all oall of equationf equation

CAT#C3553_HalfTitlePage 8/12/03 2:33 PM Page 1

HANDBOOK OF

NONLINEAR PARTIALDIFFERENTIAL

EQUATIONS

2004 by Chapman & Hall/CRC

CAT#C3553_TitlePage 8/12/03 2:32 PM Page 1

CHAPMAN & HALL/CRCA CRC Press Company

Boca Raton London New York Washington, D.C.

HANDBOOK OF

NONLINEAR PARTIALDIFFERENTIAL

EQUATIONS

Andrei D. PolyaninValentin F. Zaitsev


This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted withpermission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publishreliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materialsor for the consequences of their use.

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No claim to original U.S. Government worksInternational Standard Book Number 1-58488-355-3

Library of Congress Card Number 2003058473Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Polyanin, A.D. (Andrei Dmitrievich)Handbook of nonlinear partial differential equations / by Andrei D. Polyanin, Valentin F. Zaitsev.

p. cm.Includes bibliographical references and index.ISBN 1-58488-355-3 (alk. paper)1. Differential equations, Nonlinear Numerical solutions. 2. Nonlinear mechanics Mathematics.I. Zaitsev, V.F. (Valentin F.) II. Title.

QA372.P726 2003515

.355 dc22 2003058473

C3553_disclaimer.fm Page 1 Thursday, September 18, 2003 10:56 AM


CONTENTS

Authors

Foreword

Some Notations and Remarks

1. Parabolic Equations with One Space Variable1.1. Equations with PowerLaw Nonlinearities

1.1.1. Equations of the Form wt

= a 2w

x2+ bw + cw2


= a 2w

x2+ b0 + b1w + b2w2 + b3w3


= a 2w

x2+ f (w)


= a 2w

x2+ f (x, t,w)


= a 2w

x2+ f (w) w

x+ g(w)


= a 2w

x2+ f (x, t,w) w

x+ g(x, t,w)


= a 2w

x2+ b

(wx

)2+ f (x, t,w)


= a 2w

x2+ f

(x, t,w, w

x

)


= awk 2w

x2+ f

(x, t,w, w

x

)


= a x

(wm w

x

)


= a x

(wm w

x

)+ bwk


= a x

(wm w

x

)+ bw + c1wk1 + c2wk2 + c3wk3


=x

[f (w) w

x

]+ g(w)


=x

[f (w) w

x

]+ g

(x, t,w, w

x

)

1.1.15. Other Equations1.2. Equations with Exponential Nonlinearities


= a 2w

x2+ b0 + b1ew + b2e2w


= a x

(ew w

x

)+ f (w)


=x

[f (w) w

x

]+ g(w)

1.2.4. Other Equations Explicitly Independent of x and t1.2.5. Equations Explicitly Dependent on x and/or t

1.3. Equations with Hyperbolic Nonlinearities1.3.1. Equations Involving Hyperbolic Cosine1.3.2. Equations Involving Hyperbolic Sine1.3.3. Equations Involving Hyperbolic Tangent1.3.4. Equations Involving Hyperbolic Cotangent

1.4. Equations with Logarithmic Nonlinearities1.4.1. Equations of the Form w

t= a

2wx2

+ f (x, t,w)1.4.2. Other Equations

1.5. Equations with Trigonometric Nonlinearities1.5.1. Equations Involving Cosine1.5.2. Equations Involving Sine1.5.3. Equations Involving Tangent1.5.4. Equations Involving Cotangent1.5.5. Equations Involving Inverse Trigonometric Functions

Page v


1.6. Equations Involving Arbitrary Functions1.6.1. Equations of the Form w

t= a

2wx2

+ f (x, t,w)1.6.2. Equations of the Form w

t= a

2wx2

+ f (x, t) wx

+ g(x, t,w)1.6.3. Equations of the Form w

t= a

2wx2

+ f (x, t,w) wx


t= a

2wx2

+ b(

wx

)2+ f (x, t,w)


= a 2w

x2+ b

(wx

)2+ f (x, t,w) w

x+ g(x, t,w)


= a 2w

x2+ f (x, t,w)( w

x

)2+ g(x, t,w) w

x+ h(x, t,w)


= a 2w

x2+ f

(x, t,w, w

x

)


= f (x, t) 2wx2

+ g(x, t,w, w

x

)


= aw 2w

x2+ f (x, t,w) w

x+ g(x, t,w)


