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

Area of Science:

• Mathematics
• Differential Equations
• Numerical Analysis

Background:

• Linear systems with derivatives are fundamental in modeling dynamic processes.
• Traditional methods for solving such systems can be computationally intensive, especially those involving diagonalization.
• Deformable derivatives offer a novel framework for analyzing system dynamics.

Purpose of the Study:

• To develop a matrix-based approach for solving linear systems with deformable derivatives.
• To demonstrate the utility of this method through practical examples.
• To explore the role of a new parameter in system evolution.

Main Methods:

• Transformation of the deformable derivative system into a classical matrix differential equation.
• Utilizing the Putzer algorithm and Cayley-Hamilton theorem for explicit solutions.
• Introducing a deformation-adjusted system matrix.

Main Results:

• The proposed method avoids eigenvector calculations and diagonalization.
• A single parameter θ was identified, which modifies the effective evolution rate of linear constant-coefficient systems.
• The parameter θ can tune system response to mimic delays without significant computational overhead.

Conclusions:

• The matrix-based approach provides an efficient alternative for solving linear systems with deformable derivatives.
• The deformable framework with parameter θ offers a computationally light method for simulating delayed responses.
• This method shows promise for applications where fractional models are computationally prohibitive.