Linear Equations and Problems of Mathematical Physics... Professor Polyanin has made important contributions to exact and approximate analytical methods in the theory of differential equ
Trang 1CONTENTS xxi
T5 Ordinary Differential Equations 1207
T5.1 First-Order Equations 1207
T5.2 Second-Order Linear Equations 1212
T5.2.1 Equations Involving Power Functions 1213
T5.2.2 Equations Involving Exponential and Other Functions 1220
T5.2.3 Equations Involving Arbitrary Functions 1222
T5.3 Second-Order Nonlinear Equations 1223
T5.3.1 Equations of the Form y xx = f (x, y) 1223
T5.3.2 Equations of the Form f (x, y)y xx = g(x, y, y x ) 1225
References for Chapter T5 1228
T6 Systems of Ordinary Differential Equations 1229
T6.1 Linear Systems of Two Equations 1229
T6.1.1 Systems of First-Order Equations 1229
T6.1.2 Systems of Second-Order Equations 1232
T6.2 Linear Systems of Three and More Equations 1237
T6.3 Nonlinear Systems of Two Equations 1239
T6.3.1 Systems of First-Order Equations 1239
T6.3.2 Systems of Second-Order Equations 1240
T6.4 Nonlinear Systems of Three or More Equations 1244
References for Chapter T6 1246
T7 First-Order Partial Differential Equations 1247
T7.1 Linear Equations 1247
T7.1.1 Equations of the Form f (x, y) ∂w ∂x + g(x, y) ∂w ∂y =0 1247
T7.1.2 Equations of the Form f (x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y) 1248
T7.1.3 Equations of the Form f (x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y)w + r(x, y) 1250
T7.2 Quasilinear Equations 1252
T7.2.1 Equations of the Form f (x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y, w) 1252
T7.2.2 Equations of the Form ∂w ∂x + f (x, y, w) ∂w ∂y =0 1254
T7.2.3 Equations of the Form ∂w ∂x + f (x, y, w) ∂w ∂y = g(x, y, w) 1256
T7.3 Nonlinear Equations 1258
T7.3.1 Equations Quadratic in One Derivative 1258
T7.3.2 Equations Quadratic in Two Derivatives 1259
T7.3.3 Equations with Arbitrary Nonlinearities in Derivatives 1261
References for Chapter T7 1265
T8 Linear Equations and Problems of Mathematical Physics 1267
T8.1 Parabolic Equations 1267
T8.1.1 Heat Equation ∂w ∂t = a ∂ ∂x2w2 1267
T8.1.2 Nonhomogeneous Heat Equation ∂w ∂t = a ∂ ∂x2w2 +Φ(x, t) 1268
T8.1.3 Equation of the Form ∂w ∂t = a ∂ ∂x2w2 + b ∂w ∂x + cw + Φ(x, t) 1270
T8.1.4 Heat Equation with Axial Symmetry ∂w ∂t = a∂2w ∂r2 + 1r ∂w ∂r 1270
T8.1.5 Equation of the Form ∂w ∂t = a∂2w ∂r2 + 1r ∂w ∂r +Φ(r, t) 1271
T8.1.6 Heat Equation with Central Symmetry ∂w ∂t = a∂2w ∂r2 + 2r ∂w ∂r 1272
T8.1.7 Equation of the Form ∂w ∂t = a∂2w ∂r2 + 2r ∂w ∂r +Φ(r, t) 1273
T8.1.8 Equation of the Form ∂w ∂t = ∂ ∂x2w2 + 1–x2β ∂w ∂x 1274
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T8.1.9 Equations of the Diffusion (Thermal) Boundary Layer 1276
T8.1.10 Schr¨odinger Equation i∂w ∂t = –2m2 ∂ ∂x2w2 + U (x)w 1276
T8.2 Hyperbolic Equations 1278
T8.2.1 Wave Equation∂ ∂t2w2 = a2∂ ∂x2w2 1278
T8.2.2 Equation of the Form ∂ ∂t2w2 = a2∂ ∂x2w2 +Φ(x, t) 1279
T8.2.3 Klein–Gordon Equation ∂ ∂t2w2 = a2∂ ∂x2w2 – bw 1280
T8.2.4 Equation of the Form ∂ ∂t2w2 = a2∂ ∂x2w2 – bw + Φ(x, t) 1281
T8.2.5 Equation of the Form ∂ ∂t2w2 = a2∂2w ∂r2 + 1r ∂w ∂r +Φ(r, t) 1282
T8.2.6 Equation of the Form ∂ ∂t2w2 = a2∂2w ∂r2 + 2r ∂w ∂r +Φ(r, t) 1283
T8.2.7 Equations of the Form ∂ ∂t2w2 + k ∂w ∂t = a2∂ ∂x2w2 + b ∂w ∂x + cw + Φ(x, t) 1284
T8.3 Elliptic Equations 1284
T8.3.1 Laplace EquationΔw =0 1284
T8.3.2 Poisson EquationΔw + Φ(x) =0 1287
