Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

Abstract

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation
$$\begin{array}{r}{\gamma }^{iv}(\omega )+{\rho }_1\gamma {}^{‴}(\omega )+{\rho }_2\gamma {}^{″}(\omega )+{\rho }_3{\gamma }^{\prime }(\omega )+{\rho }_4\gamma (\omega )=\chi (\omega ),\end{array}$$
where $\gamma \in c^4[\alpha ,\beta ],\chi \in [\alpha ,\beta ]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.