Non-homogeneous Linear Set-Valued Differential Equations with Variable Matrix Coefficients

TL;DR

This paper studies initial value problems for non-homogeneous linear set-valued differential equations in two-dimensional space, using generalized derivatives including Hukuhara, BG, and PS derivatives, and provides formulas and examples.

Contribution

It introduces solution formulas for set-valued differential equations with variable matrix coefficients using multiple generalized derivatives.

Findings

Derived general and constructive solution formulas.

Extended the theory to include various generalized derivatives.

Provided illustrative examples demonstrating the solutions.

Abstract

We investigate the initial value problems for non-homogeneous linear differential equations whose solutions are set-valued maps taking values in the space of nonempty compact convex subsets of $\mathbb{R}^2$, denoted by $K_{c}(\mathbb{R}^2)$. The differential formulation is based on the generalized derivative that includes the Hukuhara derivative, as well as its extensions, Bede-Gal (BG), and Plotnikov-Skripnik (PS) derivatives, and we obtain some general as well as constructive formulas for the solutions. Several illustrative examples are provided.