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Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations with Engineering Applications

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Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations with Engineering Applications Computer Facilitated Generalized Coordinate Transformations of Partial Differential Equations With Engineering Applications A. ELKAMEL,1 F.H. BELLAMINE,1,2 V.R. SUBRAMANIAN3 1Department of Chemical Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 2National Institute of Applied Science and Technology in Tunis, Centre Urbain Nord, B.P. No. 676, 1080 Tunis Cedex, Tunisia 3Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505 Received 16 February 2008; accepted 2 December 2008 ABSTRACT: Partial differential equations (PDEs) play an important role in describing many physical, industrial, and biological processes. Their solutions could be considerably facilitated by using appropriate coordinate transformations. There are many coordinate systems besides the well-known Cartesian, polar, and spherical coordinates. In this article, we illustrate how to make such transformations using Maple. Such a use has the advantage of easing the manipulation and derivation of analytical expressions. We illustrate this by considering a number of engineering problems governed by PDEs in different coordinate systems such as the bipolar, elliptic cylindrical, and prolate spheroidal. In our opinion, the use of Maple or similar computer algebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommon transformations more frequently at the very least for situations where the transformations provide smarter and easier solutions. ß2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365À376, 2011; View this article online at wileyonlinelibrary.com; DOI 10.1002/cae.20318 Keywords: partial differential equations; symbolic computation; Maple; coordinate transformations INTRODUCTION usual Cartesian, polar, and spherical coordinates. For example, Figure 1 shows two identical pipes imbedded in a concrete slab. To Partial differential equations (PDEs) are mathematical models find the steady-state temperature, the most suitable coordinate describing physical laws such as chemical processes, electrostatic system is the bipolar coordinate system as will be demonstrated in distributions, heat flow, and fluid motion. The solution to a number more details in Application of the Bipolar Coordinate System of PDEs is shaped by the boundaries of the geometry, and thus the Section. In addition, the separation of variables is a common method coordinate system selected is influenced by these boundaries. By for solving linear PDEs. The separation is different for different choosing a curvilinear coordinate system (1, 2, 3) such that the coordinate systems. In other words, we find out in what coordinate boundary surface is one of the coordinate surfaces, it is possible to system an equation will be amenable to a separation of variables express the solution of the PDE in terms of these new coordinates, solution. The properties of the solution can be related to the that is, 1, 2, 3. There are many coordinate systems besides the characteristics of the equations and the geometry of the selected coordinate system. It is possible to prove using the theory of analytic functions of the complex variable that there are a number of two- Correspondence to A. Elkamel ([email protected]). and three-dimensional separable coordinate systems. The coordinate ß 2009 Wiley Periodicals Inc. system is defined by relationships between the rectangular 365 366 ELKAMEL, BELLAMINE, AND SUBRAMANIAN Figure 1 Two identical pipes imbedded in an infinite concrete slab. coordinates (x, y, z) and the coordinates (1, 2, 3). The new coordinate 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 circular boundaries 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 as illustrated in Figure 2. Coordinate systems more general than Figure 3 Elliptic cylindrical coordinates. [Color figure can be viewed in the polar are the elliptic coordinates consisting of ellipses and the online issue, which is available at wileyonlinelibrary.com.] hyperbolas. These coordinates are suitable for elliptic boundaries or ones consisting of hyperbolas as illustrated in Figure 3. Parabolic cylindrical coordinates, shown in Figure 4, are two coordinates, bi-spherical coordinates, and toroidal coordinates. orthogonal families of parabolas, with axes along the x-axis. The names are generally descriptive of the coordinate systems. These coordinates are suitable, for example, for a boundary For example, the circular cylinder coordinates involve coordinate consisting of the negative half of the x-axis. Generally speaking, surfaces, which are cylinder coaxial. Maple implements, the separable coordinate systems for two dimensions consisted of respectively, about 15 coordinate systems in two dimensions conic sections; that is, ellipses and hyperbolas or their