approach is that it doesn’t need eigenvector calculation or diagonalization. To illustrate the method, we look at two scenarios: a radioactive decay-growth scenario and a mortgage payback issue. The findings demonstrate that, for linear constant-coefficient systems, the deformable framework introduces a single parameter $\theta$ that modifies the effective evolution rate. Although the present analysis is confined to two elementary applications, the numerical experiments suggest that $\theta$ can be tuned to mimic delayed responses without increasing computational complexity – a feature that may prove useful in domains where fractional models are too heavy.