T8.3.3 Helmholtz EquationΔw + λw = –Φ(x) 1289
T8.4 Fourth-Order Linear Equations 1294
T8.4.1 Equation of the Form ∂ ∂t2w2 + a2∂ ∂x4w4 =0 1294
T8.4.2 Equation of the Form ∂ ∂t2w2 + a2∂ ∂x4w4 =Φ(x, t) 1295
T8.4.3 Biharmonic EquationΔΔw =0 1297
T8.4.4 Nonhomogeneous Biharmonic EquationΔΔw = Φ(x, y) 1298
References for Chapter T8 1299
T9 Nonlinear Mathematical Physics Equations 1301
T9.1 Parabolic Equations 1301
T9.1.1 Nonlinear Heat Equations of the Form ∂w ∂t = ∂ ∂x2w2 + f (w) 1301
T9.1.2 Equations of the Form ∂w ∂t = ∂x ∂ f (w) ∂w ∂x + g(w) 1303
T9.1.3 Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases 1307
T9.1.4 Nonlinear Schr¨odinger Equations 1309
T9.2 Hyperbolic Equations 1312
T9.2.1 Nonlinear Wave Equations of the Form ∂ ∂t2w2 = a ∂ ∂x2w2 + f (w) 1312
T9.2.2 Other Nonlinear Wave Equations 1316
T9.3 Elliptic Equations 1318
T9.3.1 Nonlinear Heat Equations of the Form ∂ ∂x2w2 + ∂ ∂y2w2 = f (w) 1318
T9.3.2 Equations of the Form ∂x ∂ f (x) ∂w ∂x + ∂y ∂ g (y) ∂w ∂y = f (w) 1321
T9.3.3 Equations of the Form ∂x ∂ f (w) ∂w ∂x + ∂y ∂ g (w) ∂w ∂y = h(w) 1322
T9.4 Other Second-Order Equations 1324
T9.4.1 Equations of Transonic Gas Flow 1324
T9.4.2 Monge–Amp`ere Equations 1326
T9.5 Higher-Order Equations 1327
T9.5.1 Third-Order Equations 1327
T9.5.2 Fourth-Order Equations 1332
References for Chapter T9 1335
T10 Systems of Partial Differential Equations 1337
T10.1 Nonlinear Systems of Two First-Order Equations 1337
T10.2 Linear Systems of Two Second-Order Equations 1341
Trang 3CONTENTS xxiii
T10.3 Nonlinear Systems of Two Second-Order Equations 1343
T10.3.1 Systems of the Form ∂u ∂t = a ∂ ∂x2u2 + F (u, w), ∂w ∂t = b ∂ ∂x2w2 + G(u, w) 1343
T10.3.2 Systems of the Form ∂u ∂t = x a n ∂ ∂x x n ∂u ∂x + F (u, w), ∂w ∂t = x b n ∂ ∂x x n ∂w ∂x + G(u, w) 1357
T10.3.3 Systems of the Form Δu = F (u, w), Δw = G(u, w) 1364
T10.3.4 Systems of the Form ∂ ∂t2u2 = x a n ∂ ∂x x n ∂u ∂x + F (u, w), ∂2w ∂t2 = x b n ∂x ∂ x n ∂w ∂x + G(u, w) 1368
T10.3.5 Other Systems 1373
T10.4 Systems of General Form 1374
T10.4.1 Linear Systems 1374
T10.4.2 Nonlinear Systems of Two Equations Involving the First Derivatives in t 1374
T10.4.3 Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378 T10.4.4 Nonlinear Systems of Many Equations Involving the First Derivatives in t 1381 References for Chapter T10 1382
T11 Integral Equations 1385
T11.1 Linear Equations of the First Kind with Variable Limit of Integration 1385
T11.2 Linear Equations of the Second Kind with Variable Limit of Integration 1391
T11.3 Linear Equations of the First Kind with Constant Limits of Integration 1396
T11.4 Linear Equations of the Second Kind with Constant Limits of Integration 1401
References for Chapter T11 1406
T12 Functional Equations 1409
T12.1 Linear Functional Equations in One Independent Variable 1409
T12.1.1 Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments 1409
T12.1.2 Other Linear Functional Equations 1421
T12.2 Nonlinear Functional Equations in One Independent Variable 1428
T12.2.1 Functional Equations with Quadratic Nonlinearity 1428
T12.2.2 Functional Equations with Power Nonlinearity 1433
T12.2.3 Nonlinear Functional Equation of General Form 1434
T12.3 Functional Equations in Several Independent Variables 1438
T12.3.1 Linear Functional Equations 1438
T12.3.2 Nonlinear Functional Equations 1443
References for Chapter T12 1450
Supplement Some Useful Electronic Mathematical Resources 1451
