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A Technique for Solving the Singular Integral Equations of Potential Theory

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A Technique for Solving the Singular Integral Equations of Potential Theory Old Dominion University ODU Digital Commons Mathematics & Statistics Theses & Dissertations Mathematics & Statistics Spring 2007 A Technique for Solving the Singular Integral Equations of Potential Theory Brian George Burns Old Dominion University Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_etds Part of the Mathematics Commons Recommended Citation Burns, Brian G.. "A Technique for Solving the Singular Integral Equations of Potential Theory" (2007). Doctor of Philosophy (PhD), Dissertation, Mathematics & Statistics, Old Dominion University, DOI: 10.25777/q9kv-0x34 https://digitalcommons.odu.edu/mathstat_etds/6 This Dissertation is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Theses & Dissertations by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY by Brian George Burns M.Sci. 1998, University of Glasgow M.Sc. 1999, University of Stirling M.S. 2005, Old Dominion University A Dissertation Submitted to the Faculty of Old Dominion University in Partial Fulfillment of the Requirement for the Degree of DOCTOR OF PHILOSOPHY COMPUTATIONAL AND APPLIED MATHEMATICS OLD DOMINION UNIVERSITY May 2007 John Tweed (Director) Ion Melrose John/A. Adam Richard D. Noren Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ABSTRACT A TECHNIQUE FOR SOLVING THE SINGULAR INTEGRAL EQUATIONS OF POTENTIAL THEORY Brian George Burns Old Dominion University, 2007 Director: Dr. John Tweed The singular integral equations of Potential Theory are investigated using ideas from both classical and contemporary mathematics. The goal of this semi-analytic ap­ proach is to produce numerical schemes that are both general and computationally simple. Previous works based on classical methods have yielded solutions only for very special cases while contemporary methods such as finite differences, finite el­ ements and boundary element techniques are computationally extensive. Since the two-dimensional integral equations of interest exhibit structural invariance under a wide class of conformal mappings initial emphasis is placed on circular domains. By Fourier expansion with respect to the angular variable, such two-dimensional integral equations yield simultaneous systems of one-dimensional integral equations that, in many cases, uncouple. Integral transform techniques and classical function theory are used to identify the eigenfunctions associated with the dominant parts of the one­ dimensional singular equations. Hilbert spaces spanned by these eigenfunctions are then constructed and an operator theory developed for the general class of integral equations. Numerical algorithms are derived for both Galerkin and collocation solu­ tion techniques with convergence proved in the Galerkin case and collocation method verified experimentally. A generalization of the Hilbert space theory is then applied to the two-dimensional case with eigenfunctions created by combining the angular Fourier terms with the radial eigenfunctions of the dominant one-dimensional parts. Numerical algorithms based Galerkin and collocation methods are again derived and used to solve the two-dimensional equations. The techniques developed are used to solve a number of both previously known and new problems in Electrostatics and Fracture Mechanics. Simple layer potential representations yield weakly singular in­ tegral equations for the induced charge on disc shaped conductors that are placed in an electrostatic field. Similarly, double layer potentials yield hyper-singular integral equations for the crack opening displacement of penny shaped cracks in an elastic Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. solid under various loading conditions. Conformal mapping techniques for solving problems on non-circular domains are also briefly discussed as are extensions to fields that are governed by the Helmholtz Equation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS Thanks and appreciation go out to all of my committee members for being part of this project and taking the time to read this rather long and detailed document. So appreciation and thanks to Professor James Cox of the Physics Department for being my external committee member. Thanks to Dr. Richard Noren for becoming the final department member. Thanks to Professor Adam for taking time from his busy schedule as a Virginian Outstanding Faculty to provide not only creditability to my thesis but also a significant source of humor. Thanks to Dr. Melrose for his service as a teacher, a supervisor and especially being the inspiration and encouragement behind my coming to the US and Old Dominion in particular. Special thanks go out to Professor Tweed who has been an inspirational advisor and mentor whose patience has held strong at all times despite being tested by my stubbornness and occasional lack of attention to detail. I could not have asked for any more in an advisor than what Professor Tweed provided. I must also thank Professor Hideaki Kaneko and in particular Professor Mark Dorrepaal for their support in finishing my dissertation both financial and in their efforts freeing up my teaching commitments to facilitate my dissertation completion. Special thanks too to Dr. Humberto Rocha for all his help and support both while a colleague at Old Dominion and since his return to Portugal. In particular his help with Latex and formatting was invaluable. I would also like to thank my family for all their moral and occasional financial support through my hardships as a graduate student. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vi TABLE OF CONTENTS Page List of Tables ................................................................................................................. ix List of Figures .............................................................................................................. xii CHAPTERS I Introduction .......................................................................................................... 1 II Preliminary Results ............................................................................................. 6 II. 1 Weber-Schafheitlin Integrals .................................................................... 6 II.2 Layer Potential Solutions to Laplacian Boundary Value Problems . 11 11.2.1 The Three-Dimensional Green’s Functions ................................ 12 11.2.2 The Single Layer Potential Operator .......................................... 12 11.2.3 The Double Layer Potential O perator ....................................... 14 11.2.4 The Dirichlet Problem .................................................................... 15 11.2.5 The Neumann Problem ................................................................ 16 III Hilbert Spaces I ................................................................................................... 18 III.l Introduction .............................................................................................. 18 111.2 The Hilbert Spaces L?! ( 0 ,1 ) ..................................................................... 18 111.3 Operators on the L% (0,1) Hilbert Spaces ............................................. 23 111.3.1 The L" Operators .......................................................................... 24 111.3.2 The Differential (®m) and the Hyper-Singular (BmL^) Oper­ ators ................................................................................................. 37 111.4 The (0,1) Hilbert Spaces ................................................................. 40 IV A Weakly-Singular Integral Equation ............................................................. 49 IV. 1 Introduction .............................................................................................. 49 IV. 2 The Dominant Weakly-Singular E quation ........................................... 49 IV. 3 The General E q u a tio n .............................................................................. 50 IV.4 Numerical Solutions .................................................................................. 58 IV.4.1 Q uadrature........................................................................................ 58 IV.4.2 The Galerkin M ethod .................................................................... 59 IV.4.3 The Collocation Method .............................................................. 60 IV.4.4 Numerical T ests .............................................................................. 61 V A Hyper-Singular Integral Equation ................................................................ 63 V.l Introduction .............................................................................................. 63 V.2 The Dominant Hyper-Singular Equation .............................................. 63 V.3 The General Hyper-Singular Equation ................................................. 64 V.4 Numerical Solutions .................................................................................. 6 8 V.4.1 Q uadrature.......................................................................................
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