Piecewise linear approximation of nonlinear ordinary differential equations

Abstract

The study of linear ordinary differential equations (ODEs) is an important component of the undergraduate engineering curriculum. However, most of the interesting behaviour of nature is described by nonlinear ODEs whose solutions are analytically intractable. We present a simple method based on the idea that the curve of the nonlinear terms of the dependent variable can be replaced by an approximate curve consisting of a set of line segments tangent to the original curve. This enables us to replace a nonlinear ODE with a finite set of linear inhomogeneous ODEs for which analytic solutions are possible. We apply this method to the cooling of a body under the combined effects of convection and radiation and demonstrate very accurate solutions with a relatively few number of line segments. Furthermore, we discuss how a number of key and usually disparate concepts of calculus are needed to apply this method, including continuity and differentiability, Taylor polynomials and optimisation. References S. Theodorakis and E. Svoukis. Piecewise linear emulation of propagating fronts as a method for determining their speeds. Physical Review E 68, 2003, 027201. doi:10.1103/PhysRevE.68.027201 D. D. Ganji, M. J. Hosseini and J. Shayegh. Some nonlinear heat transfer equations solved by three approximate methods. International Communications in Heat and Mass Transfer 34, 2007, 1003--1016. doi:10.1016/j.icheatmasstransfer.2007.05.010 H. D. Young. University Physics. 8th edition, Addison-Wesley Publishing Company, Massachusetts, 1992. F. P. Incropera, D. P. Dewitt, T. L. Bergman and A. S. Lavine. Fundamentals of Heat and Mass Transfer. 6th edition, John Wiley and Sons, New Jersey, 2007.