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: 365 376, 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 1 3) 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 Þf ¼ 0; þ g ¼ 0 ð1Þ dr2 r dr r2 d 2 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)(u2 v2), 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 cos u cosh v cos u Spherical coordinates (r, , y) x ¼ r cos sin ; y ¼ r sin sin ; z ¼ r cos rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 a2Þð 2 a2Þ ð 2 b2Þð 2 b2Þ Conical coordinates ( , , ) x ¼ ; y ¼ ; z ¼ ab a a2 b2 b b2 a2 1 Parabolic coordinates (u, v, ) x ¼ uvcos ; y ¼ uvsin ; z ¼ ðu2 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 ð 2 a2Þð 2 a2Þð 2 a2Þ ð 2 b2Þð 2 b2Þð 2 b2Þ x ¼ 1 2 3 ; y ¼ 1 2 3 ; ða2 c2Þða2 b2Þ ðb2 c2Þðb2 a2Þ Ellipsoidal coordinates sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( , , ) ð 2 c2Þð 2 c2Þð 2 c2Þ 1 2 3 z ¼ 1 2 3 ðc2 a2Þðc2 b2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 a2Þð 2 a2Þð 2 a2Þ ð 2 b2Þð 2 b2Þð 2 b2Þ x ¼ 1 2 3 ; y ¼ 1 2 3 ; ð 2 2Þ ð 2 2Þ Paraboloidal coordinates a b b a ( , , ) 1 1 2 3 z ¼ ð 2 þ 2 þ 2 a2 b2Þ 2 1 2 3 sin sin sinh Bispherical coordinates x ¼ a cos ; y ¼ a sin ; z ¼ a cosh cos cosh cos cosh cos ( , , ) sinh sinh sin Toroidal coordinates x ¼ a cos ; y ¼ a sin ; z ¼ a cosh cos cosh cos cosh 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 1 2 (or 1 2 3 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. Maple facilitates the manipulation and derivation of APPLICATION OF THE BIPOLAR analytical expressions, and can be used to perform tedious COORDINATE SYSTEM algebra, complicated integrals, and differential equations [1 3]. A secondary objective of this paper is to expose the student There are a number of real applications for bipolar coordinates to different skills in using Maple to perform the algebra and work such as pairs of ducts, pipes, transmission lines, and bubbles with differential equations calculations and solutions in different [4 9]. We will illustrate the use of the bipolar orthogonal coordinate systems. coordinate system in this section by the following example. Two For the sake of readers not familiar with Maple, a brief identical circular pipes of identical radius R are imbedded in introduction will follow. Maple is a powerful symbolic computa- an infinite concrete slab as shown in Figure 1. The uniform tional tool used to perform analytical derivations and numerical temperature of both pipes is T0. We want to solve the temperature calculations. It is easy to use, and its commands are often distribution in the concrete slab by solving the following straightforward to know even for a first-time user. In this paper, differential Laplace equation: 368 ELKAMEL, BELLAMINE, AND SUBRAMANIAN
Table 2 Scale Factors for Common Coordinate Systems
Circular cylindrical h1 ¼ 1, h2 ¼ , h3 ¼ 1 coordinates ( , , z) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Elliptic cylindrical h1 ¼ d sinh u þ sin ; h2 ¼ h1; h3 ¼ 1 coordinates (u, , z) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Parabolic cylindrical h1 ¼ d u þ v ; h2 ¼ h1; h3 ¼ 1 coordinates (u, v, z) Bipolar coordinates ( , , z) h ¼ ; h ¼ h ; h ¼ 1 1 cosh cos 2 1 3 Spherical coordinate (r, y, ) h ¼1, h ¼ r sin y, h ¼ r 1 2 s3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 2Þ ð 2 2Þ Conical coordinates ( , , ) h ¼ 1; h ¼ ; h ¼ 1 2 ð 2 a2Þðb2 2Þ 3 ð 2 a2Þð 2 b2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Parabolic