Computer FacilitatedGeneralized CoordinateTransformations of PartialDifferential Equations WithEngineering ApplicationsA.ELKAMEL,1F.H.BELLAMINE,1,2V.R.SUBRAMANIAN31DepartmentofChemicalEngineering,UniversityofWaterloo,200UniversityAvenueWest,Waterloo,Ontario,CanadaN2L3G12NationalInstituteofAppliedScienceandTechnologyinTunis,CentreUrbainNord,B.P.No.676,1080TunisCedex,Tunisia3DepartmentofChemicalEngineering,TennesseeTechnologicalUniversity,Cookeville,Tennessee38505Received16February2008;accepted2December2008ABSTRACT: Partial differential equations (PDEs) play an important role in describing many physical,industrial, andbiological processes. Their solutions couldbeconsiderably facilitatedby usingappropriatecoordinatetransformations. Therearemanycoordinatesystemsbesidesthewell-knownCartesian, 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 byconsideringanumber of engineeringproblems governedby PDEs indifferent coordinatesystems suchasthebipolar, ellipticcylindrical, andprolatespheroidal. Inour opinion, theuseof Mapleor similar computeralgebraic systems (e.g. Mathematica, Reduce, etc.) will help researchers and students use uncommontransformationsmorefrequentlyattheveryleastforsituationswherethetransformationsprovidesmarterandeasier solutions. 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 19: 365376, 2011; View this article online atwileyonlinelibrary.com;DOI10.1002/cae.20318Keywords: partialdifferentialequations;symboliccomputation;Maple;coordinatetransformationsINTRODUCTIONPartial differential equations (PDEs) are mathematical modelsdescribingphysical lawssuchaschemical processes, electrostaticdistributions, heat ow, and uid motion. The solution to a numberof PDEs is shaped by the boundaries of the geometry, and thus thecoordinatesystemselectedisinuencedbytheseboundaries. Bychoosingacurvilinearcoordinatesystem(1,2,3)suchthattheboundary surface is one of the coordinate surfaces, it is possible toexpress the solution of the PDE in terms of these new coordinates,thatis, 1, 2, 3. Therearemanycoordinatesystemsbesidestheusual 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 isthebipolarcoordinate systemas willbedemonstratedinmore details in Application of the Bipolar Coordinate SystemSection. In addition, the separation of variables is a common methodfor solvinglinear PDEs. Theseparationisdifferent for differentcoordinate systems. In other words, we find out in what coordinatesystemanequationwill beamenabletoaseparationofvariablessolution. The properties of the solution can be related to thecharacteristicsof theequationsandthegeometryof theselectedcoordinate system. It is possible to prove using the theory of analyticfunctionsofthecomplexvariablethatthereareanumberoftwo-and three-dimensional separable coordinate systems. The coordinatesystem is defined by relationships between the rectangularCorrespondencetoA.Elkamel(aelkamel@cape.uwaterloo.ca).2009Wiley PeriodicalsInc.365coordinates (x, y, z) and the coordinates (1, 2, 3). The newcoordinateaxesaregivenbytheequations1(x, y, z) constant,2(x, y, z) constant, and 3(x, y, z) constant.For example, polar coordinates are useful for circularboundariesoronesconsistingoftwolinesmeetingat anangle(see Fig. 2). The families r constant and constant are,respectively, the concentric circles and the radial lines asillustratedinFigure2. Coordinate systems moregeneral thanthepolar aretheellipticcoordinatesconsistingof ellipsesandhyperbolas. These coordinates are suitable for elliptic boundariesor ones consisting of hyperbolas as illustrated in Figure 3.Parabolic cylindrical coordinates, showninFigure 4, are twoorthogonal families of parabolas, with axes along the x-axis.These coordinates are suitable, for example, for a boundaryconsisting of the negative halfof the x-axis. Generallyspeaking,the separable coordinate systems for two dimensions consisted ofconic sections; that is, ellipses and hyperbolas or their