Mathematical Methods in Interdisciplinary Sciences

11Affine‐Contractor Approach to Handle Nonlinear Dynamical Problems in Uncertain Environment

Nisha Rani Mahato, Saudamini Rout, and Snehashish Chakraverty

Department of Mathematics, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

11.1 Introduction

Nonlinear dynamical problems form backbone in several fields of science and engineering, viz., structural mechanics, fluid dynamics, control theory, robotics, seismology, circuit analysis, etc. In this regard, nonlinear ordinary differential equations (NODEs) are taken into consideration in this chapter. The general form of NODE governed from different science and engineering problems is given as

(11.1)

subject to the n initial conditions (ICs), y(t) = a₁ and y⁽ⁱ⁾(t) = a_i + 1 for i = 1, 2, …, n. Here, L(y) and N(y) are the respective linear and nonlinear parts of the NODE 11.1, and F(t) is the external force vector applied on the nonlinear problem. In actual practice, the material properties of the modeled NODEs are taken as crisp (exact) values for ease handling of the nonlinear systems. Few literature studies regarding the computation of solutions for nonlinear dynamical problems having crisp parameters using various semianalytical methods are discussed initially. He [1] proposed a nonlinear analytical technique, viz., homotopy perturbation method (HPM) for solving NODEs with crisp parameters. Further, ...