Computer Facilitated Generalized Coordinate .

Computer FacilitatedGeneralized CoordinateTransformations of PartialDifferential Equations WithEngineering ApplicationsA. ELKAMEL,1 F.H. BELLAMINE,1,2 V.R. SUBRAMANIAN31Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario,Canada N2L 3G12National Institute of Applied Science and Technology in Tunis, Centre Urbain Nord, B.P. No. 676, 1080 Tunis Cedex,Tunisia3Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505Received 16 February 2008; accepted 2 December 2008ABSTRACT: Partial differential equations (PDEs) play an important role in describing many physical,industrial, and biological processes. Their solutions could be considerably facilitated by using appropriatecoordinate transformations. There are many coordinate systems besides the well-known Cartesian, polar, andspherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use hasthe advantage of easing the manipulation and derivation of analytical expressions. We illustrate this byconsidering a number of engineering problems governed by PDEs in different coordinate systems such asthe bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computeralgebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommontransformations more frequently at the very least for situations where the transformations provide smarter andeasier solutions. ß2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365 376, 2011; View this article online atwileyonlinelibrary.com; DOI 10.1002/cae.20318Keywords: partial differential equations; symbolic computation; Maple; coordinate transformationsINTRODUCTIONPartial differential equations (PDEs) are mathematical modelsdescribing physical laws such as chemical processes, electrostaticdistributions, heat flow, and fluid motion. The solution to a numberof PDEs is shaped by the boundaries of the geometry, and thus thecoordinate system selected is influenced by these boundaries. Bychoosing a curvilinear coordinate system ( 1, 2, 3) such that theboundary surface is one of the coordinate surfaces, it is possible toexpress the solution of the PDE in terms of these new coordinates,that is, 1, 2, 3. There are many coordinate systems besides theCorrespondence to A. Elkamel (aelkamel@cape.uwaterloo.ca).ß 2009 Wiley Periodicals Inc.usual Cartesian, polar, and spherical coordinates. For example,Figure 1 shows two identical pipes imbedded in a concrete slab. Tofind the steady-state temperature, the most suitable coordinatesystem is the bipolar coordinate system as will be demonstrated inmore details in Application of the Bipolar Coordinate SystemSection. In addition, the separation of variables is a common methodfor solving linear PDEs. The separation is different for differentcoordinate systems. In other words, we find out in what coordinatesystem an equation will be amenable to a separation of variablessolution. The properties of the solution can be related to thecharacteristics of the equations and the geometry of the selectedcoordinate system. It is possible to prove using the theory of analyticfunctions of the complex variable that there are a number of twoand three-dimensional separable coordinate systems. The coordinatesystem is defined by relationships between the rectangular365

366Figure 1ELKAMEL, BELLAMINE, AND SUBRAMANIANTwo identical pipes imbedded in an infinite concrete slab.coordinates (x, y, z) and the coordinates ( 1, 2, 3). The newcoordinate axes are given by the equations 1(x, y, z) ¼ constant, 2(x, y, z) ¼ constant, and 3(x, y, z) ¼ constant.For example, polar coordinates are useful for circularboundaries or ones consisting of two lines meeting at an angle(see Fig. 2). The families r ¼ constant and ’ ¼ constant are,respectively, the concentric circles and the radial lines asillustrated in Figure 2. Coordinate systems more general thanthe polar are the elliptic coordinates consisting of ellipses andhyperbolas. These coordinates are suitable for elliptic boundariesor ones consisting of hyperbolas as illustrated in Figure 3.Parabolic cylindrical coordinates, shown in Figure 4, are twoorthogonal families of parabolas, with axes along the x-axis.These coordinates are suitable, for example, for a boundaryconsisting of the negative half of the x-axis. Generally speaking,the separable coordinate systems for two dimensions consisted ofconic sections; that is, ellipses and hyperbolas or their degenerateforms (lines, parabolas, circles).For three dimensions, the separable coordinate systems arequadratic surfaces or their degenerate forms. A commoncoordinate system is the spherical coordinates. The coordinatesurfaces are spheres having centers at the origin, cones havingvertices at the origin, and planes through the z-axis. Robertson’scondition, which relates scale factors and the properties ofStäckel determinant, places a limit on the number of possiblecoordinate systems. Table 1 lists a number of common coordinatesystems. These coordinate systems are: rectangular coordinates,circular cylindrical coordinates, elliptic cylinder coordinates,parabolic cylinder coordinates, spherical coordinates, bipolarcoordinates, conical coordinates, parabolic coordinates, prolatespheroidal coordinates, oblate spheroidal coordinates, ellipsoidalFigure 2 Circular cylindrical coordinates. [Color figure can be viewedin the online issue, which is available at wileyonlinelibrary.com.]Figure 3 Elliptic cylindrical coordinates. [Color figure can be viewed inthe online issue, which is available at wileyonlinelibrary.com.]coordinates, bi-spherical coordinates, and toroidal coordinates.The names are generally descriptive of the coordinate systems.For example, the circular cylinder coordinates involve coordinatesurfaces, which are cylinder coaxial. Maple implements,respectively, about 15 coordinate systems in two dimensionsand 31 in three dimensions.Let us illustrate one useful application of using thesecoordinate systems. For example, if the boundary conditionsrequire the use of polar coordinates (shown in Fig. 1), theequation D2’ þ k2’ ¼ 0 (k is a constant), for example, can be split(making use of Tables 1 3) into two ordinary differentialequations, each for a single independent variabled2 f 1 df1þþ ðr 2 k2 2 Þf ¼ 0;dr 2 r dr r 2d2 gþ 2 g ¼ 0d 2ð1Þwhere ’(r, ) ¼ f(r)g( ) and is the separation constant (whichmust be integer in the case of polar coordinates since ’ is aperiodic coordinate). We will apply the method of separation ofvariables to the PDEs in the examples.Figure 4 Parabolic coordinates. [Color figure can be viewed in theonline issue, which is available at wileyonlinelibrary.com.]