Index 1453
Trang 5Andrei D Polyanin, D.Sc., Ph.D., is a well-known scientist
of broad interests who is active in various areas of mathe-matics, mechanics, and chemical engineering sciences He
is one of the most prominent authors in the field of reference literature on mathematics and physics.
Professor Polyanin graduated with honors from the De-partment of Mechanics and Mathematics of Moscow State University in 1974 He received his Ph.D in 1981 and his D.Sc in 1986 from the Institute for Problems in Mechanics
of the Russian (former USSR) Academy of Sciences Since
1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences;
he is also Professor of Mathematics at Bauman Moscow State Technical University He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation.
Professor Polyanin has made important contributions to exact and approximate analytical methods in the theory of differential equations, mathematical physics, integral equations, engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics.
He has obtained exact solutions for several thousand ordinary differential, partial differen-tial, and integral equations.
Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgarian as well as more than 120 research papers and three patents He has written a number of fundamental handbooks, including A D Polyanin and V F Zaitsev,
Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 and 2003; A D Polyanin and A V Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A D Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A D Polyanin, V F Zaitsev, and
A Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A D Polyanin and V F Zaitsev, Handbook of Nonlinear Partial Differential Equation, Chapman & Hall/CRC Press, 2004.
Professor Polyanin is editor of the book series Differential and Integral Equations and Their Applications, Chapman & Hall/CRC Press, London/Boca Raton, and Physical and Mathematical Reference Literature, Fizmatlit, Moscow He is also Editor-in-Chief
of the international scientific-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru), which is visited by over 1000 users a day worldwide Professor Polyanin is a member of the Editorial Board of the journal Theoretical Foundations
of Chemical Engineering.
In 1991, Professor Polyanin was awarded a Chaplygin Prize of the Russian Academy
of Sciences for his research in mechanics In 2001, he received an award from the Ministry
of Education of the Russian Federation.
Address: Institute for Problems in Mechanics, Vernadsky Ave 101 Bldg 1, 119526 Moscow, Russia
Home page: http://eqworld.ipmnet.ru/polyanin-ew.htm
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Trang 6xxvi AUTHORS
Alexander V Manzhirov, D.Sc., Ph.D., is a noted scientist
in the fields of mechanics and applied mathematics, integral equations, and their applications.
After graduating with honors from the Department of Mechanics and Mathematics of Rostov State University in
1979, Professor Manzhirov attended postgraduate courses
at Moscow Institute of Civil Engineering He received his Ph.D in 1983 from Moscow Institute of Electronic Engi-neering Industry and his D.Sc in 1993 from the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences Since 1983, Professor Manzhirov has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences, where he is currently head
of the Laboratory for Modeling in Solid Mechanics.