coordinate (u, v, ) h1 ¼ þ ; h2 ¼ h1; h3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Prolate spheroidal h1 ¼ d sinh u þ sin ; h2 ¼ h1; h3 ¼ d sinh u sin coordinates (u, v, ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Oblate spheroidal h1 ¼ d sinh u þ sin ; h2 ¼ h1; h3 ¼ d cosh u sin coordinates (u, v, ) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 2Þð 2 2Þ ð 2 2Þð 2 2Þ Ellipsoidal coordinates ¼ 1 2 1 3 ; ¼ 2 1 2 3 ; h1 2 2 2 h2 2 2 2 ( 1, 2, 3) ð a2Þð b2Þð c2Þ ð a2Þð b2Þð c2Þ s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 1 2 2 2 ð 2 2Þð 2 2Þ h ¼ 3 1 3 2 3 ð 2 2Þð 2 2Þð 2 2Þ 3 a 3 b 3 c sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð 2 2Þð 2 2Þ ð 2 2Þð 2 2Þ Paraboloidal coordinates ¼ 1 2 1 3 ; ¼ 2 1 2 3 ; h1 2 2 h2 2 2 ( 1, 2, 3) ð a2Þð b2Þ ð a2Þð b2Þ s1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 ð 2 2Þð 2 2Þ h ¼ 3 1 3 2 3 ð 2 2Þð 2 2Þ 3 a 3 b Bispherical coordinates a a h1 ¼ ; h2 ¼ ; ( , , ) ðcosh cos Þ sinh ðcosh cos Þ sin a sin h ¼ 3 ðcosh cos Þ sin
a a sinh Toroidal coordinates h ¼ ; h ¼ h ; h ¼ 1 cosh cos 2 1 3 cosh cos ( , , )
> c sinðuðx; yÞÞ restart: with(student): eq2 :¼ y ¼ > eq :¼ diff(T(x,y),x$2) þ coshðvðx; yÞÞ cosðuðx; yÞÞ diff(T(x,y),y$2) ¼ 0; We will show how tractable and convenient it is to solve the @2 @2 Laplace equation in the bipolar coordinates, whereas in terms of eq :¼ Tðx; yÞ þ Tðx; yÞ ¼ 0 @x2 @y2 x, y, and z the temperature field expression is complex. So, the bipolar coordinates are the ‘‘natural’’ coordinates for this type of The boundary conditions of this problem dictate the use of problem. bipolar coordinates (see Fig. 5). According to Table 1, the First of all, we show that the bipolar coordinate system is an following equations define bipolar coordinates: orthogonal coordinate system. This means that the two families of the coordinate surfaces u(x, y) and v(x, y) are mutually c sinh v c sin u orthogonal. The lines of intersection of these surfaces constitute x ¼ ; y ¼ ð2Þ cosh v cos u cosh v cos u two families of lines. At the point (u, v), we have unit vectors ~e1 and ~e2 each, respectively, tangent to the coordinate line of the > eq1 :¼ x c*sinh(v(x,y))/ bipolar coordinate system which goes through the point. Since the (cosh(v(x,y)) cos(u(x,y))); coordinate system is orthogonal, ~e1 and ~e2 are mutually perpendicular everywhere ð ð ; ÞÞ :¼ c sinh v x y 1 @~r 1 @~r eq1 x ð ð ; ÞÞ ð ð ; ÞÞ ~e1 ~e2 ¼ 0or ¼ 0 ð3Þ cosh v x y cos u x y h1 @u h2 @v ~r > eq2 :¼ y ¼ c*sin(u(x,y))/ where is a position vector and is given by (cosh(v(x,y)) cos(u(x,y))); ~r ¼ x~e1 þ y~e2 ð4Þ TRANSFORMATIONS OF PDES 369
Table 3 Line and Volume Elements Along With Differential Operators in Curvilinear Orthogonal Coordinate System sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 1 @ 2 @ 2 @ 2 @ 2 @ 2 @ 2 x y z 4 n n n 5 Scale factors, hn hn ¼ þ þ ¼ þ þ @ n @ n @ n @x @y @z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2ð Þ2 Line element, ds ds hn d n
Volume element, dV dV ¼ h1h2h3d 1d 2d 3 1 @ 1 @ 1 @ Gradient, r r ¼ ~e1 þ ~e2 þ ~e3 h1 @ 1 h2 @ 2 h3 @ 3 X3 @ r ~ r ~ ¼ 1 An Divergence, A A @ h1h2h3 h1h2h3 n¼1 1 hn X @ @ r ~ r ~ ¼ 1 ~ ð Þ ð Þ ; ; ; ¼ ; ; ; ; ; ; ; Curl, A A hmem @ hpAp @ hnAn m n p 1 2 3 or 2 3 1or3 1 2 h1h2h3 m;n;p n p 1 X3 @ h h h @ Laplacian, r2 r2 ¼ 1 2 3 @ 2 @ h1h2h3 n¼1 n hn n