degenerate and 31 in three dimensions. forms (lines, parabolas, circles). Let us illustrate one useful application of using these For three dimensions, the separable coordinate systems are coordinate systems. For example, if the boundary conditions quadratic surfaces or their degenerate forms. A common require the use of polar coordinates (shown in Fig. 1), the coordinate system is the spherical coordinates. The coordinate equation D2’ þ k2’ ¼ 0(k is a constant), for example, can be split surfaces are spheres having centers at the origin, cones having (making use of Tables 1À3) into two ordinary differential vertices at the origin, and planes through the z-axis. Robertson’s equations, each for a single independent variable condition, which relates scale factors and the properties of Sta¨ckel determinant, places a limit on the number of possible 2 2 d f 1 df 1 2 2 2 d g 2 coordinate systems. Table 1 lists a number of common coordinate þ þ ðr k À Þf ¼ 0; þ g ¼ 0 ð1Þ dr2 r dr r2 d2 systems. These coordinate systems are: rectangular coordinates, circular cylindrical coordinates, elliptic cylinder coordinates, where ’(r, ) ¼ f(r)g() and is the separation constant (which parabolic cylinder coordinates, spherical coordinates, bipolar must be integer in the case of polar coordinates since ’ is a coordinates, conical coordinates, parabolic coordinates, prolate periodic coordinate). We will apply the method of separation of spheroidal coordinates, oblate spheroidal coordinates, ellipsoidal variables to the PDEs in the examples. Figure 2 Circular cylindrical coordinates. [Color figure can be viewed Figure 4 Parabolic coordinates. [Color figure can be viewed in the in the online issue, which is available at wileyonlinelibrary.com.] online issue, which is available at wileyonlinelibrary.com.] TRANSFORMATIONS OF PDES 367 Table 1 Definition of Common Coordinate Systems Circular cylindrical (polar) x ¼ cos , y ¼ sin , z coordinates (, , z) Elliptic cylindrical x ¼ d cosh u cos , y ¼ d sinh u sin , z coordinates (u, , z) Parabolic cylindrical x ¼ (1/2)(u2Àv2), y ¼ uv, z coordinates (u, v, z) sinh v sin u Bipolar coordinates (u, v, z) x ¼ a ; y ¼ a ; z cosh v À cos u cosh v À cos u Spherical coordinates (r, , y) x ¼ r cos sin ; y ¼ r sin sin ; z ¼ r cos rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 À a2Þð2 À a2Þ ð2 À b2Þð2 À b2Þ Conical coordinates (, , ) x ¼ ; y ¼ ; z ¼ ab a a2 À b2 b b2 À a2 1 Parabolic coordinates (u, v, ) x ¼ uvcos; y ¼ uvsin; z ¼ ðu2 À v2Þ 2 Prolate spheroidal x ¼ d sinh u sin cos ; y ¼ d sinh u sin sin ; z ¼ d cosh u cos coordinates (u, v, ) Oblate spheroidal x ¼ d cosh u sin cos ; y ¼ d cosh u sin sin ; z ¼ d sinh u cos coordinates (u, v, ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 À a2Þð2 À a2Þð2 À a2Þ ð2 À b2Þð2 À b2Þð2 À b2Þ x ¼ 1 2 3 ; y ¼ 1 2 3 ; ða2 À c2Þða2 À b2Þ ðb2 À c2Þðb2 À a2Þ Ellipsoidal coordinates sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( , , ) ð2 À c2Þð2 À c2Þð2 À c2Þ 1 2 3 z ¼ 1 2 3 ðc2 À a2Þðc2 À b2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 À a2Þð2 À a2Þð2 À a2Þ ð2 À b2Þð2 À b2Þð2 À b2Þ x ¼ 1 2 3 ; y ¼ 1 2 3 ; ð 2 À 2Þ ð 2 À 2Þ Paraboloidal coordinates a b b a ( , , ) 1 1 2 3 z ¼ ð2 þ 2 þ 2 À a2 À b2Þ 2 1 2 3 sin sin sinh Bispherical coordinates x ¼ a cos ; y ¼ a sin ; z ¼ a cosh À cos cosh À cos cosh À cos (, , ) sinh sinh sin Toroidal coordinates x ¼ a cos ; y ¼ a sin ; z ¼ a cosh À cos cosh À cos cosh À cos (, , ) So, the basic technique is to transform a given boundary the student version of Maple is used. We recommend that the value problem in the xy plane (or x, y, z space) into a simpler one student uses ‘‘;’’ and not ‘‘:’’ at the end of a command statement in the plane 12 (or 123 space) and then write the solution of so that Maple prints the results. This helps in fixing mistakes in the original problem in terms of the solution obtained for the the program since the results are printed after every command simpler equation. This is explained in the next three sections. The statement. In addition, the user might have to manipulate the transformations will make the solution more tractable and resulting expressions from a Maple command to obtain the convenient to find. equation in the simplest or desired form. All the mathematical Nowadays, high-performance computers coupled with manipulations involved can be performed in the same program, highly efficient user-friendly symbolic computation software and Maple can be used to perform all the required steps from tools such as Maple, Mathematica, Matlab, Reduce [htt setting up the equations to interpreting plots in the same sheet. p:www.maplesoft.com, http://www. wolfram.com, http://www. Please note that equations containing ‘‘:¼’’ are results printed by mathworks.com, http://www. reduce-algebra.com] are very useful Maple. in teaching mathematical methods involving tedious algebra and manipulations. In this paper, the powerful software tool Maple is used.
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