Handbook of Exact Solutions to Mathematical Equations

This Handbook is a unique reference for scientists and engineers, containing over 3,800 nonlinear partial differential equations withsolutions.

This reference book describes the exact solutions of the following types of mathematical equations:

Algebraic and Transcendental Equations
Ordinary Differential Equations
Systems of Ordinary Differential Equations
First-Order Partial Differential Equations
Linear Equations and Problems of Mathematical Physics
Nonlinear Equations of Mathematical Physics
Systems of Partial Differential Equations
Integral Equations
Difference and Functional Equations
Ordinary Functional Differential Equations
Partial Functional Differential Equations

The book delves into equations that find practical applications in a wide array of natural and engineering sciences, including the theory of heat and mass transfer, wave theory, hydrodynamics, gas dynamics, combustion theory, elasticity theory, general mechanics, theoretical physics, nonlinear optics, biology, chemical engineering sciences, ecology, and more. Most of these equations are of a reasonably general form and dependent on free parameters or arbitrary functions.

The Handbook of Exact Solutions to Mathematical Equations generally has no analogs in world literature and contains a vast amount of new material. The exact solutions given in the book, being rigorous mathematical standards, can be used as test problems to assess the accuracy and verify the adequacy of various numerical and approximate analytical methods for solving mathematical equations, as well as to check and compare the effectiveness of exact analytical methods.

Autorentext

Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests and is active in various areas of mathematics, theory of heat and mass transfer, hydrodynamics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics. Professor Polyanin graduated with honors from the Department of Mechan- ics and Mathematics at the Lomonosov Moscow State University in 1974. Since 1975, Professor Polyanin has been working at the Ishlinsky Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences, where he defended his Ph.D. in 1981 and D.Sc. degree in 1986.

Professor Polyanin has made important contributions to the theory of differential and integral equations, mathematical physics, applied and engineering mathematics, the theory of heat and mass transfer, and hydrodynamics. He develops analytical methods for constructing solutions to mathematical equations of various types and has obtained a huge number of exact solutions of ordinary differential, partial differential, delay partial differential, integral, and functional equations.

Professor Polyanin is an author of more than 30 books and over 270 articles and holds three patents. His books include V. F. Zaitsev and A. D. Polyanin, Discrete- Group Methods for Integrating Equations of Nonlinear Mechanics, CRC Press, 1994; A. D. Polyanin and V. V. Dilman, Methods of Modeling Equations and Analogies in Chemical Engineering, CRC Press/Begell House, Boca Raton, 1994; A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 (2nd edition in 2003); A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998 (2nd edition in 2008); A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002 (2nd edition, co-authored with V. E. Nazaikinskii, in 2016); A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; A. D. Polyanin, A. M. Kutepov, et al., Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, 2002; A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, 2004 (2nd edition in 2012); A. D. Polyanin and A. V. Manzhirov, Handbook of Mathematics for Engineers and Scientists, Chapman & Hall/CRC Press, 2007; A. D. Polyanin and V. F. Zaitsev, Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, CRC Press, 2018; A. D. Polyanin and A. I. Zhurov, Separation of Variables and Exact Solutions to Nonlinear PDEs, CRC Press, 2022, and A. D. Polyanin, V. G. Sorokin, and A. I. Zhurov, Delay Ordinary and Partial Differential Equations, CRC Press, 2023.

Professor Polyanin is editor-in-chief of the international scientific educational website EqWorld- The World of Mathematical Equations and a member of the editorial boards of several journals.

Klappentext

This Handbook is a unique reference for scientists and engineers, containing over 3,800 nonlinear partial differential equations withsolutions.

Inhalt

1 Algebraic and Transcendental Equations

1.1. Algebraic Equations

1.1.1. LinearandQuadraticEquations

1.1.2. Cubic Equations

1.1.3. EquationsoftheFourthDegree

1.1.4. EquationsoftheFifthDegree

1.1.5. Algebraic Equations of Arbitrary Degree

1.1.6. Systems of Linear Algebraic Equations

1.2. Trigonometric Equations

1.2.1. Binomial Trigonometric Equations

1.2.2. Trigonometric Equations Containing Several Terms

1.2.3. Trigonometric Equations of the General Form

1.3. Other Transcendental Equations

1.3.1. Equations Containing Exponential Functions

1.3.2. Equations Containing Hyperbolic Functions

1.3.3. Equations Containing Logarithmic Functions

References for Chapter 1

2 Ordinary Differential Equations

2.1. First-Order Ordinary Differential Equations

2.1.1. Simplest First-Order ODEs

2.1.2. Riccati Equations

2.1.3. Abel Equations

2.1.4. Other First-Order ODEs Solved for the Derivative

2.1.5. ODEs Not Solved for the Derivative and ODEs Defined Parametrically

2.2. Second-Order Linear Ordinary Differential Equations

2.2.1. Preliminary Remarks and Some Formulas

2.2.2. Equations Involving Power Functions

2.2.3. Equations Involving Exponential and Other Elementary Functions

2.2.4. Equations Involving Arbitrary Functions

2.3. Second-Order Nonlinear Ordinary Differential Equations

2.3.1. Equations of the Form y*xx *= f (x, y)

2.3.2. Equations of the Form f (x, y)y*xx *= g(x, y, y*x* )

2.3.3. ODEs of General Form Containing Arbitrary Functions of Two Arguments

2.4. Higher-Order Ordinary Differential Equations

2.4.1. Higher-Order Linear Ordinary Differential Equations

2.4.2. Third-andFourth-OrderNonlinearOrdinaryDifferentialEquations

2.4.3. Higher-Order Nonlinear Ordinary Differential Equations

References for Chapter 2

3 Systems of Ordinary Differential Equations

3.1. Linear Systems of ODEs

3.1.1. Systems of Two First-Order ODEs

3.1.2. Systems of Two Second-Order ODEs

3.1.3. Other Systems of Two ODEs

3.1.4. Systems of Three and More ODEs

3.2. Nonlinear Systems of Two ODEs

3.2.1. Systems of First-Order ODEs

3.2.2. Systems of Second- and Third-Order ODEs

3.3. Nonlinear Systems of Three or More ODEs

3.3.1. Systems of Three ODEs

3.3.2. Equations of Dynamics of a Rigid Body with a Fixed Point

References for Chapter 3

4 First-Order Partial Differential Equations

4.1. Linear Partial Differential Equations in Two Independent Variables

4.1.1. Preliminary Remarks. Solution Methods

4.1.2. Equations of the Form f (x, y)u*x + g(x, y)u*y = 0

4.1.3. Equations of the Form f (x, y)u*x + g(x, y)u*y = h(x, y)

4.1.4. Equations of the Form f (x, y)u*x + g(x, y)u*y = h(x, y)u + r(x, y)

4.2. Quasilinear Partial Differential Equations in Two Independent Variables

4.2.1. Preliminary Remarks. Solution Methods

4.2.2. Equations of the Form f (x, y)u*x + g(x, y)u*y = h(x, y, u)

4.2.3. Equations of the Form ux + f (x, y, u)u*y* = 0…