degenerateforms(lines,parabolas,circles).Forthreedimensions,theseparablecoordinatesystemsarequadratic surfaces or their degenerate forms. A commoncoordinatesystemisthespherical coordinates. Thecoordinatesurfacesaresphereshavingcentersat theorigin, coneshavingverticesattheorigin,andplanesthroughthez-axis.Robertsonscondition, which relates scale factors and the properties ofStackel determinant, places alimit onthenumber of possiblecoordinate systems. Table 1 lists a number of common coordinatesystems.Thesecoordinatesystemsare:rectangularcoordinates,circular cylindrical coordinates, elliptic cylinder coordinates,parabolic cylinder coordinates, spherical coordinates, bipolarcoordinates, conical coordinates, paraboliccoordinates, prolatespheroidal coordinates, oblate spheroidal coordinates, ellipsoidalcoordinates, bi-spherical coordinates, andtoroidal coordinates.Thenamesaregenerallydescriptiveofthecoordinatesystems.For example, the circular cylinder coordinates involve coordinatesurfaces, which are cylinder coaxial. Maple implements,respectively, about 15 coordinate systems in two dimensionsand31inthreedimensions.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 D2k20 (k is a constant), for example, can be split(making use of Tables 13) into two ordinary differentialequations,eachforasingleindependentvariabled2fdr21rdfdr1r2r2k2c2f 0.d2gdc2c2g 0 1where (r, c) f(r)g(c)and cistheseparationconstant(whichmust beinteger inthe caseof polar coordinates sinceis aperiodiccoordinate).WewillapplythemethodofseparationofvariablestothePDEsintheexamples.Figure1 Twoidenticalpipesimbeddedinaninfinite concreteslab.Figure2 Circularcylindricalcoordinates.[Colorfigurecanbeviewedintheonlineissue,whichisavailableatwileyonlinelibrary.com.]Figure3 Elliptic cylindrical coordinates. [Color figure can be viewed intheonlineissue,whichisavailableatwileyonlinelibrary.com.]Figure4 Parabolic coordinates. [Color figure can be viewed in theonlineissue,whichisavailableatwileyonlinelibrary.com.]366 ELKAMEL, BELLAMINE, AND SUBRAMANIANSo, thebasictechniqueis totransformagivenboundaryvalue problem in the xy plane (or x, y, z space) into a simpler oneintheplane 12(or 123space) andthenwritethesolutionoftheoriginal probleminterms of thesolutionobtainedfor thesimpler equation. This is explained in the next three sections. Thetransformations will make the solution more tractable andconvenienttofind.Nowadays, high-performance computers coupled withhighly efcient 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. Inthispaper, thepowerful softwaretool Mapleis used. Maple facilitates the manipulation and derivation ofanalytical expressions, and can be used to performtediousalgebra,complicatedintegrals,anddifferentialequations[13].Asecondary 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 differentcoordinatesystems.For thesakeof readers not familiar withMaple, abriefintroduction will follow. Maple is a powerful symbolic computa-tionaltoolusedtoperformanalyticalderivationsandnumericalcalculations. It is easy to use, and its commands are oftenstraightforwardtoknowevenforarst-timeuser.Inthispaper,thestudent versionof Mapleisused. Werecommendthat thestudent uses ; and not : at the end of a command statementsothatMapleprintstheresults.Thishelpsinxingmistakesintheprogramsincetheresultsareprintedafter everycommandstatement. Inaddition, the user might havetomanipulate theresulting expressions froma Maple command to obtain theequationinthesimplest or desiredform. All themathematicalmanipulationsinvolvedcanbeperformedinthesameprogram,andMaplecanbeusedtoperformall therequiredstepsfromsettinguptheequationstointerpretingplotsinthesamesheet.Please note that equations containing : are results printed byMaple.APPLICATION OF THE BIPOLARCOORDINATE SYSTEMThereareanumberofreal applicationsforbipolarcoordinatessuchas pairs of ducts, pipes, transmissionlines, andbubbles[49]. We will illustrate the use of the bipolar orthogonalcoordinate system in this section by the following