Microsoft excel-based numerical solution of linear, homogeneous id transient partial differential equations

Abstract

Many transient phenomenons are mathematically described and simulated by the homogeneous, parabolic partial differential equations. Irregular and non-linear boundary conditions pose formidable difficulties to engineers to obtain closed form (exact) solutions. In this paper, a procedure is outlined to make use of Microsoft Excel software to solve these differential equations by converting them into finite difference equations employing explicit and implicit techniques. The advantages of this study are to apply universally available Microsoft Excel to solve fairly complex engineering problems. The method lends itself as a powerful tool to Engineering students to design and perform parametric analysis by employing simple, quick techniques without having to acquire specialized and sometimes expensive software packages. A case study is presented where the time history of temperature in a one-dimensional heat transfer problem is analyzed. Finite difference techniques are used to solve the differential equations. Initially the equations are solved using an explicit method and the same problem is also solved using an implicit method. Step by step procedure to generate Excel worksheet is described. Results obtained with the two numerical methods are compared with analytical results. Effects of grid size and time interval on the accuracy of the results are graphically presented.