Handbook of Exact Solutions to Mathematical Equations

1. Auflage 2024
ISBN: 978-0-367-50799-2
Verlag: Chapman and Hall/CRC

Buch, Englisch, 659 Seiten, Format (B × H): 183 mm x 260 mm, Gewicht: 1417 g

Reihe: Advances in Applied Mathematics

This Handbook is a unique reference for scientists and engineers, containing over 3,800 nonlinear partial differential equations withsolutions.Thebook concernsfirst-, second-, third-, fourth-, and higher-order nonlinear PDEs and systems of coupled PDEs.It discusses parabolic, hyperbolic, and elliptic equations as well as those of mixed and general type.

All in all, the handbook contains many more nonlinear equations of mathematical physics and other nonlinear PDEs with their exact solutions, than any other book currently available. Apart from the exact solutions, it also provides various reductions and transformations leading to ordinary differential equations, linear PDEs, or simpler PDEs.

A solution is called exact if, when substituted into the differential equation under consideration, it turns the equation into an identity. In this case, no approximations or simplifications of the equation are allowed, and no a priori assumptions are used.

Exact solutions to nonlinear equations of mathematical physics are understood as follows: solutions expressed in terms of elementary functions, functions included in the equation (this is necessary when the equation depends on arbitrary functions), and indefinite integrals; solutions expressed in terms of solutions to ordinary differential equations or systems of such equations.

Exact solutions of mathematical equations have always played and continue to play a massive role in forming a correct understanding of the qualitative features of many phenomena and processes in various fields of natural science. The book will be helpful for a wide range of researchers, university teachers, and graduate and postgraduate students specializing in the fields of applied and computational mathematics, mathematical and theoretical physics, continuum mechanics, control theory, biology, biophysics, biochemistry, medicine, chemical engineering sciences, and ecology.

Zielgruppe

Postgraduate, Professional, and Undergraduate Advanced

Autoren/Hrsg.

Polyanin, Andrei D.

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein
• Mathematik | Informatik Mathematik Mathematische Analysis
• Interdisziplinäres Wissenschaften Wissenschaften Interdisziplinär Naturwissenschaften, Technik, Medizin
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Forschungsmethodik, Wissenschaftliche Ausstattung
• Interdisziplinäres Wissenschaften Wissenschaften: Allgemeines Enzyklopädien, Nachschlagewerke, Wörterbücher

Inhaltsverzeichnis

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 yx''x = f (x, y)

2.3.2. Equations of the Form f (x, y)yx''x = g(x, y, yx' )

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)ux + g(x, y)uy = 0

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

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

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

4.2.4. Equations of the Form ux + f (x, y, u)uy = g(x, y, u)

4.3. NonlinearPartialDifferentialEquationsinTwoIndependent Variables

4.3.1. Preliminary Remarks. A Complete Integral

4.3.2. Equations Quadratic in One Derivative

4.3.3. Equations Quadratic in Two Derivatives

4.3.4. Equations with Arbitrary Nonlinearities in Derivatives

References for Chapter 4

5 Linear Equations and Problems of Mathematical Physics

5.1. Parabolic Equations

5.1.1. Heat (Diffusion) Equation ut = auxx

5.1.2. Nonhomogeneous Heat Equation ut = auxx + F(x, t)

5.1.3. Heat Type Equation of the Form ut = auxx + bux + cu + F(x, t)

5.1.4. Heat Equation with Axial Symmetry ut = a(urr + r-1ur)

5.1.5. Nonhomogeneous Heat Equation with Axial Symmetry

ut = a(urr + r-1ur) + F(r, t)

5.1.6. Heat Equation with Central Symmetry ut = a(urr + 2r-1ur)

5.1.7. Nonhomogeneous Heat Equation with Central Symmetry

ut = a(urr + 2r-1ur) + F(r, t)

5.1.8. Heat Type Equation of the Form ut = uxx + (1 - 2ß)x-1ux

5.1.9. Heat Type Equation of the Form ut = [f (x)ux]x

5.1.10.

- Equations of the Form s(x)ut = [p(x)ux]x q(x)u + F(x, t)

5.1.11.

- Liquid-Film Mass Transfer Equation (1 y2)ux = auyy

5.1.12. Equations of the Diffusion (Thermal) Boundary Layer

n2

5.1.13.

t

2m

xx Schro¨dinger Equation inu = - u + U (x)u

5.2. Hyperbolic Equations

5.2.1. Wave Equation utt = a2uxx

5.2.2. Nonhomogeneous Wave Equation utt = a2uxx + F(x, t)

5.2.3.

- Klein–Gordon Equation utt = a2uxx bu

5.2.4. Nonhomogeneous Klein–Gordon Equation

- utt = a2uxx bu + F(x, t)

5.2.5. Wave Equation w

Über Autor(innen)

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.