h1 and h2 are scale factors for the bipolar coordinates u and v.We > vx :¼ simplify(subs(diff will write about them later on Equation (2) becomes (u(x,y),x) ¼ ux,vx)); 1 @x @x @y @y cosðuðx; yÞÞ coshðvðx; yÞÞ 1 þ ¼ 0 ð5Þ h1h2 @u @v @u @v c > ¼ ¼ Equation (1) is a conformal transformation to x, y from u, v eq5 : diff(eq1,y):eq6 : diff (eq2,y):uy :¼ solve(eq5,diff coordinates. u, uy, vx, and vy are computed using Maple as ¼ follows: (u(x,y),y)):eq61 : simplify (subs(diff(u(x,y),y) ¼ uy,eq6)): > eq3 :¼ diff(eq1,x): vy :¼ solve(eq61,diff(v(x,y),y)); eq4 :¼ diff(eq2,x): ¼ sinðuðx; yÞÞ sinhðvðx; yÞÞ vx : solve(eq4,diff(v(x,y),x)): vy :¼ eq31 :¼ simplify(subs(diff(v(x,y),x) c ¼ ¼ vx,eq3)): ux : solve(eq31,diff uy :¼ simplify(subs(diff (u(x,y),x)); (v(x,y),y) ¼ vy,uy)); sinðuðx; yÞÞ sinhðvðx; yÞÞ ux :¼ cosðuðx; yÞÞ coshðvðx; yÞÞ 1 c uy :¼ c The Cauchy Riemann conditions are satisfied since:
simplify(ux vy);simplify(uy þ vx); 0 0 and thus we conclude that the bipolar coordinate system is orthogonal. Next, we obtain the scale factors hi (I ¼ 1, 2) given by
@~ ¼ r ; ¼ ¼ ð Þ hi @ 1 u and 2 v 6 i
The scale factor hi can be interpreted as follows: a change du in the bipolar coordinate system produces a displacement h1 du along the coordinate line. Now, we notice that the rate of displacement along u due to a displacement along the x-axis is
h1(@u/@x) which is the same as the rate of change of x due to a displacement h1 du. The same argument goes for the scale factor h2. The scales of the new coordinates and the change of scale Figure 5 Bipolar coordinates. [Color figure can be viewed in the online from point to point determine the important properties of the issue, which is available at wileyonlinelibrary.com.] coordinate system. The scale factors play a role in expressing the 370 ELKAMEL, BELLAMINE, AND SUBRAMANIAN
differential/integral operators, line, surface, and volume ele- > uxx :¼ diff(ux,x):uxx :¼ simplify ments, and for the sake of completeness, they are shown in (subs(diff(u(x,y),x) ¼ ux, Table 2 for common coordinate systems. After some algebraic diff(v(x,y),x) ¼ vx,uxx)): manipulations, we find that > vxx :¼ diff(vx,x):vxx :¼ simplify (subs(diff(u(x,y),x) ¼ ux, > h1square :¼ factor(simplify diff(v(x,y),x) ¼ vx,vxx)): (1/(ux^(2) þ uy^(2)))):h[1] : > uyy :¼ diff(uy,y):uyy :¼ simplify ¼ simplify(sqrt(h1square), (subs(diff(u(x,y),y) ¼ uy, sqrt,symbolic); diff(v(x,y),y) ¼ vy,uyy)): c > vyy :¼ diff(vy,y):vyy :¼ simplify h1 :¼ cosðuðx; yÞÞ coshðvðx; yÞÞ (subs(diff(u(x,y),y) ¼ uy, diff(v(x,y),y) ¼ vy,vyy)): So, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Then, we map the differential equation into the bipolar @x 2 @y 2 c h ¼ þ ¼ ð7Þ coordinates: 1 @u @u cosh v cos u > eq1 :¼ eval(subs(T(x,y) ¼ TT > h2square :¼ factor(simplify (u(x,y),v(x,y)),eq)): (1/(vx^(2) þ vy^(2)))): eq2 :¼ subs(diff(v(x,y),y$2) ¼ h[2] :¼ simplify(sqrt diff(v(x,y),x$2),diff (h2square),sqrt,symbolic); (u(x,y),y$2) ¼ diff(u(x,y),x$2), c diff(v(x,y),y) ¼ diff (u(x,y),x), h :¼ ¼ 2 coshðvðx; yÞÞ þ cosðuðx; yÞÞ diff(v(x,y),x) -diff (u(x,y),y),eq1): and so, eq3 :¼ factor(expand(eq2)): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eq4 :¼ D[2,2](TT)(u(x,y),v(x,y)) @ 2 @ 2 x y c þD[1,1](TT)(u(x,y),v(x,y)): h2 ¼ þ ¼ ð8Þ @v @v cosh v cos u eq5 :¼ subs(u(x,y) ¼ u,v(x,y) ¼ v,eq4):Eq :¼ convert(eq5,diff); Next, we need to obtain expressions for the parameter c in terms of R and L. The y-axis lies in the middle between the two @2 @2 cylinders, while the x-axis crosses the centers of the cylinders. Eq :¼ TTðu; vÞ þ TTðu; vÞ @u2 @v2 After algebraic manipulations, the relationship between the Cartesian and bipolar coordinates can be expressed as follows: which can be rewritten as
