"This Handbook is a unique reference for scientists and engineers, containing over 3,800 nonlinear partial differential equations with solutions. The book concerns first-, 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"--
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 pays special attention to equations found in various fields of natural and engineering sciences (in 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 others) and equations of a reasonably general form, which depend 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 for checking and comparing the effectiveness of exact anlytical methods.
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 yxx = f (x, y)
2.3.2. Equations of the Form f (x, y)yxx = 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 + (x, t)
5.1.3. Heat Type Equation of the Form ut = auxx + bux + cu + (x, t)
5.1.4. Heat Equation with Axial Symmetry ut = a(urr + r1ur)
5.1.5. Nonhomogeneous Heat Equation with Axial Symmetry
ut = a(urr + r1ur) + (r, t)
5.1.6. Heat Equation with Central Symmetry ut = a(urr + 2r1ur)
5.1.7. Nonhomogeneous Heat Equation with Central Symmetry
ut = a(urr + 2r1ur) + (r, t)
5.1.8. Heat Type Equation of the Form ut = uxx + (1 2)x1ux
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 + (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
SchroØdinger 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 + (x, t)
5.2.3.
KleinGordon Equation utt = a2uxx bu
5.2.4. Nonhomogeneous KleinGordon Equation
utt = a2uxx bu + (x, t)
5.2.5. Wave Equation with Axial Symmetry
utt = a2(urr + r1ur) + (r, t)
5.2.6. Wave Equation with Central Symmetry
utt = a2(urr + 2r1ur) + (r, t)
5.2.7.
Equations of the Form s(x)utt = [ p(x)ux]x q(x)u + (x, t)
5.2.8. Telegraph Type Equations utt + kut = a2uxx + bux + cu + (x, t)
5.3. Elliptic Equations
5.3.1. Laplace Equation u = 0
5.3.2. Poisson Equation u + (x, y) = 0
5.3.3.
Helmholtz Equation u + u = (x, y)
5.3.4. Convective Heat and Mass Transfer Equations
5.3.5. Equations of Heat and Mass Transfer in Anisotropic Media
5.3.6. Tricomi and Related Equations
5.4. Simplifications of Second-Order Linear Partial Differential Equations
5.4.1. Reduction of PDEs in Two Independent Variables to Canonical Forms
5.4.2. Simplifications of Linear Constant-Coefficient Partial Differential
Equations
5.5. Third-Order Linear Partial Differential Equations
5.5.1. Equations Containing the First Derivative in t and the Third
Derivative in x
5.5.2. Equations Containing the First Derivative in t and a Mixed Third
Derivative
5.5.3. Equations Containing the Second Derivative in t and a Mixed
Third Derivative
5.6. Fourth-Order Linear Partial Differential Equations
5.6.1. Equation of Transverse Vibration of an Elastic Rod
utt + a2uxxxx = 0
5.6.2. Nonhomogeneous Equation of the Form utt + a2uxxxx = (x, t)
5.6.3. Biharmonic Equation u = 0
5.6.4. Nonhomogeneous Biharmonic Equation u = (x, y)
References for
Chapter 5
6 Nonlinear Equations of Mathematical Physics
6.1. Parabolic Equations
6.1.1. Quasilinear Heat Equations with a Source of the Form ut = auxx + f
(u)
6.1.2. Burgers Type Equations and Related PDEs
6.1.3. Reaction-Diffusion Equations of the Form ut = [ f (u)ux]x + g(u)
6.1.4. Other Reaction-Diffusion and Heat PDEs with Variable Transfer
Coefficient
6.1.5. Convection-Diffusion Type PDEs
6.1.6. NonlinearSchroØdinger EquationsandRelatedPDEs
6.2. Hyperbolic Equations
6.2.1. Nonlinear KleinGordon Equations of the Form utt = auxx + f (u)
6.2.2. OtherNonlinearWaveTypeEquations
6.3. Elliptic Equations
6.3.1. Heat Equations with Nonlinear Source of the Form uxx + uyy = f (u)
6.3.2. Stationary Anisotropic Heat/Diffusion Equations of the Form [ f
(x)ux]x + [ g(y)uy]y = h(u)
6.3.3. Stationary Anisotropic Heat/Diffusion Equations of the Form
[ f (u)ux]x + [ g(u)uy]y = h(u)
6.4. Other Second-Order Equations
6.4.1. EquationsofTransonicGasFlow
6.4.2. MongeAmpe`reTypeEquations
6.5. Higher-Order Equations
6.5.1. Third-OrderEquations
6.5.2. Fourth-OrderEquations
References for
Chapter 6
7 Systems of Partial Differential Equations
7.1. Systems of Two First-Order PDEs
7.1.1. LinearSystemsofTwoFirst-OrderPDEs
7.1.2. Nonlinear Systems of the Form ux = F (u, w), wt = G(u, w)
7.1.3. Gas Dynamic Type Systems Linearizable with the Hodograph