Professor Manzhirov is also head of a branch of the Department of Applied Mathematics
at Bauman Moscow State Technical University, professor of mathematics at Moscow State University of Engineering and Computer Science, vice-chairman of Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation, executive secretary of Solid Mechanics Scientific Council of the Russian Academy of Sciences, and an expert in mathematics, mechanics, and computer science
of the Russian Foundation for Basic Research He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the European Mechanics Society
(EUROMECH), and a member of the editorial board of the journal Mechanics of Solids and the international scientific-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru).
Professor Manzhirov has made important contributions to new mathematical methods for solving problems in the fields of integral equations and their applications, mechanics of growing solids, contact mechanics, tribology, viscoelasticity, and creep theory He is the
au-thor of ten books (including Contact Problems in Mechanics of Growing Solids [in Russian], Nauka, Moscow, 1991; Handbook of Integral Equations, CRC Press, Boca Raton, 1998; Handbuch der Integralgleichungen: Exacte L¨osungen, Spektrum Akad Verlag, Heidelberg, 1999; Contact Problems in the Theory of Creep [in Russian], National Academy of Sciences
of Armenia, Erevan, 1999), more than 70 research papers, and two patents.
Professor Manzhirov is a winner of the First Competition of the Science Support Foundation 2001, Moscow.
Address: Institute for Problems in Mechanics, Vernadsky Ave 101 Bldg 1, 119526 Moscow, Russia
Home page: http://eqworld.ipmnet.ru/en/board/manzhirov.htm
Trang 7This book can be viewed as a reasonably comprehensive compendium of mathematical definitions, formulas, and theorems intended for researchers, university teachers, engineers, and students of various backgrounds in mathematics The absence of proofs and a concise presentation has permitted combining a substantial amount of reference material in a single volume.
When selecting the material, the authors have given a pronounced preference to practical aspects, namely, to formulas, methods, equations, and solutions that are most frequently used in scientific and engineering applications Hence some abstract concepts and their corollaries are not contained in this book.
• This book contains chapters on arithmetics, elementary geometry, analytic geometry, algebra, differential and integral calculus, differential geometry, elementary and special functions, functions of one complex variable, calculus of variations, probability theory, mathematical statistics, etc Special attention is paid to formulas (exact, asymptotical, and approximate), functions, methods, equations, solutions, and transformations that are of frequent use in various areas of physics, mechanics, and engineering sciences.
• The main distinction of this reference book from other general (nonspecialized) math-ematical reference books is a significantly wider and more detailed description of methods for solving equations and obtaining their exact solutions for various classes of mathematical equations (ordinary differential equations, partial differential equations, integral equations, difference equations, etc.) that underlie mathematical modeling of numerous phenomena and processes in science and technology In addition to well-known methods, some new methods that have been developing intensively in recent years are described.
• For the convenience of a wider audience with different mathematical backgrounds, the authors tried to avoid special terminology whenever possible Therefore, some of the methods and theorems are outlined in a schematic and somewhat simplified manner, which
is sufficient for them to be used successfully in most cases Many sections were written
so that they could be read independently The material within subsections is arranged in increasing order of complexity This allows the reader to get to the heart of the matter quickly.
The material in the first part of the reference book can be roughly categorized into the following three groups according to meaning:
1 The main text containing a concise, coherent survey of the most important definitions, formulas, equations, methods, and theorems.
2 Numerous specific examples clarifying the essence of the topics and methods for solving problems and equations.
3 Discussion of additional issues of interest, given in the form of remarks in small print.
For the reader’s convenience, several long mathematical tables — finite sums, series, indefinite and definite integrals, direct and inverse integral transforms (Laplace, Mellin, and Fourier transforms), and exact solutions of differential, integral, functional, and other mathematical equations—which contain a large amount of information, are presented in the second part of the book.
This handbook consists of chapters, sections, subsections, and paragraphs (the titles of the latter are not included in the table of contents) Figures and tables are numbered sep-arately in each section, while formulas (equations) and examples are numbered sepsep-arately
in each subsection When citing a formula, we use notation like (3.1.2.5), which means
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