example. Twoidentical circular pipes of identical radius Rare imbeddedinan 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 followingdifferentialLaplaceequation:Table1 DefinitionofCommonCoordinateSystemsCircularcylindrical(polar)coordinates(,, c,z)x , cos c,y , sin c,zEllipticcylindricalcoordinates(u, c,z)x d cosh u cos c,y d sinh u sin c,zParaboliccylindricalcoordinates(u,v,z)x (1/2)(u2v2),y uv,zBipolarcoordinates(u,v,z) x asinh vcosh v cos u. y asin ucosh v cos u. zSphericalcoordinates(r, c, y) x r cos csin 0. y r sin csin 0. z r cos 0Conicalcoordinates(`, j, i) x `jiab. y `aj2a2i2a2a2b2_. z `bj2b2i2b2b2a2_Paraboliccoordinates(u,v, c) x uvcosc. y uvsinc. z 12u2v2Prolatespheroidalcoordinates(u,v, c)x d sinh u sin i cos c. y d sinh u sin i sin c. z d cosh u cos iOblatespheroidalcoordinates(u,v, c)x d cosh u sin i cos c. y d cosh u sin i sin c. z d sinh u cos iEllipsoidalcoordinates(1, 2, 3)x 21 a222 a223 a2a2c2a2b2. y 21 b222 b223 b2b2c2b2a2.z 21 c222 c223 c2c2a2c2b2Paraboloidalcoordinates(1, 2, 3)x 21 a222 a223 a2a2b2. y 21 b222 b223 b2b2a2.z 1221 22 23 a2b2Bisphericalcoordinates(j, j, c)x a cos csin jcosh j cos j. y a sin csin jcosh j cos j. z asinh jcosh j cos jToroidalcoordinates(j, j, c)x a cos csinh jcosh j cos j. y a sin csinh jcosh j cos j. z asin jcosh j cos jTRANSFORMATIONSOFPDES 367 restart: with(student): eq:diff(T(x,y),x$2)diff(T(x,y),y$2)0;eq :020x2 Tx. y_ _020y2 Tx. y_ _ 0Theboundaryconditionsofthisproblemdictatetheuseofbipolar coordinates (see Fig. 5). According to Table 1, thefollowingequationsdene bipolarcoordinates:x c sinh vcosh v cos u. y c sin ucosh v cos u2 eq1:xc*sinh(v(x,y))/(cosh(v(x,y))cos(u(x,y)));eq1 : x c sinhvx. ycoshvx. y cosux. y eq2:yc*sin(u(x,y))/(cosh(v(x,y))cos(u(x,y)));eq2 : y c sinux. ycoshvx. y cosux. yWe will show how tractable and convenient it is to solve theLaplaceequationin the bipolar coordinates,whereas in termsofx, y, andzthetemperaturefieldexpressioniscomplex. So, thebipolar coordinates are the natural coordinates for this type ofproblem.First of all, we show that the bipolar coordinate system is anorthogonalcoordinatesystem. Thismeansthatthetwofamiliesof the coordinate surfaces u(x, y) and v(x, y) are mutuallyorthogonal.Thelinesofintersectionofthesesurfacesconstitutetwo families of lines. At the point (u, v), we have unit vectors e1and e2each, respectively, tangent tothecoordinatelineof thebipolar coordinate system which goes through the point. Since thecoordinate system is orthogonal, e1and e2are mutuallyperpendiculareverywheree1 e2 0 or1h10r0u1h20r0v 0 3where risaposition vectorandisgiven byr xe1ye24Table2 ScaleFactorsforCommonCoordinateSystemsCircularcylindricalcoordinates(,, c,z)h11,h2,,h31Ellipticcylindricalcoordinates(u, c,z)h1 dsinh2u sin2c_. h2 h1. h3 1Paraboliccylindricalcoordinates(u,v,z)h1 du2v2_. h2 h1. h3 1Bipolarcoordinates(j, j,z) h1 ccosh j cos j. h2 h1. h3 1Sphericalcoordinate(r, y, c) h11,h2r sin y,h3rConicalcoordinates(`, j, i) h1 1. h2 `j2i2j2a2b2j2. h3 `j2i2i2a2i2b2Paraboliccoordinate(u,v, c) h1 `2j2_. h2 h1. h3 `jProlatespheroidalcoordinates(u,v, c)h1 dsinh2u sin2i_. h2 h1. h3 d sinh u sin iOblatespheroidalcoordinates(u,v, c)h1 dsinh2u sin2i_. h2 h1. h3 d cosh u sin iEllipsoidalcoordinates(1, 2, 3)h1 212221 2321a221 b221 c2. h2 22 2122 2322 a222 b222 c2.h3 232123 2223a223 b223 c2Paraboloidalcoordinates(1, 2, 3)h1 212221 2321a221 b2. h2 22 2122 2322 a222 b2.h3 232123 2223a223 b2Bisphericalcoordinates(j, j, c)h1 acosh j cos j sinh j. h2 acosh j cos j sin j.h3 a sin jcosh j cos j sin cToroidalcoordinates(j, j, c)h1 acosh j cos j. h2 h1. h3 a sinh jcosh j cos j368 ELKAMEL, BELLAMINE, AND SUBRAMANIANh1 and h2 are scale factors for the bipolar coordinates u and v. WewillwriteaboutthemlateronEquation(2)becomes1h1h20x0u_ _0x0v_ _0y0u_ _0y0v_ _ _ _ 0 5Equation (1) is a conformal transformation to x, y from u, vcoordinates. u, uy, vx, and vyare computed using Maple asfollows: eq3:diff(eq1,x):eq4:diff(eq2,x):vx:solve(eq4,diff(v(x,y),x)):eq31:simplify(subs(diff(v(x,y),x)vx,eq3)): ux:solve(eq31,diff(u(x,y),x));ux : sinux. y sinhvx. yc vx:simplify(subs(diff(u(x,y),x)ux,vx));cosux. y coshvx. y 1c eq5:diff(eq1,y):eq6:diff(eq2,y):uy:solve(eq5,diff(u(x,y),y)):eq61:simplify(subs(diff(u(x,y),y)uy,eq6)):vy:solve(eq61,diff(v(x,y),y));vy : sinux. y sinhvx. ycuy:simplify(subs(diff(v(x,y),y)vy,uy));uy :cosux. y coshvx. y 1cTheCauchyRiemannconditionsaresatised since:simplify(uxvy);simplify(uyvx);00and thus we conclude that the bipolar coordinate systemisorthogonal.Next,weobtainthescalefactorshi(I 1,2)givenbyhi 0r0i. 1 u and 2 v 6The scale factor hi can be interpreted as follows: a change duinthebipolarcoordinatesystemproducesadisplacement h1dualong the coordinate line. Now, we notice that the rate ofdisplacement alonguduetoadisplacement alongthex-axisish1(0u/0x)whichisthesameastherateofchangeofxduetoadisplacement h1du. The same argument goes for the scale factorh2. Thescalesof thenewcoordinatesandthechangeof scalefrompoint topoint determine theimportant properties of thecoordinate system. The scale factors play a role in expressing theTable3 LineandVolumeElementsAlongWithDifferentialOperatorsinCurvilinearOrthogonalCoordinateSystemScalefactors,hnhn 0x0n_ _20y0n_ _20z0n_ _20n0x_ _20n0y_ _20n0z_ _2 ____1Lineelement,ds ds h2ndn2_Volumeelement,dV dV h1h2h3d1d2d3Gradient, r r 1h1001e11h2002e21h3003e3Divergence, r

A r

A 1h1h2h3

3n1001h1h2h3Anhn_ _Curl, r

A r

A 1h1h2h3

m.n.phmem00nhpAp 00phnAn_ _. m. n. p 1. 2. 3. or 2. 3. 1 or 3. 1. 2Laplacian, r2 r2 1h1h2h3

3n100nh1h2h3h2n00n_ _Figure5 Bipolar coordinates. [Color figure can be viewed in the onlineissue,whichisavailableatwileyonlinelibrary.com.]TRANSFORMATIONSOFPDES 369differential/integral operators, line, surface, and volume ele-ments, and for the sake of completeness, they are shown inTable2for commoncoordinatesystems. After somealgebraicmanipulations,wefindthat h1square:factor(simplify(1/(ux^(2)uy^(2)))):h[1]:simplify(sqrt(h1square),sqrt,symbolic);h1 : ccosux. y coshvx. ySo,h1 0x0u_ _20y0u_ _2ccosh v cos u7 h2square:factor(simplify(1/(vx^(2)vy^(2)))):h[2]:simplify(sqrt(h2square),sqrt,symbolic);h2: ccoshvx. y cosux. yandso,h2 0x0v_ _20y0v_ _2ccosh v cos u8Next,weneedtoobtainexpressionsfortheparameter cintermsofRandL.The y-axisliesinthemiddlebetweenthetwocylinders, whilethex-axiscrossesthecentersofthecylinders.After algebraic manipulations, the relationship between theCartesianandbipolarcoordinatescanbeexpressedasfollows:x2y c cot u2 c2csc2u 9x c coth v2y2 c2csc h2v 10This form of writing the relationship between the old setof coordinates, that is, x, y and the new set of coordinates u, vprovidesusafurtherinsightsincewecannoticethatEquations(9)and(10)arecircles. Foranarbitraryv j0, fromEquation(10) we have a circle of radius c csc hh0 and center (c coth j0, 0).Also, whenv j0, wehaveacircleof radius c csc hh0andcenter(c coth j0,0).Now,whenc csc hj0 R 11and,c coth j0 L2R 12Thenc Rsinh cosh11 L2R_ _ _ _13The obvious next step is to transformthe differentialequationfromx, ytou, vcoordinates. UsingMaple, we cansimply use the Laplacian command. However, Laplace form doesnot govern many differential equations, and so we will follow theapproachneededtomapadifferential equationfromaset ofcoordinates to another. So, first, we need to get the secondderivativesofu,vwithrespecttox,y uxx:diff(ux,x):uxx:simplify(subs(diff(u(x,y),x)ux,diff(v(x,y),x)vx,uxx)): vxx:diff(vx,x):vxx:simplify(subs(diff(u(x,y),x)ux,diff(v(x,y),x)vx,vxx)): uyy:diff(uy,y):uyy:simplify(subs(diff(u(x,y),y)uy,diff(v(x,y),y)vy,uyy)): vyy:diff(vy,y):vyy:simplify(subs(diff(u(x,y),y)uy,diff(v(x,y),y)vy,vyy)):Then, we map the differential equation into the bipolarcoordinates: eq1:eval(subs(T(x,y)TT(u(x,y),v(x,y)),eq)):eq2:subs(diff(v(x,y),y$2)diff(v(x,y),x$2),diff(u(x,y),y$2)diff(u(x,y),x$2),diff(v(x,y),y)diff (u(x,y),x),diff(v(x,y),x)-diff(u(x,y),y),eq1):eq3:factor(expand(eq2)):eq4:D[2,2](TT)(u(x,y),v(x,y))D[1,1](TT)(u(x,y),v(x,y)):eq5:subs(u(x,y)u,v(x,y)v,eq4):Eq:convert(eq5,diff);Eq :020u2 TTu. v_ _020v2 TTu. v_ _whichcanberewrittenas02T0u2 02T0v2 0 14which shows that Laplaces equation is invariant under a mappingto bipolar coordinates. Now, if T changes with respect to z as well,thenthetransformedPDEcanbeshowntobe1h1h202T0u2 02T0v2_ _02T0z2 0 15but since the temperature does not change withrespect toz,that is, 0T/0z 0, thenEquation(14) is reducedto(13). Thetransformedboundaryconditionsintheu,vcoordinatesareT T1atv j0. T T2atv j016At this point, we can use the method of separation ofvariables covered in many undergraduate and graduate textbooks[10,11] (so T(u, v) X(u)Y(v)). The separation of variables splitsthe PDE into two ODEs. We assume that the temperature T(u, v)isonlydependenton v.So,inourparticularcase,theseparationconstant is ln0. So, we end up solving the followingdifferentialequationwithitsDirichletboundaryconditions: Eq:diff(Y(v),v$2);d2dv2 Yv 0 bc1:subs(veta[0],Y(v))T[1];bc1 : Yh0 T1370 ELKAMEL, BELLAMINE, AND SUBRAMANIAN bc2:subs(veta[0], Y(v))T[2];bc2 : Yh0 T2 sol:dsolve(Eq,Y(v)):eq1:subs(veta[0],Y(eta[0])T[1],sol):eq2:subs(veta[0],Y(eta[0])T[2],sol):eq3:solve({eq1,eq2},{_C1,_C2});andthus:Yv T1T2v2h012T212T1whichisequaltoT(u,v).Generally, when we conduct the separation of variables, onegets eqs:subs(TT(u,v)X(u)*Y(v),Eq):eqs:expand(eqs):eqs:eqs/(X(u)*Y (v)):eqs:expand(eqs);eqs :d2du2 XuXud2dv2 YvYvwe obtain two ODEs in X(u) and Y(v) and the first one is eqs1:diff(X(u),u$2)lambda^2*X(u): bc1:subs(u0,D(X)(u))0:bc2:subs(uPi,D(X)(u))0;eqs1 :d2du2u_ _`2ubc1 : DX0 0bc2 : DXp 0TheODEinX(u) isaregular SturmLiouvilleboundaryvalue problem [10], with separation constant lnn (where n 1,2, . . .) withcorrespondingeigenfunctions Xn(u) sin(nu). TheODEinY(v)is eqs2:diff(Y(v),v$2)n^2*Y(v);eqs2 :d2dv2 Yv_ _n2Yv dsolve(eqs2);Yv C1 envC2 en.vInorder tosatisfytheboundaryconditions (Eq. 16), weconsiderthefollowingsolutionwhichisalinearcombinationofproducts,andforwhichtheseparationconstantisnotzero:Tu. v T1T22j0v T1T22

1n1cnenvdnenv sinnuwherecnanddnarecomputedbyMapleasfollows: an:(2/Pi)*int(T[1]*sin(n*u),u0. . .Pi):bn:(2/Pi)*int(T[2]*sin(n*u),u0. . .Pi) : assume(n::integer) : assume(n,odd):eval(an):eval(bn): eqc1:cn*exp(n*eta[0])dn*exp(n*eta[0])an:eqc2:cn*exp(n*eta[0])dn*exp(n*eta [0])bn:solc:solve({eqc1,eqc2},{cn,dn});solc : _cn 4enj0T2T1enj0nenj0_ _2enj02_ _ _ _.dn 4T1enj0T2enj0nenj0_ _2enj0_ _2_ __wherendenotesanoddnumber(i.e.n 2m1,m1,2. . .).APPLICATION OF THE ELLIPTIC CYLINDRICALCOORDINATE SYSTEMAnothertypeofanorthogonalcurvilinearcoordinatessystemisthe elliptic cylinder. The coordinate surfaces are elliptic cylinders(u constant)and hyperbolic cylinders (v constant) in the twocoordinate systems. Many problems are amenable to this type ofcoordinatesystemssuchascoils, solar, andheat concentrators,metallurgical junctions, material flawshapes, shells, fluidflowpast anobstacle[1217]. ThetransformationstotheCartesiancoordinatesfromtheellipticcylindrical coordinatesislistedinTable1,andarex a coshu cosv. y a sinhu sinvwhereaisthelengthofthesemi-majoraxisoftheellipse. Forexample, if an elliptical hole (Fig. 6) is cut in a region as we willsee in this section as an example, then it is more tractable to solvetheLaplaceequationintheellipticcylindrical coordinates. Wetake the center of the coordinate system to be that of the hole. Thescalefactorsare given by restart:with(student): x:a*cosh(u)*cos(v):y:a*sinh(u)*sin(v):h[u]:sqrt(diff(x,u)^2diff(y,u)^2):h[u]:simplify(h[u]):h[u]:subs(cosh(u)^21sinh(u)^2,cos(v)^21sin(v)^2,h[u]):h[u]:simplify(h[u],sqrt,symbolic);hu: asinv2sinhu2_eq3 :C1 T1T22j0. C2 12T212T1_ _TRANSFORMATIONSOFPDES 371Likewise,weobtainhvhv: asinv2sinhu2_The Laplace equation in elliptic cylindrical coordinates,assuminga 1,is with(linalg): eq1:laplacian(phi(u,v),[u,v],coordselliptic):eq1:numer(eq1);eq1 :020u2 cu. v_ _020v2 cu. v_ _Using separation of variables, we have