Transformation
7.2. Systems of Two Second-Order PDEs
7.2.1. LinearSystemsofTwoSecond-OrderPDEs
7.2.2. Nonlinear Parabolic Systems of the Form
ut = auxx + F (u, w), wt = bwxx + G(u, w)
7.2.3. Nonlinear Parabolic Systems of the Form
ut = axn(xnux)x + F (u, w), wt = bxn(xnwx)x + G(u, w)
7.2.4. Nonlinear Hyperbolic Systems of the Form
utt = auxx + F (u, w), wtt = bwxx + G(u, w)
7.2.5. Nonlinear Hyperbolic Systems of the Form
utt = axn(xnux)x + F (u, w), wtt = bxn(xnwx)x + G(u, w)
7.2.6. Nonlinear Elliptic Systems of the Form
u = F (u, w), w = G(u, w)
7.3. PDE Systems of General Form
7.3.1. Linear Systems
7.3.2. Nonlinear Systems of Two Equations Involving the First Derivatives
with Respect to t
7.3.3. Nonlinear Systems of Two Equations Involving the Second Derivatives
with Respect to t
References for
Chapter 7
8 Integral Equations
8.1. IntegralEquationsoftheFirstKindwithVariableLimitofIntegration
8.1.1. Linear Volterra Integral Equations of the First Kind
8.1.2. Nonlinear Volterra Integral Equations of the First Kind
8.2. Integral Equations of the Second Kind with Variable Limit of Integration
8.2.1. Linear Volterra Integral Equations of the Second Kind
8.2.2. Nonlinear Volterra Integral Equations of the Second Kind
8.3. Equations of the First Kind with Constant Limits of Integration
8.3.1. Linear Fredholm Integral Equations of the First Kind
8.3.2. Nonlinear Fredholm Integral Equations of the First Kind
8.4. Equations of the Second Kind with Constant Limits of Integration
8.4.1. Linear Fredholm Integral Equations of the Second Kind
8.4.2. Nonlinear Fredholm Integral Equations of the Second Kind
References for
Chapter 8
9 Difference and Functional Equations
9.1. Difference Equations
9.1.1. Difference Equations with Discrete Argument
9.1.2. Difference Equations with Continuous Argument
9.2. Linear Functional Equations in One Independent Variable
9.2.1. Linear Functional Equations Involving Unknown Function with
Two Different Arguments
9.2.2. Other Linear Functional Equations
9.3. Nonlinear Functional Equations in One Independent Variable
9.3.1. Functional Equations with Quadratic Nonlinearity
9.3.2. Functional Equations with Power Nonlinearity
9.3.3. Nonlinear Functional Equation of General Form
9.4. Functional Equations in Several Independent Variables
9.4.1. Linear Functional Equations
9.4.2. Nonlinear Functional Equations
References for
Chapter 9
10 Ordinary Functional Differential Equations
10.1. First-Order Linear Ordinary Functional Differential Equations
10.1.1. ODEs with Constant Delays
10.1.2. Pantograph-Type ODEs with Proportional Arguments
10.1.3. Other Ordinary Functional Differential Equations
10.2. First-Order Nonlinear Ordinary Functional Differential Equations
10.2.1. ODEs with Constant Delays
10.2.2. Pantograph-Type ODEs with Proportional Arguments
10.2.3. Other Ordinary Functional Differential Equations
10.3. Second-Order Linear Ordinary Functional Differential Equations
10.3.1. ODEs with Constant Delays
10.3.2. Pantograph-Type ODEs with Proportional Arguments
10.3.3. Other Ordinary Functional Differential Equations
10.4. Second-Order Nonlinear Ordinary Functional Differential Equations
10.4.1. ODEs with Constant Delays
10.4.2. Pantograph-Type ODEs with Proportional Arguments
10.4.3. Other Ordinary Functional Differential Equations
10.5. Higher-Order Ordinary Functional Differential Equations
10.5.1. Linear Ordinary Functional Differential Equations
10.5.2. Nonlinear Ordinary Functional Differential Equations
References for
Chapter 10
11 Partial Functional Differential Equations
11.1. Linear Partial Functional Differential Equations
11.1.1. PDEs with Constant Delay
11.1.2. PDEs with Proportional Delay
11.1.3. PDEs with Anisotropic Time Delay
11.2. Nonlinear PDEs with Constant Delays
11.2.1. Parabolic Equations
11.2.2. Hyperbolic Equations
11.3. Nonlinear PDEs with Proportional Arguments
11.3.1. Parabolic Equations
11.3.2. Hyperbolic Equations
11.4. Partial Functional Differential Equations with Arguments of General
Type
11.4.1. Parabolic Equations
11.4.2. Hyperbolic Equations
11.5. PDEs with Anisotropic Time Delay
11.5.1. Parabolic Equations
11.5.2. Hyperbolic Equations
References for
Chapter 11
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.
