The initial value problem for interval-valued second-order differential equations under generalized H-differentiability

Abstract

In this paper the interval-valued second-order differential equations under generalized Hukuhara differentiability (ISDEs) are introduced. Under suitable conditions we obtain the existence and uniqueness results of solutions to ISDEs. To prove this assertion we use idea of contraction principle and successive approximations. Furthermore, we use the method based on properties of linear transformations (LTM) to find the explicit solution of initial value problem for linear second-order differential equation with interval-valued forcing function and with interval initial values (IIVP). We apply the linear transformations method to two example problems including a vibrating mass and an electrical circuit. Finally, the method based on the analysis of a solution to real-valued second-order differential equation is investigated to solve interval-valued second-order differential equations with interval-valued coefficients, interval-valued forcing function and interval initial values. Some examples are presented to illustrate applicability of the proposed method.