Particular Solutions of the Equation of Conduction of Heat in One Dimension

The problem of the conduction of heat in one dimension is usually concerned with the propagation of a thermal disturbance along a bar or rod of uniform cross section. The solution of the problem is required for a given initial distribution of temperature, and given boundary values, usually at each end of the rod. In most cases this solution is found by assuming a series solution and then proving that the series satisfies the equation of the disturbance well as all the assigned conditions. Other methods, for example the contour integral method developed by Carslaw, also introduce this arbitrary element of choice in choosing the integrand and the contour of integration. The object of the present paper is to develop the application of Heaviside's Operational method to the solution of the problem, and to show that it leads in all cases to solutions equivalent to the known forms, although initially no assumptions are made regarding the nature of the solution.