c(u, v) c1(u,v)c2(u,v): eq2:subs(phi(u,v)phi[1](u)*phi[2](v),eq1) : eq3:expand(eq2):eq4:eq3*(1/(phi[1](u)*phi[2](v))):eq4:expand(eq4);eq4 :020u2 c1uc1u020v2c2vc2vWeendupwithtwodifferentialequationsforc1(u,v)andc2(u,v).eq41:diff(phi[1](u),u$2)p^2*phi[1](u);eq41 :020u2 c1u_ _p2c1ueq42:diff(phi[2](v),v$2)p^2*phi[2](v);eq42 :020v2 c2v_ _p2c2vThetwoboundaryconditionslabeledbc1andbc2are suchthat:bc1:subs(uinfinity,phi(u,v))phi[0]E[0]*sinh(u)*sin(v);bc1 : c1. v c0E0 sinhu sinvbc2:subs(uu[0],D[ 1](phi)(u,v))0;bc2 : D1cu0. v 0Assumingthatc(u,v) A0A1sinh u sinv A1cosh sinv,then enforcing the first boundary condition and since sin-h u cosh uforlarge u,wehaveeq5:phi(u,v)_A0_A1*sinh(u)*sin(v)_A2*cosh(u)*sin(v) : eq6:subs(uinfinity,_A0phi[0],eq5):eq7:phi[0]E[0]*sinh(u)*sin(v)phi[0](_A1_A2)*sinh(u)*sin(v);eq7 : c0E0 sinhu sinv c0A1 A2 sinhu sinvWeassumethat A0E0whichisequal tothepotential aty 0 as shown in Figure 3. The second boundary condition givesphi(u,v):_A0_A1*sinh(u)*sin(v)_A2*cosh(u)*sin(v):eq8:diff(phi(u,v),u):eq8:subs(uu[0],eq8)0;eq8 : A1 coshu0 sinvA2 sinhu0 sinv 0So,nowwecansolve fortheconstantsA1andA2:eq9:solve({eq7, eq8},{_A1,_A2});eq9 : A1 E0 sinhu0coshu0 sinhu0_A2 coshu0E0coshu0 sinhu0_SubstitutingA1andA2inthepotential c(u,v),oneobtainseq10:subs(_A0phi[0],eq5):eq11:subs(eq9,eq10);eq11 : c0E0 sinhu0 sinhu sinvcoshu0 sinhu0coshu0E0 coshu sinvcoshu0 sinhu0The electrostatic eld E is the gradient of the potential, andthusisgivenbyE(u,v):grad(rhs(eq11),[u,v],coordselliptic);Figure6 TheshapeoftheellipticholewhichisilluminatedwiththefieldE0.372 ELKAMEL, BELLAMINE, AND SUBRAMANIANA eld plot is shown Figure 7 reecting the ellipticcylindricalcoordinatesystem.APPLICATIONOF THE PROLATESPHEROIDALCOORDINATESInthreedimensions,asshowninTable1,thereareanumberofthree-dimensional coordinates. In here, we will give as anexamplethecaseof theprolatespheroidal coordinates systemillustratedinFigure8.Thistypeofcoordinatessystemhasbeenusedextensivelyinmanyelds. For example, raindrops, dustgrains inplasma, moltenregions inlaser welding, groundrodconnection, conducting electrodes, ventricle, diatomic molecules,hydrogen molecular ions, and biological cells [11,1824]aremodeledasprolatespheroidal objects. Asanexample, theshape of a football ball is a prolate spheroid. The transformationto the Cartesian coordinates from the prolate spheroidalcoordinatesislistedinTable1.We willrepeathereforthesakeofclarity:x d sinhu sinv cosc.y d sinhu sinv sinc. z d coshu coswwheredis thefocal lengthfor theprolatespheroidal system.However, weneedtobeawarethat thereareother equivalenttransformations.When cosh(u), j cos v,thenone getsx d211 j2_cosc.y d211 j2_sinc. z djwhere1 1, 1 j 1,0 c2p.To compute the volume and surface area in the prolatespheroidal coordinate system, the Jacobian matrix associatedwith the coordinate transformation must be calculated. TheJacobian matrix elements are the partial derivatives of thetransformationfromprolatespherical toCartesiancoordinates.So,theinnitesimalchangein volumeisthedeterminantoftheJacobianmatrix:restart:with(student):with(linalg):with (PDEtools):T:[d*sinh(u)*sin(v)*cos(phi),d*sinh(u)*sin(v)*sin(phi),d*cosh(u)*cos(v)]:J:jacobian(T,[u,v,phi]):dV:simplify(det(J))*d(u)*d(v)*d(phi);dV: d3sinv sinhucoshu2cosv2 du dv dcSimilarly, for the computation of surfaces, we need tocompute the two-dimensional Jacobian matrix in the desireddirection.Eu. v : E0 sinhu0 coshu sinvcoshu0 sinhu0coshu0 E0 sinhu sinvcoshu0 sinhu0sinhu2sinv2_ .__E0 sinhu0 sinhu cosvcoshu0 sinhu0coshu0 E0 coshu cosvcoshu0 sinhu0sinhu2sinv2_ Figure7 FieldplotofE(u, v/E0)whichreflectstheellipticcoordinatesystem.Figure8 Prolatesphericalcoordinates.