Abstract
The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.
Highlights
The heat equation in one dimension We begin the study of partial differential equations with the problem of heat flow in uniform bar of length 8 units situated on the x – axis with one end at origin and the other at 8 units.We assume that the thermal properties of the bar are independent of x and t
Heat equation in one dimension can be solved by analytical approach
One dimensional heat equation with this method we will apply separation of variables which is the solutions to homogeneous PDE [5, 1, 4, 12]
Summary
We assume that the thermal properties of the bar are independent of x and t. The temperature , at time t at point x units from the origin satisfies the partial differential equation. We assume that the bar is perfectly insulated except possibly at its end points and that the temperature is constant on each cross section and depends only on x and t. This is the heat equation with its boundary and initial condition which governs the heat distribution in the above uniform bar of length 8 units, see e.g. International Journal of Applied Mathematics and Theoretical Physics 2021; 7(2): 53-61
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