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Thesis (Ph.D.)University of South Florida, 2005.
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ABSTRACT: USFKAD is an encoded expert system for the eigenfunction expansion of solutions to the wave, diffusion, and Laplace equations: both homogeneous and nonhomogenous; one, two, or three dimensions; Cartesian, cylindrical, or spherical coordinates; Dirichlet, Neumann, Robin, or singular boundary conditions; in time, frequency, or Laplace domain. The user follows a menu to enter his/her choices and the output is a LaTeX file containing the formula for the solution together with the transcendental equation for the eigenvalues (if necessary) and the projection formulas for the coefficients. The file is suitable for insertion into a book or journal article, and as a teaching aid. Virtually all cases are covered, including the Mellin, spherical harmonic, Bessel, modified Bessel, spherical Bessel, Dini, Hankel, Weber, MacDonald, and KantorovichLebedev expansions, mixed spectrum, and rigid body modes.
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Adviser: Arthur David Snider, PhD.
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Eigen function.
Analytic solutions.
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Seperation of variables.
Symbolic computing.
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Dissertations, Academic
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x Electrical Engineering
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t USF Electronic Theses and Dissertations.
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u http://digital.lib.usf.edu/?e14.1144
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USFKAD: An Expert System Fo r Partial Differential Equations by Sami M. Kadamani A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Arthur David Snider, Ph.D. Kenneth A. Buckle, Ph.D. Stanley C. Kranc, Ph.D. Wilfrido A. Moreno, Ph.D. Mohamed Elhamdadi, Ph.D. Date of Approval: March 28, 2005 Keywords: eigen function, analytic solutions, pd e, separation of variables, and symbolic computing Copyright 2005, Sami M. Kadamani
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DEDICATION This dissertation is dedicated to my wife Daid, and my two children Serene and Tareq. Thank you for your patience, I would not have been able to accomplish this goal without your support.
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ACKNOWLEDGEMENTS My deepest thanks to my advisor Dr. David Arthur Snider, for his belief in me and for both his guidance and support. I would like to extend my gratitude to Chairperson Sam Sakmar, Ph.D. and the members of the examin ing committee: Kenneth A. Buckle, Ph.D., Stanley C. Kranc, Ph.D., Wilfrido A. Moreno, Ph.D., and Mohamed Elhamdadi, Ph.D. In addition I would like to extend my thanks to my friends Husam Elrabi and Krista Morehead, whose input was invaluable.
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i TABLE OF CONTENTS LIST OF TABLES iii LIST OF FIGURES iv ABSTRACT v CHAPTER 1: INTRODUCTION 1 Need 1 Feasibility 3 CHAPTER 2: EXISTING SOFTWARE 5 CHAPTER 3: THE SEPARATION OF VARIABLES PROCEDURE AND ITS DECISION TREE STRUCTURE 8 CHAPTER 4: DISCUSSION AND FLOW CHART FOR USFKAD 14 Decision Tree 15 Fundamental Concept 19 Example 1 22 OnScreen Inquires 26 CHAPTER 5: APPLICATION EXAMPLES 28 Example 1 28 Example 2 30 Example 3 31 Example 4 33 Example 5 34 Example 6 35 CHAPTER 6: RECOMMENDATIONS FOR FUTURE DEVELOPMENTS 37 REFERENCES 38 APPENDICES 39 APPENDIX A: COMPARISON OF SOLVING PARTIAL DIFFERENTIAL EQUATIONS USING THE TRADITIONAL METHOD AND USFKAD 40
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ii Solution 1 43 Solution 2 47 APPENDIX B: READ ME 50 APPENDIX C: TO USE USFKAD.PDF 53 ABOUT THE AUTHOR End Page
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iii LIST OF TABLES Table 1: PDE Equation Type Matrix 21 Table 2: PDE BC Matrix 21 Table 3: Matrix Example 23
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iv LIST OF FIGURES Figure 1: USFKAD Flow Chart 1620 Figure 2: PDE Problem 40 Figures 3: (ad) Decomposition of th e PDE Problem 40
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v USFKAD: AN EXPERT SYSTEM FOR PARTIAL DIFFERENTIAL EQUATIONS Sami M. Kadamani ABSTRACT USFKAD is an encoded expert system for the eigenfunction expansion of solutions to the wave, diffusion, and Laplace equations: both homogeneous and nonhomogenous; one, two, or three dimensions ; Cartesian, cylindrical, or spherical coordinates; Dirichlet, Neumann, Robin, or singular boundary conditions; in time, frequency, or Laplace domain. The user follows a menu to en ter his/her choices and the output is a LaTeX file cont aining the formula for the solution together with the transcendental equation for the eigenvalues (i f necessary) and the projection formulas for the coefficients. The file is suitable for inse rtion into a book or jour nal article, and as a teaching aid. Virtually all cases are covere d, including the Mellin, spherical harmonic, Bessel, modified Bessel, spherical Besse l, Dini, Hankel, Weber, MacDonald, and KantorovichLebedev expansions, mixed spectrum, and rigid body modes.
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1 CHAPTER 1 INTRODUCTION Need Every practicing engineer whose spec ialty involves modeling of physical phenomena, such as electromagnetic fields, te mperature, sound, stress and strain, fluid flow, diffusion, etc., has to deal with the ma thematical syntax of the discipline the partial differential equation (PDE). For example, the electrical engineering undergraduate classes in electromagnetic, semiconductor processing, thermal issues in electronic packaging, etc. should be able to call on this mathematical concept, at least peripherally, to provide the students some familiarity w ith the technical issues involved in the quantitative models. However, this subject (P DEs) is vast, compli cated, and compromises have to be made in incorporating it into the undergraduate's curriculum. A 2semester course that deals honestly and rigorously with the subject is out of the question. The compromises presently employed at undergraduate institutions are: (1) A short treatment of PDEs that relies completely on numerical solvers; or (2) A brief tutorial that covers the basics of the separation of variables technique. Each of these is unsatisfactory. (1) is inferior to (2) because, even with the graphic capabilities of today's hardware and so ftware, it is extremely difficult for an inexperienced undergraduate user to tell, fr om a vast assemblage of tabulations and graphs, how the solutions will respond to changes in the boundary conditions or the
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physical dimensions issues of prime impor tance to engineering. For example consider the frequency of the resonant mode of a r ectangular cavity with sides X, Y and Z given by: 221/1/1/ cXYZ 2 These are not a terribly complicated formula, but contemplate trying to deduce it from graphs! The eigenfunction expansions yielded by (2) do reveal these dependencies (and are exact). The drawback of this solution proced ure is the lack of time to impart expertise in its implementation except for a few elemen tary cases rectangul ar geometries and ideal conductors, for instance. The electrical world of cables, motors, and antennas is replete with cylindrical and s pherical devices made of lous y materials, whose analyses entail Bessel functions and transcendental eigenvalue equatio ns. The presentday curriculum has no room for the mastery of th e ``special functions" that occupied the toolbox of the 1950s engineer. On the other hand, usually it is well within the capability of senior undergraduates to verify most features of an eige nfunction solution expansion. Therefore an expert system USFKAD, for partial differential equations (a smart software tool) that can automatically cull, from a library of eigenfunctions, the assemblage constituting the solutions to explicit problems, together with relevant graphics, would be a powerful enabler for undergraduate engineering training: 1. It would allow engineering analysis/d esign to proceed efficiently without being sidetracked by concerns of mathematical solvability. 2. As such, it would cut across many engineering disciplines. 2
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3 3. It could be used to give a perspectiv e on the separationofvariables technique itself, by enabling reverseengineering of the explicit solution formulas. (This will be elaborated below.) 4. In fact it would be a research tool that the engineer could continue to use in his professional career. Eigenfuncti on expansions are integral to the modematching procedure that is used in contemporary computational electromagnetism. And indeed, virtually every technical paper describing a new numerical solver compares its resu lts with eigenfunction expansions, as testament to its accuracy. Feasibility USFKAD, the subject of this dissert ation, focuses on the theme that the mathematical structure, afforded by superposition, of the eigenfunction method for solving the separable PDEs of engineering can be expressed by a compact, universal, inviolate, and reasonably lucid algorithm; its fo rmidability lies only in the details of its implementation that is, in the enormous variety of eigenfunctions that must be employed for the curvilinear geometries. Thus it becomes feasible to contemplate a smart computer program that exploits this structur e to judiciously select, from a library of eigenfunctions, the assemblage constituting the solutions to problems with explicit initial/boundary conditions. In the subsequent chapters, more discussion and a list of examples of PDE solutions will be presented that progressively demonstrate the decisi ontree nature of the general separation of variables procedure. This will exemplify the thesis that by reverse
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4 engineering explicit solution fo rmulas one can experience a tutorial intercourse with the procedure itself.
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5 CHAPTER 2 EXISITING SOFTWARE Software for obtaining (analytic) solutions to ordinary differential equations exists in several forms, including Mathemati ca [6] and MAPLE [7]. It has not received universal adoption because extens ive training in ordinary diffe rential equations is already part of the required curriculum for all SM ET (Science, Mathematics, Engineering, and Technology) students. For partial differential equations, MAPLE's pdsolve [Solution 1a] is a step in the right direc tion, but its arcane solution format provides little assistance for a nonexpert in fitting th e initial and boundary conditi ons that determine such dependencies. An example of its output, the electrostatic potential inside a sphere with charges distributed on the surface, is displaye d as Solution 1a below. It is expressed (correctly) in terms of hypergeometric and complex signum functions. But comparing this with the more recognizable solution di splay using USFKAD as shown in Solution 1b, one can clearly see the obvi ous simplification and straight forwardness of USFKAD. Solution 1a: Output from pdesolve
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() F,, rtp_C1r() / 12 14 _c 1_C3 () 2() sin t2 () / 12 _c 2hypergeom ,, 1 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 1 4 1 2 _c2 1 2 1 2 () cos2 t 1 2 _C5 () sin _c2p r _C1r() / 12 14 _c 1_C3 () 2() sin t2 () / 12 _c 2hypergeom ,, 1 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 1 4 1 2 _c2 1 2 1 2 () cos2 t 1 2 _C6() cos _c2p r 2 _C1r() / 12 14 _c 1_C4 () 2() sin t2 () / 12 _c 2() csgn()cos t () cos t hypergeom 3 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 3 4 1 2 _c2 3 2 ,, 1 2 () cos2 t 1 2 _C5 () sin _c2p r 2 _C1r() / 12 14 _c 1_C4 () 2() sin t2 () / 12 _c 2() csgn()cos t () cos t hypergeom ,, 3 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 3 4 1 2 _c2 3 2 1 2 () cos2 t 1 2 _C6 () cos _c2p r _C2r() / 12 14 _c 1_C3 () 2() sin t2 () / 12 _c 2hypergeom ,, 1 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 1 4 1 2 _c2 1 2 1 2 () cos2 t 1 2 _C5 () sin _c2p r _C2r() / 12 14 _c 1_C3 () 2() sin t2 () / 12 _c 2hypergeom ,, 1 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 1 4 1 2 _c2 1 2 1 2 () cos2 t 1 2 _C6 () cos _c2p r 2 _C2r() / 12 14 _c 1_C4 () 2() sin t2 () / 12 _c 2() csgn()cos t () cos t hypergeom 3 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 3 4 1 2 _c2 3 2 ,, 1 2 () cos2 t 1 2 _C5 () sin _c2p r 2 _C2r() / 12 14 _c 1_C4 () 2() sin t2 () / 12 _c 2() csgn()cos t () cos t hypergeom ,, 3 4 1 2 _c2 1 4 14 _c1 1 4 14 _c1 3 4 1 2 _c2 3 2 1 2 () cos2 t 1 2 _C6() cos _c2p r 6
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Solution 1b: Output from USFKAD 7mA 01 1(,)emmYr ll l ll With 2* 00sin (,) (,) 1mm rrb rAddY Mf M b ll l
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CHAPTER 3 THE SEPARATION OF VARIABLE PROCEDURE AND ITS DECISION TREE STRUCTURE The three most common partial differentia l equations encountered in engineering physics are: 2 2 2 2 21Poisson's Equation 2Diffusion Equation () 3Wave Equation f f i t f t These equations are solvable by the separa tion of variables process in the common coordinate systems: Cartesian, cy lindrical (polar), and spherical. The Separation of Variables Technique is the most important analytical method in engineering analysis for solving partial di fferential equations. The present exposition borrows heavily from [1]; see also [4 and 5]. The successful execution of this procedure for a given boundary value problem can be leng thy and tedious, because it involves three different mathematical procedures: 1. the use of superposition to decompose a complicated problem into a set of simpler ones; 2. the separation of the partial different ial equation into a set of ordinary differential equations in an appr opriate coordinate system; and 8
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3. the construction of eigenfunction expans ions (which are generalizations of the Fourier series) which satisfy the boundary conditions. We shall briefly describe the solution pro cedure in such a way as to elucidate the algorithmic (decisiontree) nature. Then we describe a new code, USFKAD, which implements the algorithm in C++, genera ting a LaTeX output file for the solutions. Appendix A solution 1 shows part of a st raightforward and uncomplicated problem solved using the traditional method and appe ndix A solution 2 using USFKAD. It could be seen that how cumbersome, time consumi ng, tedious, and treacherous it is to solve a problem manually. The requisites for guaranteed success fo r the separation of variables are: 1 All boundary/initial conditions are applied at edges (s urfaces, manifolds) where one of the independent variables is constant. 2 All boundary/initial conditions are li near, taking the generic format g n ( ii) n is the coordinate that is cons tant on the boundary in question. If = 0 ( ii ) is called a Dirchlet condition (heat sink, electrical ground); if = 0 ( ii ) is called a Neumann condition (ideal insulation, magne tic wall); otherwise it is a Robin condition (imperfect insulation). Also singular boundary conditions (singular points of the differential equation, boundaries at infin ity, Sommerfeld radiation conditions, etc.) may be accommodated. 9
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The given functions f or g ( i ), ( ii ) are called nonhomogeneities; any equation is homogenous when f or g is zero. By linearit y, any system can be decomposed into a superposition of systems in each of which only one equation is nonhomogenous. Solutions to the homogenous forms of each of the three partial deferential equations can be found in the form of products of functions depending on one independent variable only, in the appropriate co ordinate systems [1]. In a wide range of engineering situations thes e onedimensional functions are solutions of one of the following four types of ordinary differential equations: 1 Constant coefficient (Cartesian coordinates x, y, z, time t, angle ) 2 Equidimensional (CauchyEuler) (polar coordinate radius) 3 Bessel (Cylindrical or sphe rical coordinate radius) 4 Legendre (azimuthal angle ) Associated with the ordinary differential equation for each independent variable (other than time) are boundary conditions. After th e decomposition step all of these boundary conditions are homogenous with one excep tion. For the unexceptional ordinary differential equations, the ordinary differential equation is second order and contains an unspecified parameter called the separation constant. The ordinary differential equation, separation constant, and boundary conditions constitute a SturmLouisville eignenvalue problem. The steps for solving such a problem are [1]: 10
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1. With the differential equation expressed in the generic form 210()"()'()() axyaxyaxygxy (1) write down the general solution as the sum of two independent particular solutions with undetermined coefficients: 11 22(;)(;) y CyxCyx (2) The constant will appear as a parameter in the formulas. Satisfy one of the boundary conditions: ()'()0,()'()0 yayaybyb (3) by the choice of and In other words, use the relation 1C 2C '' 11 22 11 22[(;)(;)][(;)(;)]0 CyaCyaCyaCya (4) to express in terms of or vice versa. The resulting solution is a constant multiple of (5) 1C 2C '' 221112(;)[(;)(;)](;)[(;)(;)](;) yxyayayxyayayx 3. The remaining boundary condition (;)'(;)0 ybyb (6) Is regarded as an equation for Insert (5) and solve it for the eigenvalues { n }, which will form a sequence going to according to whether and g have opposite or the same signs. The n ()(;) 2a th eigen function n x yxn should have n1 interior zeros (assuming that the eigen values are enumerated from n=1). 11
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12" For the Dirichlet condition as an exam ple of step (2): with differential equations (0)0 and boundary conditions: '()0 X and the general solution is: 12cossin. CXC X Applying the boundary condition yields 12(0)0,(1)(0)0, CC sin. CX For the Robin condition as another example: with differential equation and boundary conditions: 0 x '(0)(0)0 and '()()0 XX xX the general solution is 12cossin CXC X ; applying the boundary condition 0120 1221'(0)(0)(0)(1)[(1)(0)] 0 x xxyCCC CC C yields 0[cossin].xCX X The ordinary differential equation in th e final independent variable has a nonhomogenous boundary/initial c ondition and no unspecified parameters; it is not an eigenvalue problem, but its basic so lution, satisfying the homogenous boundary condition only, is found as above. The general separation of va riables solution for a threedimensional problem with a single boundary nonhomogeneity looks like: 1. Coordinates , {Cartesian, cylindrical, or spherical} ,,,, ,,()()()() A HNT t 2. T is a sinusoid in time (wave equation) or exponential (d iffusion equation) or not present (timeindependent Po issons equation, or frequency or Laplace domain).
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3. Two of the factors H N each satisfy a second order ordinary differential equation with a para meter and one homogenous boundary condition (holding for all values of the parameter), while the specified values for the parameter enfor ce the second boundary condition. The remaining factor satisfies only one homogenous boundary condition, and its parameter is fixed by the othe rs (through the partial differential equation). 4. The coefficients ,,A are determined from the boundary/initial conditions by orthogonality. 13
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14 CHAPTER 4 DISCUSSION AND FLOW CHART FOR USFKAD The coding of the expert system USFKAD was completed and successfully tested. The C++ [8] source contains about 5,000 lines of code and occupies 211 KB of computer memory. The executable file occupies 1300 KB. USFKAD utilizes the Separation of Variables technique. USFKAD has been encoded as an expert system for the eigenfunction expansion of solutions to the wave, diffusion, a nd Laplace equations: both homogeneous and nonhomogeneous; one, two, or three dimensions ; Cartesian, cylindrical, or spherical coordinate systems; Dirichlet, Neumann, R obin, or singular boundary conditions; in time, frequency, or Laplace domain. The user follo ws a menu to enter his/her choices and the output is a LaTeX file cont aining the formula for the solution together with the transcendental equation for the eigenvalues (if necessary) and the orthogonalprojection formulas for the coefficients. The output file is suitable, among other things, for insertion into a book or journal article and as a teaching aid. Virt ually all cases are covered, including the Mellin, spherical harmonic, Bessel, modified Be ssel, spherical Bessel, Dini, Hankel, Weber, MacDonald, and KantorovichLebedev expansions, mixed spectrum, and rigid body modes. The enabling attribute of this expert syst em is the observation that a decisiontree algorithm can be constructed to assemble th e eigenfunctions needed for any particular
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15 problem. Input to the algorithm would be the pa rticular PDE, the solution domain (time, Laplace, or frequency), and the geometry a nd boundary conditions. The software begins by scanning the boundaries looking for a nonhom ogeneous boundary condition. It then assembles the (noneigenfunction) factor for this direction and th e eigenfunction factors for the other directions, writes this to a file, and moves on to the next boundary. Finally it assembles and writes the Green's function te rms if the PDE is nonhomogeneous, and the transient or oscillatory term s. By passing generic variab le names between subroutines and exploiting the similarity of the l ogic for the Laplace, frequency, and timeindependent cases, the program needs only a bout 200 eigenfunction subroutines to cover all the possibilities. Each such subroutine co ntains the requisite normalization constants and the transcendental equati ons defining the eigenvalues. Decision Tree The flow chart below (Figure 1) depict s and describes the logic behind USFKAD. Following the chart, one can be guided to reach a successful, complete, and comprehensive solution to any appropriate pa rtial differential equation along with its specific boundary conditions. This flow ch art represents the brain of USFKAD.
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16 Type of Partial Differential Equations Laplace Diffusion Wave Figure 1: USFKAD Flow Chart
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17 Laplace Â“homogeneousÂ”? Â“NonhomogeneousÂ”(Poisson Equation)? Number of dimensions 2 or 3? Coordinate System? Rectangular, cylindrical (polar), or spherical Boundary Conditions? Next Boundary Next Boundary or Next variable Homogeneous or Nonhomogeneous? Dirichlet, Neum. Or Robin? Periodic? Regular? Singular? Construct. Nonhomogeneous Homogeneous Done Use Green function code append nonhomogeneous 123...BC Figure 1: Continued
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18 Diffusion Â“homogeneousÂ”or Â“nonhomogeneousÂ” (Sources) Â“Are any of the nonhomogeneities(source or boundary conditions) time dependent?Â” Yes No Â“Must use Laplacetransform in time formulationÂ” Timedomain or Laplacetransform in time? Time? Laplace? # dimensions, coordinate system, BCs, Same as Laplace # dimensions, coordinate system, BCs Same as Laplace Construct BC. Use initialvalue code append IC Homogeneous? Use Green function code append nonhomogeneous NonHomogeneous? Construct 1+ 2+ ...+ BC All Â’s and FÂ’s are capitalized, and s is included as a variable: Fx= 0(s; x,y, z) instead of fx= 0(x,y, z) Use Green Function code; append ICor IC+nonhomogeneous Figure 1: Continued
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19 Wave Time? No Yes Homogeneous or Nonhomogeneous(sources) Are any ofthe nonhomogeneities (source or boundary conditions) time dependent? Frequency? Laplace? Must use Laplace or Frequency Domain Proceed exactly like diffusion equation,time domain. The form for IChas an extra term to account for the initial value of #dim, cord. Syst., BCÂ’s same as Laplace Proceed exactly like diffusion equation, Laplace Transform domain. The form ICis slightly different. # Dim? Coordinate System? Boundary Conditions: Lower, Upper Periodic Singular Regular (Continued on next page) Time, Laplace or Frequency Domain? t Figure 1: Continued
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20 Periodic Singular Regular Homogeneous Next Variable Construct 1+ 2+...+ BC Next Boundary Condition or Next Variable Incoming or Outgoing Wave Same as Laplace Dir., Neum., Robin, #N Nonhomogeneous Done Use GreenÂ’s function code, Append nonhomogeneous Figure 1: Continued
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21 Fundamental Concept The fundamental concept of USFKAD is contained in the following two matrices: the PDE type matrix (table 1) and the PDE boundary condition (BC) ma trix (table 2) as illustrated below. Table 1: PDE Equation Type Matrix PDE Equation Type Matrix Zero One Two Three Four Five Six Seven eqtype PDE Dimension Coordinate System Coordina te 1 Coordinate 2 Coordinate 3 Time Homogeneous or Nonhomogeneous? 0=Laplace 1,2,3 0=rectangular x= 0; y=1 z=2 t=8 H=0, N=1 1=Diffusion, t 1=polar/ cylindrical theta=3; r2d=4 2=Diffusion, s 2=spherical phi=8 theta3=9 r3D=6 3=Wave, t theta= 3; z=2 Rho=5 4=Wave, s 5=Wave, omega Table 2: PDE BC Matrix PDE Boundary Condition Matrix Zero One Two Three Four Five Six Seven Eight Nine Row #0 R or S D,N,R N,H CC R or S D,N,R,O,I N,H x 0 (zero) X Row #1 R or S D,N,R N,H CC R or S D,N,R,O,I N,H y 0 (zero) Y Row #2 R or S D,N,R N,H CC R or S D,N,R,O,I N,H z 0 (zero) Z Row #3 R D,N,R,P N,H CC R D,N,R,P N,H theta 0 (zero) THETA Row #4 R or S D,N,R N,H CE R or S D,N,R,O,I N,H r (2d) a b Row #5 R or S D,N,R N,H BB R or S D,N,R,O,I N,H rho a b Row #6 R or S D,N,R N,H SB R or S D,N,R,O,I N,H r3D a b Row #7 R D N,H CC R D N,H t 0 (zero) 0 (zero) Row #8 S S H LG S S H phi 0 (zero) pi Row #9 R or S P H SH R or S P H theta3d 0 (zero) 2 pi
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22 The PDE type matrix (table 1) identifie s all types of PDEÂ’s including the number of dimensions, coordinate system, coordinates, whether it is time dependent or not, and whether it is homogeneous or not. Now once the PDE Equation type has been selected the Boundary conditions must be identified. The BC matrix virtually account s for all types of PDEÂ’s. Using the PDE type matrix and the BC matrix, we came up with 200 different possible PDE cases or scenarios that virtually cover and provide so lutions for all PDE problems. To accomplish this there are 200 subroutines in USFKAD code which provides solutions for each possible PDE problems. Example 1 Consider LaplaceÂ’s equati on in a box, Homogeneous Di richlet conditions at x=0 and X, homogeneous Dirichlet at y=0, nonhomo geneous Dirichlet at y=Y. homogeneous Neumann condition at z=0, nonhomogeneous Neuma nn at z=Z. As the user inputs these data, USFKAD fills in the BC matrix as shown on table 3.
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23 Table 3: Matrix Example Column #0 #1 #2 #3 #4 #5 #6 #7 #8 #9 eqtype PDE dim coord coord 1 coord 2 coord 3 time H, N? 0=Laplace 3 0=rect x=0; y=1 z=2 H=0 Boundary Conditions "mtrx" Row #0 Regular D H CC R D H x 0 (zero) X Row #1 Regular D H CC R D N y 0 (zero) Y Row #2 Regular N H CC R N N z 0 (zero) Z Row #3 Row #4 Row #5 Row #6 Row #7 Row #8 Row #9 Note that final solution will be z Y k X j Y y k X x j b y Z k X j Z z k X x j ajk k j jk k j 2 2 1 1 2 2 0 1) ( ) ( cosh sin sin ) ( ) ( sinh cos sin with XZ Y y Z X jkdx dz Z z k X x j z x f dz Z z k dx X x j a00 0 2 0 2. cos sin ) ( cos 1 sin 1
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24 XY Z z Y X jkdx dy Y y k X x j y x f dy Y y k dx X x j b00 0 2 0 2. sin sin ) ( sin 1 sin 1 USFKAD scans down column 2, and then colu mn 6, until it sees an N (nonhomogeneous) in row 1, column 6. It begins to assemble the solution to the sub problem where all BCs are homogeneous except at y=Y. Saving y for last it looks at the xrow, row #0. It sees the specifications 1. Constant coefficien t DE (from column 3), 2. Dirichlet BC at the low end (from column 1), 3. Dirichlet BC at the high end (from column 5), 4. Homogeneous BC at both ends (because y has the only nonhomogeneous). So USFKAD assembles the word "CCDDHH" from these columns and sends this word and the symbol "x" to the eigenf unction searcher. The latter returns the eigenfunction as "sin", the eigenvalues as "x = /X, 2/X, Â…", and the superposition type as "x". (If the eigenvalues formed a continuu m, the superposition type would be dx "). It also returns the formula for Xdx X x j0 2sin 1 and the weight factor for this DE ("1", hence blank, in this case.) USFKAD writes each of these returned formulas to separate lists, and looks for the next variable, z. It reads off the char acteristics from row #2 of the BC matrix and sends "z, CCNNHH" to the eigenfunction sorter which returns the appropriate formulas as before. They are concatenated with the earlier formulas in the corresponding lists.
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25 Then USFKAD fetches the symbols for the y factor by sending "y, CCDDHN, x, z", as read from row #1 and the previous list s, to the eigenfunction sorter (of course this factor is not literally an eigenfunction). The latter returns the sinh function and the 22 x zsymbol, and USFKAD concatenates these with the other lists and merges all the lists together, calling the result 1. Next USFKAD continues to scan column s 2 and 6, looking for the next "N" (nonhomogeneous BC) marker. It finds "N" in row #2, column #6, so it assembles x, y eigenfunctions and zfactors as before. If the PDE itself is nonhomogeneous (as marked in column 7 of eqtype), USFKAD assembles a sum for the eigenfuncti on expansion of the Green's function (3, example 1, chapter 5). In the time domain (as marked in colu mn 0 of eqtype), USFKAD assembles the time factors from the eigenvalues symbols it has accumulated (example 4, chapter 5). In the frequency or Laplace domain (as marked in column 0 of eqtype), USFKAD sends the eigenvalues (x, Â…) and the transform variable (s or ) (in accordance with column #0 of eqtype) to the subroutine s earcher for the nonhomoge neous factor. These nonhomogeneousfactor subroutines simply inco rporate them into the formula display ( 2 xsin example 5 and 222 x y in example 6, chapter 5). OnScreen Inquires USFKAD is invoked by answering a list of pr inted onscreen inquiries as follows: Select the Partial Differential Equation:
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26 0 Laplace or Poisson: Lapl acian Psi + f(interior) = 0; l Diffusion, Time Domain: d Psi/dt = Laplacian Psi + f(interior); 2 Diffusion, splane: s Psi Psi(t= 0) = Laplacian Psi + F(interior); 3 Wave, Time Domain: d2 Psi/dt2 = Laplacian Psi + f(interior); 4 Wave, splane: s2 Psi sPsi(t=o) Psi'(t=0) = Laplacian Psi + F(interior); 5 Wave, frequency domain: omega2 Psi = Laplacian Psi + F(interior) Is the PDE homogeneous (enter 0) or nonhomogeneous (enter 1)? Enter 1, 2, or 3 for 1, 2, or 3 dimensions. Select the Coordinate System: 0 Rectangular, 1 Cylindrical or Polar, 2 Spherical. Select the boundary condition at the lower (upper) end for the coordinate x (y, z, r, rho, and theta): Enter 1 for Dirichlet, Homogeneous; Enter 2 for Dirichlet, Nonhomogeneous; Enter 3 for Neumann, Homogeneous; Enter 4 for Neumann, Nonhomogeneous; Enter 5 for Robin, Homogeneous; Enter 6 for Robin, Nonhomogeneous; Enter 7 for Periodic Boundary Conditions; Enter 8 for Singular Boundary Condition; Enter 9 for Singular, Sommerfeld Outgoing Wave Condition; Enter 10 for Singular, Sommerfeld Incoming Wave Condition. The output of USFKAD is a LaTeX file It requires subsequent processing by TeX software, which must be re sident on the user's computer.
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27 The above procedure and methodology ar e further illustrated through the six different examples listed in the next chapter. Further details on the le xicon of the software in expressing the physical dimensions and boundary conditi ons appear in Appendices B and C, respectively.
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CHAPTER 5 APPLICATION EXAMPLES This chapter contains a list of examples of partial differential equation solutions that progressively demonstrate the decision tree nature of the general separation of variable procedure. Physical interpretations of these mathematical problems are stated here in terms of heat and sound phenomena; they all have electrical counterparts. Example 1 Steady state heat flow in a rectangle with edge and interior heat sources (nonhomogeneous Laplace/Poisson equation in tw o dimensions, rectangular coordinates, Dirichlet conditions on two sides, Neumann conditions on two sides): 2 int 0(,) (0,)0,(,)() (,0)(),(,)0erior xX yfxy y Xyfy xfxxY yy The solution is as follows: 12 3 1cos(;)()x x yxxxyA with 28
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0; sinh() 0 1 0; 2 1 0; 1 sinh23 0,,,,... (;) ()cos ()x x xx x x x x xx Yyif yxYyotherwise X xx if X otherwise X if Y otherwise YXXX y AdxxNMf N M y ox 2sincosh()yyyyxA y with 0 00 ; 1 sinh23 ,,,... 2 ()sin ()y y y yyy Y yyx if otherwise XYYY X A dyyMfy Y M 3cossin(,)xy x yxxyA y with int 22 00 1 0; 2 .cos23 0,,,,... 23 ,,,... (,) 2 (,) sinx x xx y XY erior xy x y xy if X otherwise XXXX YYY f xy Ad xd yxNy Y N Here we see some of the featur es of separation of variables: 29
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1. The basic decomposition of the problem into three subproblems, each of which contains only one nonhomogeneous equation. 2. Each subsolution expressed as a sum of terms containing an eigenfunction factor, satisfying homogeneneous boundary c onditions at each end, and a noneigenfunction factor, satisfying a homogeneous boundary condition at one end only (note the exceptional form of the latter factor in 2 and of the normalization constants for the cosines DC term when 0x ); 3. Coefficients computed by orthogona lity to give the corresponding nonhomogeneity; 4. The construction of the Greens func tion out of the same eigenfunctions. Example 2 Steady state heat flow in a cube with facial heat s ources and imperfect facial insulation (homogeneous Laplace equation in th ree dimensions, rectangular coordinates, homogeneous Dirichlet conditi ons on four sides, nonhomogeneneous Dirichlet condition on one side, homogeneous Robin condition on one side): 20 (0,,)0,(,,)0 (,,0)0,(,,)(,) (,0,)0,(,,),,0zZ yYyzXyz xyxyZfxy zz xzxYzxYz y The solution is 22sin(;)cosh (,)xy x yyxyxyxy zA 30
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with 2200 2 (23 ,,,... (;)sin ( ) sincos0 100,(;0). 2 (,)sin(;) (,)y xy y Yx yyyy Yyy Yy y XY xy xyy zZ YXXX y ywherearethenonzerosolutionspossiblyimaginaryto YY alsoifYincludeyy Ad xd yxyNMfxy X N 3 22 22 22 22223 0; 1)/() 00 ; 1 sinhy Yy xy xy xyxyif Y otherwise if otherwise ZM This example illustrates the straightforw ard extension of the procedure to three dimensions and the transcendental equa tion that the Robin boundary condition invokes for the eigenvalues. Example 3 Steady state heat flow in a cylindrical sector with facial heat sources (homogenous Laplace equation in the three di mensions inside a partial cylinder, nonhomogenous Dirichlet condition on the top and one flat side, homogenous Dirichlet conditions on the bottom and the curved wall, and a homogenous Neumann condition on the other flat side): 31
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20 ,,0,,0, ,0,0,,,,, (,,)(,)zZbz zzf Zf z The solution to this system is expressed 12 :::: 10: :z :sin[()()()()] x cosh (,) withzz p z z z z ziziziziz zpzdzbIIb A ::: ::: :00 22 : 2: : :23 ,,,... 2sinh 21 (,)sin()()()() (,) [()]cosh cos()(;)(,)zzz zz zz Zb zz z ziziziz iZ z zZZZ Ad zdzb IIb f ZI b JzA 1 z :zi with 35 ,,,. 222 .. For each value of :, :, j b where are the positive roots of :,{ j } 0 :,() Jj : :0; : sin .(;)zif z zotherwisez : : : : ::: 22 0 1, 1 0, 1 sin22 (,)cos() (,) ()b zZ a if z otherwise ZAddJ Mf bJj M 32 This example illustrates th e singular boundary condition for = 0 and the Lebedev eigen functions in included by the nonhomogenous condition for The singular boundary condition includes a continuous, rather th an discrete, spectrum. Other than [1], the Lebedev expansions do not appear in any English language mathematics textbook except [3], where their correctness is betrayed by a persistent systematic error in the tabulations. Their omission is probably due to their intimidating nomenclature (note the
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analytic condition of the subscript into the complex plane). Ignoring them is, however, criminal, because (as we see) they occur in r ealistic problems; indeed, in the analysis of edge diffraction [2] they are crucial. Example 4 Sound wave inside a sphere (homogenous wave equations inside a sphere, timeindependent nonhomogenous Dirichlet conditions on the surface): 2 2 2(,,,),rbt rbtf The solution to this system is expressed #1 #2 0(,)steadystatetransient transienttransienttransient s teadystate m m mYr l l A l l ll with 2* 00 #1 01sin (,)(,) 1 ,()cosmm r r b r tranient m r rmr mpAddYMf M b YjrtA ll l l lll ll with ,/rpsl b where s l ,p is the pth positive zero of j l 22* 000 32 1, #2 012 sin ,,;0 () sin ,( )b mr m r steadystate p r transient m r mr mp rAddr d r Yjrr bjs t YjrA ll l ll l lll ll with 33
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34b ,/rpsl where s l ,p is the pth positive zero of j l 22* 000 32 1,2 sin ()b mr m r pAddr d r Yjr bjs ll l ll,,r;0 t This example demonstrates the decomposition of the solution into a steadystate component and a transient component. The tim e functions can be treated just like the noneigenfunctions in the previ ous solutions, except that th ey satisfy initial conditions instead of boundary conditions. Example 5 Transient heat flow in a rectangle with transient interior and edge heat sources (nonhomogenous diffusion equations, two dime nsions, rectangular coordinates, homogenous Dirichlet conditions on two sides, homogenous Neumann condition on one side, timedependent nonhomogenous Neumann conditions on one side): 2 int,, 0,0,,0 ,00,,,erior yYfxyt t yXy x xYfxt yy With int,;erior F xys and ;yY F xs denoting the Laplace transforms of int,,erior f xyt and (,)yY f xt respectively, the Laplacetransformed solution to this system is expressed 12 2 1sincosh ;x x xxxsyA s with 23 ,,,...x X XX 202 ;sinxX xxy Y s ; A sdxxMF xs X
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22 22 200 1 sinh sincos,;x xyx s xx xyxyifs M otherwise ssY xyAs ; with int 00 2223 ,,,... 23 0,,,,... (,;),,0 2 ,; sincos 1 0; 2 .y yx y erior XY xy xy xy yXXX YYY Fxysxyt AsdxdyxyN Xs if Y N otherwise Y This example demonstrates how the logic that solved Example 1 can be retooled to solve Laplace domain problems; the transformed PDE is equivalent to a nonhomogenous Laplace (Poisson) equation with the eigenvalues shifted and the initial condition wedded with the nonhomogeneity. Example 6 Wave launched from one end of a rect angular waveguide (homogeneous wave equation in a semiinfinite rectangular wa veguide, homogenous Di richlet conditions on the walls, nonhomogenous Dirichlet on the e nd face, frequency domain, outgoing wave at infinity): 2 2 2 00,,,,,0,,,0, ,,0,,zt yzXyzxzxYz xyfxyt 35
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36y With denoting the Fourier transforms of 0;,zFx 0,,z f xyt the Fourier transforms of the solution to this system is expressed 222sinsin ;,xy xyiz x yxxyeA y with 00 023 ,,,... 23 ,,,... 22 ;, sinsin;,x y XY xy y yzXXX YYY Ad x d yxyF x y X Y Again, the solution logic of Example 1 is reworked with a shift of the eigenvalues.
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37 CHAPTER 6 RECOMMENDATIONS FOR FUTURE DEVELOPMENTS To further enhance and build on the success of USFKAD, the following recommendations are offered for future development of this expert system: 1. Offer a choice of output formats (pos tscript, Mathematics Markup Language) 2. Enable transport of the output to programs like MAPLE, MathCad, or Mathematica for number crunching. 3. Offer graphical supplements to the outputs (such as sa mple eigenfunction graphs). 4. Develop graphic pointandcl ick input options to rende r it more convenient and easy to use.
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REFERENCES 1. Snider, A.D., Partial Differential Equations: Sources and Solutions, PrenticeHall, Upper Saddle River, NJ, 1999. 2. Felsen, L.B., and Marcuvitz, N., Radiation and Scattering of Waves, IEEE Press, Piscataway, NJ 1994. 3. Polyanin, A.D., Linear Partial Differential Equations for Engineers and Scientists Chapman and Hall/Chemical Rubber Co., Boca Raton, 2001. 4. ChebTerrab, E.S., and von Bulow, K., A Computational Approach for the Analytical Solving of Partial Di fferential Equations, Computer Physics Communications v. 90, 102116, 1995. 5. Kalnins, E.G. and Miller, Jr., W., Variable Separable in Mathematical Physics: From Intuitive Concept to Computational Tool (Working paper) 6. Wolfram Research, Inc., Mathematica, Version 5.1, Champaign, IL (2004). 7. Waterloo Maple, Inc., Maple V release 3 Canada, 2001. 8. Borland International, Borland C++ 5.02 Scotts Valley, CA, 1994. 38
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39 APPENDIECES
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APPENDIX A: COMPARISON OF SO LVING PARTIAL DIFFERENTIAL EQUATIONS USING THE TRADITONAL METHOD AND USFKAD y 4() f x 40 1() f y x 20 3() f y x x 2() f x Figure 2: PDE Problem Consider the boundary value problem shown in Figure 2. We take advantage of the linearity of the equations to simplify the analysis. Suppose we compute the following solutions to the following four sub problem s shown in Figure 3 ad: Decomposition of Problems. y 10 1 1() f y x 2 10 10 x x 10 (a) Figure 3: (ad) Decompos ition of the PDE Problem
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Appendix A: y Continued 20 41 20 x 2 20 20 x x 22() f x (b) y 30 30 x 2 30 3 3() f y x x 30 (c) y 44 f x 40 x 2 40 40 x x 40 (d) Figure 3: Continued
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Appendix A: Continued Problem 1: Find 1(,) x y such that inside the square (1) 2 1(,)0 xy 1 1(0,)() yfy x on the left edge (0
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Appendix A: Continued 3(0,)0 y x on the left edge (0
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Appendix A: Continued 22 2 4 22(,)0()()()() "()()()"() xy XxY y XxY y xy XxYyXxYy "()"() ()() X xY y X xY y (22) The separated equation (22) implies: "() () Xx Xx or "()()0XxXx (23) For some constant (separation constant) and "()()0YyYy (24) The general solution to the harmonic oscill ator equation (23), can be expressed as: 12()coshsinh Xxaxax if 0 (25) 12() X xbb x if 0 (26) 12()cossin Xxcxcx if 0 (27) As for the solutions for (24), they are: 12()cossin Yydydy if 0 (28) if 12() Yyeey 0 (29) 12()coshsinh Yygygy if 0 (30) From (2)(5): '()0,'(0)0XX (31) (32) (0)0Y 44
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Appendix A: Continued 12'()sinh cosh Xxaxax if 0 (33) 2'() X xb if 0 X (34) 12'()sin cos Xxcxcx if 0 (35) Applying the boundary condition '(0)0X we get 2'(0)0 Xa if 0 (36) if 2'(0)0 Xb 0 (37) 2'(0)0 Xc if 0 (38) So 1()cosh Xxax if 0 (39) 1() X xb if 0 (40) 1()cos Xxcx if 0 (41) Imposing '() 0X implies: 1'()sinh0 Xa if 0 (42) if '0 X 0 (43) 1's i n h 0 Xc0 if (44) Since we are not interested in trivial solutions, or can not equal zero, so we need to satisfy (42)(44) by the selection of 11,, ab 1c The choice 0 is acceptable 1() X xb (=constant). No positive value for yield solutions, because the sin function in (42) never vanishes. However, the sin function vanishes whenever 1,2,3...,n and the corresponding solution (39)(41) is cos nx. So we have 1c 45
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Appendix A: Continued nontrivial solutions for th e x factor in (21) if applying (32) to (28)(30): 20,1,4,...,,... n if 1(0)0 Yd 0 if 100 Ye 0 if 1(0)0 Yg 0 and 2()sinhnYygny 02() Yyey cossinhnnxny 40 1(,) cossinhn n x yayanxny Where 04 4 2 0012 (),0 ()0cos sinhnafxdxafx nx nx You see how long and cumbersome it was to do only problem 4. As for problems 3, 2, and 1, we will get the following after a lot of manipulation: Problem 3: 3 1(,)coshsinn n x ybnxny where 3 02 ()sin sinhnbf y nn n y d y Problem 2: 20 1(,)()cossinh()n n x ycycnxny where 46
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Appendix A: Continued 02 2 01 () cfx dx 02 02 ()cos sinhncf x n n x d x Problem 1: 1 1(,)cosh()sinn n x ydnx ny where 1 02 ()sin. sinhndf y nn n y d y The final answer will be 1234(,)(,)(,)(,)(,) x yxyxyxyx y Solution 2 Complete solution using USFKAD. 1234 1sincoshyyyyXxA y where 23 ,,,...yYYY 002 sinyY yy x A dyyMfy Y 47
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Appendix A: Continued 00 1 sinhyy yyif M otherwise X ; 2cos;x x yxxyA x where 23 0,,,,...x X XX 0 ; sinh .x yx xYyif y Yyotherwise 00cosxxX xx y A dxxNMfx 1 0; 2 .xxif X N otherwise X 1 0; 1 sinhxx xif Y M otherwise Y 3sincoshyyyyxA y where 23 ,,,...yYYY 02 sinyY yyx X A dyyMfy Y 00 1 sinhyy yyif M otherwise X ; 4cos;x x yxxyA x where 48
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Appendix A: Continued 23 0,,,,...x X XX 0 ; sinh .x yx xyif y yotherwise 0cosxxX xx y Y A dxxNMfx 1 0; 2 .xxif X N otherwise X 1 0; 1 sinhxx xif Y M otherwise Y If you set X=Y= you will find that both the traditional method and USFKAD solutions are identical. 49
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5 0 APPENDIX B: READ ME Welcome to USFKAD, the software for solving partial differential equations analytically, by separation of va riables. If you find this softwa re is of value to you, please consider making a donation to USF students via the Allen Gondeck scholarship fund, through Prof. A. D. Snider, University of South Florida, ENB 118, Tampa FL 33620. (Thank you.) The task of constructing complete soluti ons by separation of variables is quite tedious, and the software can do this fo r you only if you follow the format/notation conventions precisely. The current version handles (hom ogeneous or nonhomogeneous) (mixed) Dirichlet, Neumann, constantcoefficient Robin boundary conditions, or singular boundary conditions for the (possibly) nonhomogeneous Poisson, diffusion, or wave equations, in the time, frequency, or Laplace domains. You will have to label the dimensions of your domain to conform to one of the following conventions: 1. 0 < x,y,z, < X,Y,Z 2. 0 < x,y,z < (Do not use < x,y,z < X,Y,Z ) 3. < x,y,z < 4. 0 < < 2 (periodic) 5. 0 < a < r < b <
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5 1 Appendix B: Continued 6. 0 < a < r < 7. 0 < r < b < 8. 0 < r < 9. 0 < < 2 and 0 < < (spherical coordinates) Dirichlet boundary conditions take the form (0, y,z) = f x =0 ( y,z) ; ( X y,z) = f x=X ( y,z) (and similarly for y z r and ). Neumann boundary conditions take the form (0, y,z)/ x = f x=0 ( y,z) ; ( X ,y,z)/ x = f x=X ( y,z) (and similarly for y z r , and ). Note that the relevant partial will not in general, be the external normal derivative. Robin boundary conditions take the form x=0 (0, y,z ) + (0, y,z)/ x = f x=0 ( y,z ) ; X ( X ,y,z) + ( X ,y,z)/ x = f x=X ( y,z) (and similarly for y z r , and ). Note that the relevant partial will not in general, be the external normal derivative. The coefficient is presumed constant.
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5 2 Appendix B: Continued To use the software, doubleclick on the .exe file, and follow the menu instructions carefully Give the name of your output file a .tex subscript. The software does not alert the user if inconsistent parameters are input it simply fails to produce an output file. Run latex on the output file, and eith er view the .dvi result or dvips it to postscript, and print ou t. You may make format changes to the .tex file if you wish. Please inform A. D. Snider by email ( snider@eng.usf.edu ) if you feel the software has returned an incorrect answer, or if you desire elaboration of the answer; include a complete problem statement and the output .tex file, and your comments. Enjoy! Sami Kadamani Dave Snider
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53 APPENDIX C: ToUseUSFKAD.pdf To use USFKAD: Create a folder on your hard drive, place it in root of the drive, and name it USFKAD. If you have several hard drives, you may use any drive you want. For this tutorial, ddrive is used. Location of the folder, will be d:\USFKAD> Open Windows Explorer, browse to d:\USFKAD and double click USFKAD program file. The Command Prompt will open, and you get a warning message. Click OK to proceed. Now you have to create the output file, and in this example it is named: testing.tex Remember the extension tex Type in your filename, in this case testing.tex Follow the instructions in USFKAD, and for this tutorial we used the following input: 0, 0, 2, 0, 1, 1, 2 and 2 This will create the file tes ting.tex in the folder d:\USFKAD Next step will be to create th e dvi file. Click on Start, Run.. Type cmd in the Run window. This will open the command prompt. In the command promt, browse to your folder. d: will take you to the d drive, then cd usfkad, will change the directory to where the testing.tex file is. Type d: cd usfkad to get to d drive, and the usfkad folder Once you are in the correct folder type latex to open the appl ication. Then press enter to proceed after the welcome note. Open LaTeX by typing latex Type the file name, in this case testing. There is no need for the file extension, but it has to be tex when you created it. Type the filename, in this case testing This will give you a dvi output file. If ther e are any problems, you will see it on this screen The status of creating the dvi file Appendix C: Continued Next step will be to view dvi file. Click on Start, Run..
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54 Type yap in the Run window. The pr ogram used to view the dvi file. In Yap window, choose open file, and select testing.dvi Open the testing.dvi file. Yap will show the final output Output from the testing.te x file crated with USFKAD
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ABOUT THE AUTHOR Sami M. Kadamani is currently an in structor at Hillsborough Community College in Tampa, Florida. He completed his Bachel or of Science in Electrical Engineering from the University of Delaware in 1983, and his Ma ster of Science in Electrical Engineering from North Carolina A & T State University in 1986. In 1986 Mr. Kadamani moved to Tampa, Florida to pursue his Doctoral of Ph ilosophy degree in Engineering Science at the University of South Florida, which he completed in May 2005.