[Colorfigurecanbeviewedintheonlineissue,whichisavailableatwileyonlinelibrary.com.]TRANSFORMATIONSOFPDES 373Inthis section, wewant toobtainanexpressionfor theequilibriumtemperaturedistributionofametalspheroid.Letussupposethat becauseof aninternal heatingsystemwithinthespheroid( i),itssurfacetemperatureis 10 30 cos2jTheprolatespheroidmadeofmetalisimmersedinalargecontainer lled with insulating powder. The ambient temperatureis 208C. So, basically, we need to solve for the Laplace equationinprolatespheroidalcoordinates:eq1:laplacian(psi(xi,eta,phi),[xi,eta,phi],coordsprolatespheroidal(a)):eq2:expand(eq1):eq2 :coth00c. j. c_ _a2sinh2sinj20202c. j. ca2sinh2sinj2cotj00jc. j. c_ _a2sinh2sinj2020j2c. j. ca2sinh2sinj2020c2c. j. ca2sinh2sinj2 sinj2sinh2The system of prolate spheroidal coordinate is separable. Sowriting the dependent variable C(, j, c) as the product of threefunctions X()H(j)F(c) will split the Laplacian equation in threeordinarydifferentialequations:eq3:subs(psi(xi,eta,phi)Xi(xi)*Eta(eta)*Phi(phi),eq2):eq4:eq3*(1/(Xi(xi)*Eta(eta)*Phi(phi))):eq4:expand(eq4):eq4:expand(a^2*eq4);Thethreedifferentialequationsare:ode1 :020c2Fc_ _q2Fc 0eq42 :020j2 Hj_ _cotj00jHj_ _ pp 1 q2sinj2_ _Hj 0eq43 :0202X_ _coth00X_ _ pp 1 q2sinh2_ _X 0qandparetheintroducedseparationvariables.ThedifferentialequationsinH(j)and X()aretransformedusingthefollowingchangeofvariables:l : coshmj : cosjThen,we willgetthefollowingdifferentialequations:eq430:changevar(Xi(xi)f(lambda(xi)),eq43):eq431:numer(simplify(subs(lambda(xi)cosh(xi),eq430))):eq432:subs(cosh(xi)lambda,eq431):eq432:convert(eq432,diff):eq432a:collect(eq432,diff(f(lambda),lambda$2)):eq432b:collect(eq432,diff(f(lambda),lambda)):eq432c:collect(eq432,f(lambda)):a:factor(1lambda^42*lambda^2):b:factor(2*lambda^32*lambda):fac:1lambda^2:a1:simplify(a/fac):b1:simplify(b/fac):c:simplify((p^2*lambda^2p^2p*lambda^2pq^2)/fac):c1:collect(c,p^2):ode3:a1*diff(f(lambda),lambda$2)b1*diff(f(lambda),lambda)c1*f(lambda);ode3 : 1 l2020l2 f l_ _2l00lf l_ _ pp 1 q2l21_ _f landthesameMapletechniqueisusedfortheequationinH(j),ode2 : 1 m2020m2 gm_ _2m00mgm_ _ pp 1 q2m21_ _gmSo, we need to solve the differential equations labeled aboveby ode1, ode2, and ode3. We rename, respectively, f and g to X()andH(j).Phi:rhs(dsolve(ode1,Phi(phi)));alias(PLegendreP,QLegendreQ):f:rhs(dsolve(ode3,f(lambda)));g:rhs(dsolve(ode2,g(mu))):Xi:subs(lambdacosh(xi),f):Eta:subs(mucos(eta),g);F : C1 sinqc C2 cosqcX : C1 Qp. q. cosh C2 Pp. q. coshH : C1 Pp. q. cosj C2 Qp. q. cosjwhere_C1and_C2areconstants.Wenoticethatboth X()andH(j) solutionsare functionsof associatedLegendre functionsofthefirstandsecondkind.The solution C(, j, c) has axial symmetry about the z-axisand so it is independent of c and is given by C(, j,c) X()H(j). Let the boundary conditions be C(, 0,j) M(j) 10 30 cos2j, and hence mcos j ranges only374 ELKAMEL, BELLAMINE, AND SUBRAMANIANbetween 1 and 1. The associated Legendre functions ofthesecondkindQn(cos j)areunboundedform1, andsothesolution C(, j) depends only on the associated Legendrefunctions of the first kind illustrated in Figure 9 and has the formPn(cos j)Pn(cosh ),inotherwords:N:infinity;Psi(xi):sum(A[i]*P(i,mu)*(P(i,cosh(mu))/P(i,cosh(xi0))),i0. . .N);C :

1i0Ai Pi. m Pi. coshPi. cosh0EnforcingtheboundaryconditionthatC(, 0, j) M(j),oneobtainsPsi(xi):sum(A[i]*P(i,mu)*(P(i,cosh(xi))/P(i,cosh(xi0))),i0. . .N):M(mu):subs(xixi0,Psi(xi));Mm :

1i0AiPi. mBut,whenwetruncateNto5,Mm : A0A1m A2P2. m A3P3. mA4P4. m A5P5. mandsotocomputeAn, wewill havetoperformthefollowingcomputation:An n 12_ __11MmPnm dmfor i from 0 to 10 do A[i]evalf((i0.5)*int((1030*mu^2)*P(i,mu),mu-1. . .1)); od;A0 20.0. A1 0. A2 20.000000000.1. A3 0 0.1. A4 0.450 10110.1. A5 0 0.1Then,substituting the coefcients An into the solution C(,j),one getssol1:subs(A[0]20,A[2]20,A[1]0,A[3]0,A[5]0,A[4]0,mucos(eta),Psi(xi));sol1 : 20 20P2. cosjP2. coshP2. cosh0whosesolution C(, j)isillustratedinFigure10for 2.CONCLUSIONIn this paper, examples of solving problems in coordinate systemsother than the most famous Cartesian, polar, and sphericalcoordinatesaregiven. Inthisarticle, wegaveexamplesinthetwo-dimensional bipolar and elliptic cylindrical coordinates, andthe three-dimensional prolate spheroidal coordinates. Mapleproved to be very useful tool to performthe