TRANSFORMATIONS OF PDES367Table 1 Definition of Common Coordinate SystemsCircular cylindrical (polar)x ¼ cos , y ¼ sin , zcoordinates ( , , z)Elliptic cylindricalx ¼ d cosh u cos , y ¼ d sinh u sin , zcoordinates (u, , z)Parabolic cylindricalx ¼ (1/2)(u2 v2), y ¼ uv, zcoordinates (u, v, z)sinh vsin uBipolar coordinates (u, v, z)x¼a; y¼a; zcosh v cos ucosh v cos uSpherical coordinates (r, , y)x ¼ r cos sin ; y ¼ r sin sin ; z ¼ r cos ffiffiffiffiffiffiffiffiffiffi ð 2 a2 Þð 2 a2 Þ ð 2 b2 Þð 2 b2 Þ;z¼Conical coordinates ( , , )x¼; y¼abaa2 b2bb2 a21 22Parabolic coordinates (u, v, )x ¼ uvcos ; y ¼ uvsin ; z ¼ ðu v Þ2Prolate spheroidalcoordinates (u, v, )x ¼ d sinh u sin cos ;y ¼ d sinh u sin sin ;z ¼ d cosh u cos Oblate spheroidalcoordinates (u, v, )x ¼ d cosh u sin cos ;y ¼ d cosh u sin sin ;z ¼ d sinh u cos Ellipsoidal coordinates( 1, 2, 3)Paraboloidal coordinates( 1, 2, ffiffiffiffiffiffið 12 a2 Þð 22 a2 Þð 32 a2 Þ;x¼ða2 c2 Þða2 b2 ffiffiffiffiffið 12 c2 Þð 22 c2 Þð 32 c2 Þz¼ðc2 a2 Þðc2 b2 ffiffiffiffiffiffið 12 a2 Þð 22 a2 Þð 32 a2 Þx¼;ða2 b2 ffiffiffiffiffiffið 12 b2 Þð 22 b2 Þð 32 b2 Þy¼;ðb2 c2 Þðb2 a2 ffiffiffiffiffiffiffið 12 b2 Þð 22 b2 Þð 32 b2 Þ;ðb2 a2 Þ1z ¼ ð 12 þ 22 þ 32 a2 b2 Þ2Bispherical coordinates( , , )x ¼ a cos sin;cosh cosy ¼ a sin sin;cosh cosz¼asinh cosh cosToroidal coordinates( , , )x ¼ a cos sinh ;cosh cosy ¼ a sin sinh ;cosh cosz¼asincosh cosSo, the basic technique is to transform a given boundaryvalue problem in the xy plane (or x, y, z space) into a simpler onein the plane 1 2 (or 1 2 3 space) and then write the solution ofthe original problem in terms of the solution obtained for thesimpler equation. This is explained in the next three sections. Thetransformations will make the solution more tractable andconvenient to find.Nowadays, high-performance computers coupled withhighly efficient user-friendly symbolic computation softwaretools such as Maple, Mathematica, Matlab, Reduce [http:www.maplesoft.com, http://www. wolfram.com, http://www.mathworks.com, http://www. reduce-algebra.com] are very usefulin teaching mathematical methods involving tedious algebra andmanipulations. In this paper, the powerful software tool Mapleis used. Maple facilitates the manipulation and derivation ofanalytical expressions, and can be used to perform tediousalgebra, complicated integrals, and differential equations [1 3].A secondary objective of this paper is to expose the studentto different skills in using Maple to perform the algebra and workwith differential equations calculations and solutions in differentcoordinate systems.For the sake of readers not familiar with Maple, a briefintroduction will follow. Maple is a powerful symbolic computational tool used to perform analytical derivations and numericalcalculations. It is easy to use, and its commands are oftenstraightforward to know even for a first-time user. In this paper,the student version of Maple is used. We recommend that thestudent uses ‘‘;’’ and not ‘‘:’’ at the end of a command statementso that Maple prints the results. This helps in fixing mistakes inthe program since the results are printed after every commandstatement. In addition, the user might have to manipulate theresulting expressions from a Maple command to obtain theequation in the simplest or desired form. All the mathematicalmanipulations involved can be performed in the same program,and Maple can be used to perform all the required steps fromsetting up the equations to interpreting plots in the same sheet.Please note that equations containing ‘‘:¼’’ are results printed byMaple.APPLICATION OF THE BIPOLARCOORDINATE SYSTEMThere are a number of real applications for bipolar coordinatessuch as pairs of ducts, pipes, transmission lines, and bubbles[4 9]. We will illustrate the use of the bipolar orthogonalcoordinate system in this section by the following example. Twoidentical circular pipes of identical radius R are imbedded inan infinite concrete slab as shown in Figure 1. The uniformtemperature of both pipes is T0. We want to solve the temperaturedistribution in the concrete slab by solving the followingdifferential Laplace equation: