Handbook of integral equations

796 Pages·1998·6.143 MB·English

by  Andrei D. Polyanin, Alexander V. Manzhirov

HANDBOOK OF
INTEGRAL
EQUATIONS
© 1998 by CRC Press LLC
Andrei D. Polyanin
and
Alexander V. Manzhirov

HANDBOOK OF
INTEGRAL
EQUATIONS
Pageiv
Library of Congress Cataloging-in-Publication Data
Polyanin, A. D. (Andrei Dmitrievich)
Handbook of integral equations/Andrei D. Polyanin, Alexander
V. Manzhirov.
p. cm.
Includes bibliographical references (p. - ) and index.
ISBN 0-8493-2876-4 (alk. paper)
1. Integral equations—Handbooks, manuals, etc. I. Manzhirov. A.
V. (Aleksandr Vladimirovich) II. Title.
QA431.P65 1998 98-10762
515’.45—dc21 CIP
This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with
permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish
reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials
or for the consequences of their use.
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Visit the CRC Press Web site at www.crcpress.com
© 1998 by CRC Press LLC
No claim to original U.S. Government works
International Standard Book Number 0-8493-2876-4
Library of Congress Card Number 98-10762
Printed in the United States of America 3 4 5 6 7 8 9 0
Printed on acid-free paper
ANNOTATION
Morethan2100integralequationswithsolutionsaregiveninthefirstpartofthebook. Alot
of new exact solutions to linear and nonlinear equations are included. Special attention is paid to
equations of general form, which depend on arbitrary functions. The other equations contain one
or more free parameters (it is the reader’s option to fix these parameters). Totally, the number of
equationsdescribedisanorderofmagnitudegreaterthaninanyotherbookavailable.
A number of integral equations are considered which are encountered in various fields of
mechanics and theoretical physics (elasticity, plasticity, hydrodynamics, heat and mass transfer,
electrodynamics,etc.).
The second part of the book presents exact, approximate analytical and numerical methods
for solving linear and nonlinear integral equations. Apart from the classical methods, some new
methodsarealsodescribed. Eachsectionprovidesexamplesofapplicationstospecificequations.
The handbook has no analogs in the world literature and is intended for a wide audience
of researchers, college and university teachers, engineers, and students in the various fields of
mathematics,mechanics,physics,chemistry,andqueuingtheory.
Pageiii
FOREWORD
Integral equations are encountered in various fields of science and numerous applications (in
elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory,
electrostatics, electrodynamics, biomechanics, gametheory, control, queuingtheory, electricalen-
gineering,economics,medicine,etc.).
Exact (closed-form) solutions of integral equations play an important role in the proper un-
derstanding of qualitative features of many phenomena and processes in various areas of natural
science. Lotsofequationsofphysics,chemistryandbiologycontainfunctionsorparameterswhich
areobtainedfromexperimentsandhencearenotstrictlyfixed. Therefore,itisexpedienttochoose
the structure of these functions so that it would be easier to analyze and solve the equation. As a
possibleselectioncriterion,onemayadopttherequirementthatthemodelintegralequationadmita
solutioninaclosedform. Exactsolutionscanbeusedtoverifytheconsistencyandestimateerrors
ofvariousnumerical,asymptotic,andapproximatemethods.
More than 2100 integral equations and their solutions are given in the first part of the book
(Chapters1–6). Alotofnewexactsolutionstolinearandnonlinearequationsareincluded. Special
attention is paid to equations of general form, which depend on arbitrary functions. The other
equations contain one or more free parameters (the book actually deals with families of integral
equations); it is the reader’s option to fix these parameters. Totally, the number of equations
described in this handbook is an order of magnitude greater than in any other book currently
available.
Thesecondpartofthebook(Chapters7–14)presentsexact,approximateanalytical,andnumer-
icalmethodsforsolvinglinearandnonlinearintegralequations. Apartfromtheclassicalmethods,
some new methods are also described. When selecting the material, the authors have given a
pronouncedpreferencetopracticalaspectsofthematter; thatis, tomethodsthatalloweffectively
“constructing” the solution. For the reader’s better understanding of the methods, each section is
suppliedwithexamplesofspecificequations. Somesectionsmaybeusedbylecturersofcolleges
anduniversitiesasabasisforcoursesonintegralequationsandmathematicalphysicsequationsfor
graduateandpostgraduatestudents.
Fortheconvenienceofawideaudiencewithdifferentmathematicalbackgrounds, theauthors
triedtodotheirbest,whereverpossible,toavoidspecialterminology. Therefore,someofthemethods
are outlined in a schematic and somewhat simplified manner, with necessary references made to
books where these methods are considered in more detail. For some nonlinear equations, only
solutionsofthesimplestformaregiven. Thebookdoesnotcovertwo-,three-andmultidimensional
integralequations.
The handbook consists of chapters, sections and subsections. Equations and formulas are
numbered separately in each section. The equations within a section are arranged in increasing
orderofcomplexity. Theextensivetableofcontentsprovidesrapidaccesstothedesiredequations.
Forthereader’sconvenience,themainmaterialisfollowedbyanumberofsupplements,where
somepropertiesofelementaryandspecialfunctionsaredescribed,tablesofindefiniteanddefinite
integralsaregiven,aswellastablesofLaplace,Mellin,andothertransforms,whichareusedinthe
book.
Thefirstandsecondpartsofthebook,justasmanysections,werewrittensothattheycouldbe
readindependentlyfromeachother. Thisallowsthereadertoquicklygettotheheartofthematter.
Pagev
We would like to express our deep gratitude to Rolf Sulanke and Alexei Zhurov for fruitful
discussions and valuable remarks. We also appreciate the help of Vladimir Nazaikinskii and
AlexanderShternintranslatingthesecondpartofthisbook,andarethankfultoInnaShingarevafor
herassistanceinpreparingthecamera-readycopyofthebook.
The authors hope that the handbook will prove helpful for a wide audience of researchers,
collegeanduniversityteachers,engineers,andstudentsinvariousfieldsofmathematics,mechanics,
physics,chemistry,biology,economics,andengineeringsciences.
A.D.Polyanin
A.V.Manzhirov
Pagevi
SOME REMARKS AND NOTATION
1. In Chapters 1–11 and 14, in the original integral equations, the independent variable is
denotedbyx,theintegrationvariablebyt,andtheunknownfunctionbyy =y(x).
2. Forafunctionofonevariablef =f(x),weusethefollowingnotationforthederivatives:
df d2f d3f d4f dnf
f(cid:1) = , f(cid:1)(cid:1) = , f(cid:1)(cid:1)(cid:1) = , f(cid:1)(cid:1)(cid:1)(cid:1) = , and f(n) = for n≥5.
x dx xx dx2 xxx dx3 xxxx dx4 x dxn
Occasionally,weusethesimilarnotationforpartialderivativesofafunctionoftwovariables,
∂
(cid:1)
forexample,K (x,t)= K(x,t).
x ∂x
(cid:2) (cid:3)
d n
3. Insomecases,weusetheoperatornotation f(x) g(x),whichisdefinedrecursivelyby
dx
(cid:4) (cid:5) (cid:6)(cid:4) (cid:5) (cid:7)
d n d d n–1
f(x) g(x)=f(x) f(x) g(x) .
dx dx dx
4. ItisindicatedinthebeginningofChapters1–6thatf =f(x),g =g(x),K =K(x),etc.are
arbitraryfunctions,andA,B,etc.arefreeparameters. Thismeansthat:
(a) f =f(x),g =g(x),K =K(x),etc.areassumedtobecontinuousreal-valuedfunctionsofreal
arguments;*
(b) if the solution contains derivatives of these functions, then the functions are assumed to be
sufficientlydifferentiable;**
(c) ifthesolutioncontainsintegralswiththesefunctions(incombinationwithotherfunctions),then
theintegralsaresupposedtoconverge;
(d) thefreeparametersA,B,etc.mayassumeanyrealvaluesforwhichtheexpressionsoccurring
A
intheequationandthesolutionmakesense(forexample,ifasolutioncontainsafactor ,
1–A
thenitisimpliedthatA≠1;asarule,thisisnotspecifiedinthetext).
5. The notations Rez and Imz stand, respectively, for the real and the imaginary part of a
complexquantityz.
6. Inthefirstpartofthebook(Chapters1–6)whenreferencingaparticularequation,weusea
notationlike2.3.15,whichimpliesequation15fromSection2.3.
7. Tohighlightportionsofthetext,thefollowingsymbolsareusedinthebook:
(cid:1) indicatesimportantinformationpertainingtoagroupofequations(Chapters1–6);
indicatestheliteratureusedinthepreparationofthetextinspecificequations(Chapters1–6)or
sections(Chapters7–14).
* Lesssevererestrictionsonthesefunctionsarepresentedinthesecondpartofthebook.
** Restrictions(b)and(c)imposedonf =f(x),g=g(x),K=K(x),etc.arenotmentionedinthetext.
Pagevii
AUTHORS
AndreiD.Polyanin,D.Sc.,Ph.D.,isanotedscientistofbroadinterests,whoworksinvarious
fieldsofmathematics,mechanics,andchemicalengineeringscience.
A.D.PolyaningraduatedfromtheDepartmentofMechanicsand
Mathematics of the Moscow State University in 1974. He received
hisPh.D.degreein1981andD.Sc.degreein1986attheInstitutefor
Problems in Mechanics of the Russian (former USSR) Academy of
Sciences. Since1975, A.D.Polyaninhasbeenamemberofthestaff
oftheInstituteforProblemsinMechanicsoftheRussianAcademyof
Sciences.
ProfessorPolyaninhasmadeimportantcontributionstodeveloping
newexactandapproximateanalyticalmethodsofthetheoryofdiffer-
entialequations,mathematicalphysics,integralequations,engineering
mathematics, nonlinear mechanics, theory of heat and mass transfer,
and chemical hydrodynamics. He obtained exact solutions for sev-
eralthousandsofordinarydifferential,partialdifferential,andintegral
equations.
ProfessorPolyaninisanauthorof17booksinEnglish,Russian,German,andBulgarian. His
publicationsalsoincludemorethan110researchpapersandthreepatents. Oneofhismostsignificant
books is A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential
Equations,CRCPress,1995.
In1991,A.D.PolyaninwasawardedaChaplyginPrizeoftheUSSRAcademyofSciencesfor
hisresearchinmechanics.
AlexanderV.Manzhirov,D.Sc.,Ph.D.,isaprominentscientistinthefieldsofmechanicsand
appliedmathematics,integralequations,andtheirapplications.
AftergraduatingfromtheDepartmentofMechanicsandMathemat-
icsoftheRostovStateUniversityin1979,A.V.Manzhirovattendeda
postgraduatecourseattheMoscowInstituteofCivilEngineering. He
receivedhisPh.D.degreein1983attheMoscowInstituteofElectronic
Engineering Industry and his D.Sc. degree in 1993 at the Institute for
Problems in Mechanics of the Russian (former USSR) Academy of
Sciences. Since1983,A.V.Manzhirovhasbeenamemberofthestaff
of the Institute for Problems in Mechanics of the Russian Academy
of Sciences. He is also a Professor of Mathematics at the Bauman
Moscow State Technical University and a Professor of Mathematics
attheMoscowStateAcademyofEngineeringandComputerScience.
Professor Manzhirov is a member of the editorial board of the jour-
nal “Mechanics of Solids” and a member of the European Mechanics
Society(EUROMECH).
ProfessorManzhirovhasmadeimportantcontributionstonewmathematicalmethodsforsolving
problemsinthefieldsofintegralequations,mechanicsofsolidswithaccretion,contactmechanics,
andthetheoryofviscoelasticityandcreep. Heisanauthorof3books, 60scientificpublications,
andtwopatents.
Pageix
CONTENTS
Annotation
Foreword
Some Remarks and Notation
Part I. Exact Solutions of Integral Equations
1. Linear Equations of the First Kind With Variable Limit of Integration
1.1. Equations Whose Kernels Contain Power-Law Functions
1.1-1. Kernels Linear in the Arguments x and t
1.1-2. Kernels Quadratic in the Arguments x and t
1.1-3. Kernels Cubic in the Arguments x and t
1.1-4. Kernels Containing Higher-Order Polynomials in x and t
1.1-5. Kernels Containing Rational Functions
1.1-6. Kernels Containing Square Roots
1.1-7. Kernels Containing Arbitrary Powers
1.2. Equations Whose Kernels Contain Exponential Functions
1.2-1. Kernels Containing Exponential Functions
1.2-2. Kernels Containing Power-Law and Exponential Functions
1.3. Equations Whose Kernels Contain Hyperbolic Functions
1.3-1. Kernels Containing Hyperbolic Cosine
1.3-2. Kernels Containing Hyperbolic Sine
1.3-3. Kernels Containing Hyperbolic Tangent
1.3-4. Kernels Containing Hyperbolic Cotangent
1.3-5. Kernels Containing Combinations of Hyperbolic Functions
1.4. Equations Whose Kernels Contain Logarithmic Functions
1.4-1. Kernels Containing Logarithmic Functions
1.4-2. Kernels Containing Power-Law and Logarithmic Functions
1.5. Equations Whose Kernels Contain Trigonometric Functions
1.5-1. Kernels Containing Cosine
1.5-2. Kernels Containing Sine
1.5-3. Kernels Containing Tangent
1.5-4. Kernels Containing Cotangent
1.5-5. Kernels Containing Combinations of Trigonometric Functions
1.6. Equations Whose Kernels Contain Inverse Trigonometric Functions
1.6-1. Kernels Containing Arccosine
1.6-2. Kernels Containing Arcsine
1.6-3. Kernels Containing Arctangent
1.6-4. Kernels Containing Arccotangent
Pageix
1.7. Equations Whose Kernels Contain Combinations of Elementary Functions
1.7-1. Kernels Containing Exponential and Hyperbolic Functions
1.7-2. Kernels Containing Exponential and Logarithmic Functions
1.7-3. Kernels Containing Exponential and Trigonometric Functions
1.7-4. Kernels Containing Hyperbolic and Logarithmic Functions
1.7-5. Kernels Containing Hyperbolic and Trigonometric Functions
1.7-6. Kernels Containing Logarithmic and Trigonometric Functions
1.8. Equations Whose Kernels Contain Special Functions
1.8-1. Kernels Containing Bessel Functions
1.8-2. Kernels Containing Modified Bessel Functions
1.8-3. Kernels Containing Associated Legendre Functions
1.8-4. Kernels Containing Hypergeometric Functions
1.9. Equations Whose Kernels Contain Arbitrary Functions
1.9-1. Equations With Degenerate Kernel: K(x, t)=g (x)h (t)+g (x)h (t)
1 1 2 2
1.9-2. Equations With Difference Kernel: K(x, t)=K(x – t)
1.9-3. Other Equations
1.10. Some Formulas and Transformations
2. Linear Equations of the Second Kind With Variable Limit of Integration
2.1. Equations Whose Kernels Contain Power-Law Functions
2.1-1. Kernels Linear in the Arguments x and t
2.1-2. Kernels Quadratic in the Arguments x and t
2.1-3. Kernels Cubic in the Arguments x and t
2.1-4. Kernels Containing Higher-Order Polynomials in x and t
2.1-5. Kernels Containing Rational Functions
2.1-6. Kernels Containing Square Roots and Fractional Powers
2.1-7. Kernels Containing Arbitrary Powers
2.2. Equations Whose Kernels Contain Exponential Functions
2.2-1. Kernels Containing Exponential Functions
2.2-2. Kernels Containing Power-Law and Exponential Functions
2.3. Equations Whose Kernels Contain Hyperbolic Functions
2.3-1. Kernels Containing Hyperbolic Cosine
2.3-2. Kernels Containing Hyperbolic Sine
2.3-3. Kernels Containing Hyperbolic Tangent
2.3-4. Kernels Containing Hyperbolic Cotangent
2.3-5. Kernels Containing Combinations of Hyperbolic Functions
2.4. Equations Whose Kernels Contain Logarithmic Functions
2.4-1. Kernels Containing Logarithmic Functions
2.4-2. Kernels Containing Power-Law and Logarithmic Functions
2.5. Equations Whose Kernels Contain Trigonometric Functions
2.5-1. Kernels Containing Cosine
2.5-2. Kernels Containing Sine
2.5-3. Kernels Containing Tangent
2.5-4. Kernels Containing Cotangent
2.5-5. Kernels Containing Combinations of Trigonometric Functions
2.6. Equations Whose Kernels Contain Inverse Trigonometric Functions
2.6-1. Kernels Containing Arccosine
2.6-2. Kernels Containing Arcsine
2.6-3. Kernels Containing Arctangent
2.6-4. Kernels Containing Arccotangent
Pagex