x2 þðy c cot uÞ2 ¼ c2 csc2 u ð9Þ @2T @2T þ ¼ 0 ð14Þ 2 @u2 @v2 ðx c coth vÞ þ y2 ¼ c2 csc h2v ð10Þ which shows that Laplace’s equation is invariant under a mapping This form of writing the relationship between the ‘‘old’’ set to bipolar coordinates. Now, if T changes with respect to z as well, of coordinates, that is, x, y and the ‘‘new’’ set of coordinates u, v then the transformed PDE can be shown to be provides us a further insight since we can notice that Equations (9) and (10) are circles. For an arbitrary v ¼ 0, from Equation h 1 @2T @2T @2T (10) we have a circle of radius c csc h 0 and center (c coth 0, 0). þ þ ¼ 0 ð15Þ h h @u2 @v2 @z2 Also, when v ¼ 0, we have a circle of radius c csc hh0 and 1 2 center ( c coth 0, 0). Now, when but since the temperature does not change with respect to z, ¼ ð Þ @ @ ¼ c csc h 0 R 11 that is, T/ z 0, then Equation (14) is reduced to (13). The transformed boundary conditions in the u, v coordinates are and, T ¼ T at v ¼ ; T ¼ T at v ¼ ð16Þ L 1 0 2 0 c coth ¼ þ R ð12Þ 0 2 At this point, we can use the method of separation of variables covered in many undergraduate and graduate textbooks Then [10,11] (so T(u, v) ¼ X(u)Y(v)). The separation of variables splits L the PDE into two ODEs. We assume that the temperature T(u, v) c ¼ R sinh cosh 1 1 þ ð13Þ 2R is only dependent on v. So, in our particular case, the separation constant is ln ¼ 0. So, we end up solving the following The obvious next step is to transform the differential differential equation with its Dirichlet boundary conditions: equation from x, y to u, v coordinates. Using Maple, we can simply use the Laplacian command. However, Laplace form does d2 not govern many differential equations, and so we will follow the > Eq: ¼ diff(Y(v),v$2); YðvÞ¼0 v2 approach needed to map a differential equation from a set of d > ¼ ¼ ¼ coordinates to another. So, first, we need to get the second bc1 : subs(v eta[0],Y(v)) T[1]; ¼ ð Þ¼ derivatives of u, v with respect to x, y bc1 : Y h0 T1 TRANSFORMATIONS OF PDES 371
> bc2 :¼ subs(v ¼ eta[0], Y(v)) ¼ T[2]; products, and for which the separation constant is not zero: bc2 :¼ Yð h Þ¼T þ 0 2 ð ; Þ¼T1 T2 þ T1 T2 T u v v 2 0 2 > sol :¼ dsolve(Eq,Y(v)): X1 ¼ ¼ nv nv eq1 : subs(v eta[0], þ ðcne þ dne Þ sinðnuÞ Y(eta[0]) ¼ T[1],sol): n¼1 eq2 :¼ subs(v ¼ eta[0], where c and d are computed by Maple as follows: n n T1 T2 1 1 eq3 :¼ C1 ¼ ; C2 ¼ T2 þ T1 > an :¼ (2/Pi)*int(T[1]*sin(n*u), 2 0 2 2 u ¼ 0...Pi):bn :¼ (2/Pi)*int(T[2] *sin(n*u),u ¼ 0...Pi) : assume(n: Y( eta[0]) ¼ T[2],sol): :integer) : assume(n,odd): eq3 :¼ solve({eq1,eq2}, eval(an):eval(bn): eqc1 :¼ cn*exp {_C1,_C2}); (n*eta[0]) þ dn*exp( n*eta[0]) ¼ an: and thus: eqc2 :¼ cn*exp( n*eta[0]) þ dn *exp(n*eta [0]) ¼ bn:solc :¼ solve ðT1 T2Þv 1 1 YðvÞ¼ þ T2 þ T1 ({eqc1,eqc2},{cn,dn}); 2h0 2 2 ( ð nþ Þ ðnþ Þ 4ð e 0 T T e 0 which is equal to T(u, v). solc : ¼ cn ¼ ÀÁ2 1 ; ð nþ Þ 2 ðn Þ2 Generally, when we conduct the separation of variables, one n e 0 þ e 0 gets ) ð nþ Þ ðnþ Þ 4ð T e 0 T e 0 dn ¼ ÀÁ1 ÀÁ2 ð nþ Þ 2 ðn Þ 2 > eqs :¼ subs(TT(u,v) ¼ X(u)*Y n e 0 þ e 0 (v),Eq):eqs :¼ expand(eqs): eqs :¼ eqs/(X(u)*Y (v)): where n denotes an odd number (i.e. n ¼ 2m 1, m ¼ 1, 2...). eqs :¼ expand(eqs);