required trans-formationsfromonecoordinatesystemtoanother, tosimplifyexpressions,tomanipulatethemathematicalexpressions,andtoplot the solutions. The techniques used in this paper apply to othercoordinate systems such as the parabolic cylindrical coordinates,conical coordinates, parabolic coordinates, oblate spheroidalcoordinates, ellipsoidalcoordinates, bisphericalcoordinates, andtoroidal coordinates. The boundaries involved in solving the PDEprovideus withtheclueof thecoordinatesystemtobeused.TransformingthePDEtooneof thesecoordinates makes thesolution more tractable. For example, we saw that for the case ofboundaries consisting of two circles, the most appropriatecoordinate system to use is the bipolar coordinates. Theaforementioned coordinates are the most commonly used becausetheyareseparablecoordinates. Inother words, themethodofseparationofvariablescanbeusedtosolvethePDE.However,Figure9 Legendre polynomials of the first kind P0(x) . . . P4(x).[Color figurecanbeviewedintheonlineissue, whichis availableatwileyonlinelibrary.com.]Figure10 Temperature j for 2.TRANSFORMATIONSOFPDES 375one needs to be aware that there are other coordinate systems suchas the hyperbolodoidal, the exponential, andthe three-dimen-sional bipolar coordinates, which are not considered as separablecoordinates,andthey arecurrentlythefocusofourendeavors.REFERENCES[1] V. R. Subramanian, Computer facilitated mathematical methodsinchemical engineering-similaritysolution, ChemEngEduc40(2006),307312.[2] D. Richards, Advancedmathematical methodswithMaple, Cam-bridgeUniversity Press,Cambridge,2002.[3] F. H. BellameandF. Fgaier, Computer facilitatedmathematicaltechniques of differential equations governing engineeringprocesses, World Rev Sci Technol Sust Dev, 5 (2008), 234252.[4] J. T. Chen, M. H. Tsai, andC. S. 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Using Maple to obtain analytic expressions inphysicalchemistry,JChemEduc74(1997),1491 1493.BIOGRAPHIESAli Elkamel is a professor of ChemicalEngineering at the University of Waterloo.He holds a BS in Chemical Engineering and aBSinMathematicsfromColoradoSchool ofMines, anMSinChemicalEngineeringfromthe University of Colorado-Boulder, and a PhDin Chemical Engineering fromPurdue Uni-versity. The goal ofhis research program istodeveloptheoryandapplicationsforPSE.Theapplicationsfocusonenergyproductionplanning,pollutionpreven-tion, and product and process design. He is also interested inintegratingcomputingandprocesssystemsengineeringinchemicalengineering education. Prior to joining the University of Waterloo, heserved at Purdue University (Indiana), P&G (Italy), KuwaitUniversity, and the University of Wisconsin-Madison. He hascontributed more than 200 publications in refereed journals andinternationalconferenceproceedings.Fethi Bellamineis aprofessor at UniversityNovember7,InstituteofAppliedScienceandTechnologies, Tunis, Tunisia. He holds BS,MS, and PhD degrees in Electrical EngineeringfromColoradoStateandUniversityof Colo-rado at Boulder, respectively. From1995to2002, he served as a senior developmentengineer for Lucent Technologies, AlcatelNetworks, andNESA. His researchinterestsare in the areas of modeling and simulation, numerical methods, andsoftcomputing.Dr. Venkat Subramanian is an associateprofessor in the Department of ChemicalEngineering at the Tennessee TechnologicalUniversity. He received a BS degree inchemical and electrochemical engineeringfromthe Central Electrochemical ResearchInstitute in India and he received his PhDdegree in Chemical Engineering from theUniversity of South Carolina. His researchinterestsincludeenergysystemsengineering, multiscalesimulationand design of energetic materials, nonlinear model predictive control,batteries and fuel cells. He is the principal investigator of theModeling, Analysis and Process-Control Laboratory for Electro-chemical Systems (MAPLE Lab, http://iweb.tntech.edu/vsubramanian).376 ELKAMEL, BELLAMINE, AND SUBRAMANIAN