ON THE DERIVATION AND SOLUTION OF THE ONE DIMENSIONAL HEAT EQUATIONS

Abstract

The derivation and solution of the one dimensional heat equations were studied by separation and Fourier series. Heat equation is the fundamental partial parts of di erential equations which governs the temperature u in body space. By Fourier's law of heat transfer heat ow in the direction of decreasing temperature, and the rate of ow is proportional to gradient. Heat equation is di usion equations because it models chemical di usions process of one substance or gas to another. Di usion equations is a linear second order partial di erential equation u t Dux x = f where u = u(x; t) x is a real space variable, t is a time variables and D is a di usion coe cient. In heat equations consider the temperature in long thin metal of constant cross section and homogeneous materials, which oriented along x axis and is perfectly insulated laterally. So that heat ow in the x direction only. It has been shown that the heat equation is the one dimensional heat equations. @u @t (x; t) = c 2 @ 2 u @x 2 (x; t) 0 < x < ` and t > 0; (0.0.1)