d2 d2 ð Þ YðvÞ APPLICATION OF THE ELLIPTIC CYLINDRICAL 2 X u 2 eqs :¼ du þ dv COORDINATE SYSTEM XðuÞ YðvÞ we obtain two ODEs in X(u)andY(v) and the first one is Another type of an orthogonal curvilinear coordinates system is the elliptic cylinder. The coordinate surfaces are elliptic cylinders > eqs1 :¼ diff(X(u),u$2) (u ¼ constant) and hyperbolic cylinders (v ¼ constant) in the two þ lambda^2*X(u): bc1 :¼ subs coordinate systems. Many problems are amenable to this type of (u ¼ 0,D(X)(u)) ¼ 0:bc2 : coordinate systems such as coils, solar, and heat concentrators, ¼ subs(u ¼ Pi,D(X)(u)) ¼ 0; metallurgical junctions, material flaw shapes, shells, fluid flow past an obstacle [12 17]. The transformations to the Cartesian d2 eqs1 :¼ ðuÞ þ 2 ðuÞ coordinates from the elliptic cylindrical coordinates is listed in du2 Table 1, and are x ¼ a coshðuÞ cosðvÞ; y ¼ a sinhðuÞ sinðvÞ bc1 :¼ DðXÞð0Þ¼0 where a is the length of the semi-major axis of the ellipse. For bc2 :¼ DðXÞðpÞ¼0 example, if an elliptical hole (Fig. 6) is cut in a region as we will see in this section as an example, then it is more tractable to solve The ODE in X(u) is a regular Sturm Liouville boundary the Laplace equation in the elliptic cylindrical coordinates. We l ¼ ¼ value problem [10], with separation constant n n (where n 1, take the center of the coordinate system to be that of the hole. The ... ¼ 2, ) with corresponding eigenfunctions Xn(u) sin(nu). The scale factors are given by ODE in Y(v)is > restart:with(student): > ¼ eqs2 : diff(Y(v),v$2) > x:¼ a*cosh(u)*cos(v): n^2*Y(v); y:¼ a*sinh(u)*sin(v):h[u] :¼ sqrt (diff(x,u)^2 þ diff(y,u)^2):h[u] d2 ¼ ¼ eqs2 :¼ YðvÞ n2YðvÞ : simplify(h[u]):h[u] : subs dv2 (cosh(u)^2 ¼ 1 þ sinh(u)^2, cos(v)^2 ¼ 1 sin(v)^2,h[u]): h[u] :¼ simplify(h[u],sqrt, > dsolve(eqs2); symbolic); YðvÞ¼ C1eð nvÞ þ C2eðn;vÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In order to satisfy the boundary conditions (Eq. 16), we :¼ ð Þ2 þ ð Þ2 consider the following solution which is a linear combination of hu a sin v sinh u 372 ELKAMEL, BELLAMINE, AND SUBRAMANIAN
Likewise, we obtain hv bc1 :¼ subs(u ¼ infinity, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi phi(u,v)) ¼ phi[0] þ E 2 2 hv :¼ a sinðvÞ þ sinhðuÞ [0]*sinh(u)*sin(v); :¼ ð1; Þ¼ þ ð Þ ð Þ The Laplace equation in elliptic cylindrical coordinates, bc1 v 0 E0 sinh u sin v ¼ assuming a 1, is bc2 :¼ subs(u ¼ u[0], D[ 1](phi)(u,v)) ¼ 0; > with(linalg): :¼ ð Þð ; Þ¼ > eq1 :¼ laplacian(phi(u,v), bc2 D1 u0 v 0 ¼ [u,v],coords elliptic): Assuming that (u, v) ¼ A þ A sinh u sin v þ A cosh sin v, ¼ 0 1 1 eq1 : numer(eq1); then enforcing the first boundary condition and since sin- h u ¼ cosh u for large u, we have @2 @2 eq1 :¼ ðu; vÞ þ ðu; vÞ @u2 @v2 eq5 :¼ phi(u,v) ¼ _A0 þ _A1* sinh(u)*sin(v) þ _A2*cosh ¼ Using separation of variables, we have (u, v) 1(u, (u)*sin(v) : eq6 :¼ subs v) 2(u, v): (u ¼ infinity,_A0 ¼ phi[0], eq5):eq7 :¼ phi[0] þ E[0]* > eq2 :¼ subs(phi(u,v) ¼ phi[1] sinh(u)*sin(v) ¼ phi[0] þ (u)*phi[2](v),eq1) : eq3 :¼ (_A1 þ _A2)*sinh(u)*sin(v); expand(eq2):eq4 :¼ eq3* eq7 : ¼ þ E sinhðuÞ sinðvÞ¼ (1/(phi[1](u)*phi[2](v))): 0 0 0 þð þ Þ ð Þ ð Þ eq4 :¼ expand(eq4); A1 A2 sinh u sin v
We assume that A0 ¼ E0 which is equal to the potential at @2 @2 ¼ ð Þ ð Þ y 0 as shown in Figure 3. The second boundary condition gives 1 u 2 v :¼ @u2 þ @v2 eq4 ð Þ ð Þ 1 u 2 v phi(u,v) :¼ _A0 þ _A1*sinh (u)*sin(v) þ _A2*cosh(u)*sin(v): We end up with two differential equations for 1(u, v) and eq8 :¼ diff(phi(u,v),u): 2(u, v). eq8 :¼ subs(u ¼ u[0],eq8) ¼ 0; eq41 :¼ diff(phi[1](u),u$2) : ¼ ð Þ ð Þ p^2*phi[1](u); eq8 A1 cosh u0 sin v þ A2 sinhðu Þ sinðvÞ¼0 @2 0 :¼ ð Þ 2 ð Þ eq41 2 1 u p 1 u @u So, now we can solve for the constants A1 and A2:
eq42 :¼ diff(phi[2](v),v$2) þ p^2*phi[2](v); eq9 :¼ solve({eq7, eq8}, {_A1,_A2}); @2 eq42 :¼ ðvÞ p2 ðvÞ @v2 2 2 ð Þ : ¼ ¼ E0 sinh u0 The two boundary conditions labeled bc1 and bc2 are such eq9 A1 coshðu0Þ sinhðu0Þ that: coshðu ÞE A2 ¼ 0 0 coshðu0Þ sinhðu0Þ
Substituting A1 and A2 in the potential (u, v), one obtains
eq10 :¼ subs(_A0 ¼ phi[0],eq5): eq11 :¼ subs(eq9,eq10); E sinhðu0Þ sinhðuÞ sinðvÞ eq11 : ¼ 0 0 coshðu0Þ sinhðu0Þ ð Þ ð Þ ð Þ þ cosh u0 E0 cosh u sin v coshðu0Þ sinhðu0Þ
The electrostatic field E is the gradient of the potential, and thus is given by
¼ Figure 6 The shape of the elliptic hole which is illuminated with the E(u,v) : grad(rhs(eq11), ¼ field E0. [u,v],coords elliptic); TRANSFORMATIONS OF PDES 373
2 ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ E0 sinh u0 cosh u sin v þ cosh u0 E0 sinh u sin v 6 coshðu0Þ sinhðu0Þ coshðu0Þ sinhðu0Þ Eðu; vÞ : ¼ 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; sinhðuÞ2 þ sinðvÞ2 ð Þ ð Þ ð Þ ð Þ ð Þ ð Þ E0 sinh u0 sinh u cos v þ cosh u0 E0 cosh u cos v ð Þ ð Þ ð Þ ð Þ cosh u0 sinhqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu0 cosh u0 sinh u0 sinhðuÞ2 þ sinðvÞ2
A field plot is shown Figure 7 reflecting the elliptic transformations. When ¼ cosh(u), ¼ cos v, then one gets cylindrical coordinate system. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ d ð 2 1Þð1 2Þ cosð Þ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 APPLICATION OF THE PROLATE y ¼ d ð 1Þð1 2Þ sinð Þ; z ¼ d SPHEROIDAL COORDINATES where 1 1, 1 1, 0 2p. To compute the volume and surface area in the prolate In three dimensions, as shown in Table 1, there are a number of spheroidal coordinate system, the Jacobian matrix associated three-dimensional coordinates. In here, we will give as an with the coordinate transformation must be calculated. The example the case of the prolate spheroidal coordinates system Jacobian matrix elements are the partial derivatives of the illustrated in Figure 8. This type of coordinates system has been transformation from prolate spherical to Cartesian coordinates. used extensively in many fields. For example, raindrops, dust So, the infinitesimal change in volume is the determinant of the grains in plasma, molten regions in laser welding, ground rod Jacobian matrix: connection, conducting electrodes, ventricle, diatomic molecules, hydrogen molecular ions, and biological cells [11,18 24] restart:with(student): are modeled as prolate spheroidal objects. As an example, the with(linalg):with (PDEtools): shape of a football ball is a prolate spheroid. The transformation T:¼ [d*sinh(u)*sin(v) to the Cartesian coordinates from the prolate spheroidal *cos(phi),d*sinh(u)*sin(v) coordinates is listed in Table 1. We will repeat here for the sake *sin(phi),d*cosh(u) of clarity: *cos(v)]:J :¼ jacobian x ¼ d sinhðuÞ sinðvÞ cosð Þ; (T,[u,v,phi]):dV :¼simplify y ¼ d sinhðuÞ sinðvÞ sinð Þ; z ¼ d coshðuÞ cosðwÞ (det(J))*d(u)*d(v)*d(phi); where d is the focal length for the prolate spheroidal system. : ¼ 3 ð Þ ð Þð ð Þ2 However, we need to be aware that there are other equivalent dV d sin v sinh u cosh u cosðvÞ2Þ dðuÞ dðvÞ dð Þ
Similarly, for the computation of surfaces, we need to compute the two-dimensional Jacobian matrix in the desired direction.
Figure 7 Field plot of E(u, v/E0) which reflects the elliptic coordinate Figure 8 Prolate spherical coordinates. [Color figure can be viewed in system. the online issue, which is available at wileyonlinelibrary.com.] 374 ELKAMEL, BELLAMINE, AND SUBRAMANIAN
In this section, we want to obtain an expression for the change of variables: equilibrium temperature distribution of a metal spheroid. Let us lð Þ :¼ coshð Þ suppose that because of an internal heating system within the spheroid ( ¼ ), its surface temperature is i mð Þ :¼ cosð Þ ¼ 10 þ 30 cos2 Then, we will get the following differential equations: The prolate spheroid made of metal is immersed in a large eq430 :¼ changevar(Xi(xi) ¼ container filled with insulating powder. The ambient temperature f(lambda(xi)),eq43):eq431 : is 208C. So, basically, we need to solve for the Laplace equation ¼ numer(simplify(subs(lambda in prolate spheroidal coordinates: (xi) ¼ cosh(xi),eq430))):eq432 : ¼ subs(cosh(xi) ¼ lambda,eq431): eq1 :¼ laplacian(psi(xi, eq432 :¼ convert(eq432,diff): eta,phi),[xi,eta,phi], eq432a :¼ collect(eq432,diff(f coords ¼ prolatespheroidal(a)): (lambda),lambda$2)):eq432b : eq2 :¼ expand(eq1): ¼ collect(eq432,diff(f(lambda), @ lambda)):eq432c :¼ collect cothð Þ @ cð ; ; Þ eq2 :¼ (eq432,f(lambda)):a :¼ factor 2ð ð Þ2 þ ð Þ2Þ a sinh sin (1 þ lambda^4 2*lambda^2):b : @2 cð ; ; Þ ¼ factor(2*lambda^3 2*lambda): @ 2 þ fac :¼ 1 lambda^2:a1 :¼ simplify 2 2 a2ðsinhð Þ þ sinð Þ Þ ¼ (a/fac):b1 : simplify(b/fac):c : ¼ þ @ @2 simplify((p^2*lambda^2 p^2 p cotð Þ @ cð ; ; Þ cð ; ; Þ þ þ @ 2 *lambda^2 p q^2)/fac):c1 : 2 2 2 2 a2ðsinhð Þ þ sinð Þ Þ a2ðsinhð Þ þ sinð Þ Þ ¼ collect(c,p^2):ode3 :¼ a1*diff þ @2 cð ; ; Þ (f(lambda),lambda$2) b1*diff þ @ 2 (f(lambda),lambda) þ c1*f(lambda); 2ð ð Þ2 þ ð Þ2Þ ð Þ2 ð Þ2 a sinh sin sin sinh @2 ode3 :¼ð1 l2Þ f ðlÞ @l2 The system of prolate spheroidal coordinate is separable. So writing the dependent variable C( , , ) as the product of three @ q2 2l f ðlÞ þ pðp þ 1Þþ f ðlÞ functions X( )H( )F( ) will split the Laplacian equation in three @l l2 1 ordinary differential equations: and the same Maple technique is used for the equation in H( ), eq3 :¼ subs(psi(xi,eta, @2 @ ode2 :¼ð1 m2Þ gðmÞ 2m gðmÞ phi) ¼ Xi(xi)*Eta(eta) @m2 @m *Phi(phi),eq2): q2 eq4 :¼ eq3*(1/(Xi(xi) þ pðp þ 1Þþ gðmÞ m2 *Eta(eta)*Phi(phi))): 1 ¼ eq4 : expand(eq4):eq4 : So, we need to solve the differential equations labeled above ¼ expand(a^2*eq4); by ode1, ode2, and ode3. We rename, respectively, f and g to X( ) and H( ). The three differential equations are: Phi :¼ rhs(dsolve(ode1, @2 ¼ ode1 :¼ Fð Þ þ q2Fð Þ¼0 Phi(phi)));alias(P LegendreP, @ 2 Q ¼ LegendreQ):f :¼ rhs(dsolve ¼ (ode3,f(lambda)));g : rhs @2 @ (dsolve(ode2,g(mu))): eq42 :¼ Hð Þ þ cotð Þ Hð Þ @ 2 @ Xi :¼ subs(lambda ¼ cosh(xi),f) ! :Eta :¼ subs(mu ¼ cos(eta),g); q2 þ pðp þ 1Þþ Hð Þ¼0 F :¼ ð Þþ ð Þ ð Þ2 C1 sin q C2 cos q sin X :¼ ð ; ; ð ÞÞ þ ð ; ; ð ÞÞ C1Q p q cosh C2P p q cosh @2 @ H :¼ C1Pðp; q; cosð ÞÞ þ C2Qðp; q; cosð ÞÞ eq43 :¼ Xð Þ þ cothð Þ Xð Þ @ 2 @ where _C1 and _C2 are constants. We notice that both X( ) and ! H q2 ( ) solutions are functions of associated Legendre functions of pðp þ 1Þþ Xð Þ¼0 the first and second kind. ð Þ2 sinh The solution C( , , ) has axial symmetry about the z-axis and so it is independent of and is given by C( , , q and p are the introduced separation variables. The differential ) ¼ X( )H( ). Let the boundary conditions be C( , 0, equations in H( ) and X( ) are transformed using the following ) ¼ M( ) ¼ 10 þ 30 cos2 , and hence m ¼ cos ranges only TRANSFORMATIONS OF PDES 375 between 1 and 1. The associated Legendre functions of the second kind Qn(cos ) are unbounded for m ¼ 1, and so the solution C( , ) depends only on the associated Legendre functions of the first kind illustrated in Figure 9 and has the form Pn(cos )Pn(cosh ), in other words:
N:¼ infinity;Psi(xi) :¼ sum(’A[i] *P(i,mu)*(P(i,cosh(mu))/ P(i,cosh(xi0)))’,’i’¼0...N); X1 ð ; mÞ ð ; ð ÞÞ Cð Þ :¼ Ai P i P i cosh ð ; ð ÞÞ i¼0 P i cosh 0
Enforcing the boundary condition that C( , 0, ) ¼ M( ), one obtains
Psi(xi) :¼ sum(’A[i]*P (i,mu)*(P(i,cosh(xi))/P (i,cosh(xi0)))’,’i’ ¼ 0...N): ð Þ ¼ M(mu) :¼ subs(xi ¼ xi0,Psi(xi)); Figure 10 Temperature for 2. X1 ðmÞ :¼ ð ; mÞ M AiP i A0 ¼ 20:0; A1 ¼ 0; A2 ¼ 20:00000000 i¼0 11 þ0:1; A3 ¼ 0 þ 0:1; A4 ¼ 0:450 10 But, when we truncate N to 5, þ0:1; A5 ¼ 0 þ 0:1 MðmÞ : ¼ A þ A m þ A Pð2; mÞþA Pð3; mÞ 0 1 2 3 Then, substituting the coefficients A into the solution C( , þ ð ; mÞþ ð ; mÞ n A4P 4 A5P 5 ), one gets and so to compute An, we will have to perform the following computation: sol1 :¼ subs(A[0] ¼ 20,A[2] Z ¼ 20,A[1] ¼ 0,A[3] ¼ 0,A[5] 1 1 ¼ 0,A[4] ¼ 0,mu ¼ cos(eta), An ¼ n þ MðmÞPnðmÞ dm 2 1 Psi(xi)); 20Pð2; cosð ÞÞPð2; coshð ÞÞ sol1 :¼ 20 þ > for i from 0 to 10 do A[i] ¼ evalf Pð2; coshð 0ÞÞ ((i þ 0.5)*int((10 þ 30*mu^2) C ¼ *P(i,mu),mu¼-1...1)); od; whose solution ( , ) is illustrated in Figure 10 for 2.
CONCLUSION
In this paper, examples of solving problems in coordinate systems other than the most famous Cartesian, polar, and spherical coordinates are given. In this article, we gave examples in the two-dimensional bipolar and elliptic cylindrical coordinates, and the three-dimensional prolate spheroidal coordinates. Maple proved to be very useful tool to perform the required trans- formations from one coordinate system to another, to simplify expressions, to manipulate the mathematical expressions, and to plot the solutions. The techniques used in this paper apply to other coordinate systems such as the parabolic cylindrical coordinates, conical coordinates, parabolic coordinates, oblate spheroidal coordinates, ellipsoidal coordinates, bispherical coordinates, and toroidal coordinates. The boundaries involved in solving the PDE provide us with the clue of the coordinate system to be used. Transforming the PDE to one of these coordinates makes the solution more tractable. For example, we saw that for the case of boundaries consisting of two circles, the most appropriate coordinate system to use is the bipolar coordinates. The aforementioned coordinates are the most commonly used because Figure 9 Legendre polynomials of the first kind P0(x) ...P4(x). [Color figure can be viewed in the online issue, which is available at they are separable coordinates. In other words, the method of wileyonlinelibrary.com.] separation of variables can be used to solve the PDE. However, 376 ELKAMEL, BELLAMINE, AND SUBRAMANIAN one needs to be aware that there are other coordinate systems such [11] H. Zahed, J. Mahmoudi, and S. Sobhanian, Analytical study of as the hyperbolodoidal, the exponential, and the three-dimen- spheroidal dusty grains in plasma, Phys Plasmas 13 (2006), sional bipolar coordinates, which are not considered as separable 053505 053505 6. coordinates, and they are currently the focus of our endeavors. [12] R. H. 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BIOGRAPHIES
Ali Elkamel is a professor of Chemical Fethi Bellamine is a professor at University Engineering at the University of Waterloo. November 7, Institute of Applied Science and He holds a BS in Chemical Engineering and a Technologies, Tunis, Tunisia. He holds BS, BS in Mathematics from Colorado School of MS, and PhD degrees in Electrical Engineering Mines, an MS in Chemical Engineering from from Colorado State and University of Colo- the University of Colorado-Boulder, and a PhD rado at Boulder, respectively. From 1995 to in Chemical Engineering from Purdue Uni- 2002, he served as a senior development versity. The goal of his research program is to engineer for Lucent Technologies, Alcatel develop theory and applications for PSE. The Networks, and NESA. His research interests applications focus on energy production planning, pollution preven- are in the areas of modeling and simulation, numerical methods, and tion, and product and process design. He is also interested in soft computing. integrating computing and process systems engineering in chemical engineering education. Prior to joining the University of Waterloo, he Dr. Venkat Subramanian is an associate served at Purdue University (Indiana), P&G (Italy), Kuwait professor in the Department of Chemical University, and the University of Wisconsin-Madison. He has Engineering at the Tennessee Technological contributed more than 200 publications in refereed journals and University. He received a BS degree in international conference proceedings. chemical and electrochemical engineering from the Central Electrochemical Research Institute in India and he received his PhD degree in Chemical Engineering from the University of South Carolina. His research interests include energy systems engineering, multiscale simulation and design of energetic materials, nonlinear model predictive control, batteries and fuel cells. He is the principal investigator of the Modeling, Analysis and Process-Control Laboratory for Electro- chemical Systems (MAPLE Lab, http://iweb.tntech.edu/ vsubramanian).