= (aw + b) 2wx2

+ f (x, t,w)( wx

)2+ g(x, t,w) w

x+

h(x, t,w)1.6.11. Equations of the Form w

t= awm

2wx2

+ f (x, t) wx


t= a

x

(w w

x

)+ f (x, t) w

x+ g(x, t,w)


= a x

(wm w

x

)+ f (x, t) w

x+ g(x, t,w)


= a x

(ew w

x

)+f (x, t,w)


=x

[f (w) w

x

]+g

(x, t,w, w

x

)


= f (x,w) 2wx2


= f (x, t,w) 2wx2

+ g(x, t,w, w

x

)


= f(x,w, w

x

)2wx2

+ g(x, t,w, w

x

)

1.6.19. Nonlinear Equations of the Thermal (Diffusion) Boundary Layer1.7. Nonlinear Schrodinger Equations and Related Equations

1.7.1. Equations of the Form i wt

+ 2w

x2+ f (|w|)w = 0 Involving Arbitrary Parameters

1.7.2. Equations of the Form i wt

+ 1xn

x

(xn w

x

)+ f (|w|)w = 0 Involving Arbitrary

Parameters1.7.3. Other Equations Involving Arbitrary Parameters1.7.4. Equations with Cubic Nonlinearities Involving Arbitrary Functions1.7.5. Equations of General Form Involving Arbitrary Functions of a Single Argument1.7.6. Equations of General Form Involving Arbitrary Functions of Two Arguments

2. Parabolic Equations with Two or More Space Variables

2.1. Equations with Two Space Variables Involving PowerLaw Nonlinearities2.1.1. Equations of the Form w

t=

x

[f (x) w

x

]+

y

[g(y) w

y

]+ awp


= a x

(wn w

x

)+ b

y

(wk w

y

)


=x

[f (w) w

x

]+

y

[g(w) w

y

]+ h(w)

2.1.4. Other Equations

2.2. Equations with Two Space Variables Involving Exponential Nonlinearities2.2.1. Equations of the Form w

t=

x

[f (x) w

x

]+

y

[g(y) w

y

]+ aew


= a x

(ew w

x

)+ b

y

(ew w

y

)+ f (w)

2.3. Other Equations with Two Space Variables Involving Arbitrary Parameters2.3.1. Equations with Logarithmic Nonlinearities2.3.2. Equations with Trigonometrical Nonlinearities

Page vi


2.4. Equations Involving Arbitrary Functions2.4.1. Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical

Reactions2.4.2. Equations of the Form w

t=

x

[f (x) w

x

]+

y

[g(y) w

y

]+ h(w)


=x

[f (w) w

x

]+

y

[g(w) w

y

]+ h(t,w)

2.4.4. Other Equations Linear in the Highest Derivatives2.4.5. Nonlinear Diffusion Boundary Layer Equations

2.5. Equations with Three or More Space Variables2.5.1. Equations of Mass Transfer in Quiescent or Moving Media with Chemical

Reactions2.5.2. Heat Equations with PowerLaw or Exponential TemperatureDependent Thermal

Diffusivity2.5.3. Equations of Heat and Mass Transfer in Anisotropic Media2.5.4. Other Equations with Three Space Variables2.5.5. Equations with n Space Variables

2.6. Nonlinear Schrodinger Equations2.6.1. TwoDimensional Equations2.6.2. Three and nDimensional Equations

3. Hyperbolic Equations with One Space Variable3.1. Equations with PowerLaw Nonlinearities

3.1.1. Equations of the Form 2wt2

=2wx2

+ aw + bwn + cw2n1

3.1.2. Equations of the Form 2w

t2= a

2wx2

+ f (x, t,w)3.1.3. Equations of the Form

2wt2

= a 2w

x2+ f

(x, t,w, w

x

)


= f (x) 2wx2

+ g(x, t,w, w

x

)


= awn 2w

x2+ f (x,w)


t2= a

x

(wn w

x

)+f (w)

3.1.7. Other Equations3.2. Equations with Exponential Nonlinearities


t2= a

2wx2

+ bew + cew


t2= a

2wx2

+ f (x, t,w)3.2.3. Equations of the Form

2wt2

= f (x) 2wx2

+ g(x, t,w, w

x

)

3.2.4. Other Equations3.3. Other Equations Involving Arbitrary Parameters

3.3.1. Equations with Hyperbolic Nonlinearities3.3.2. Equations with Logarithmic Nonlinearities3.3.3. SineGordon Equation and Other Equations with Trigonometric Nonlinearities3.3.4. Equations of the Form

2wt2

+ a wt

=x

[f (w) w

x

]


+ f (w) wt

=x

[g(w) w

x

]

3.4. Equations Involving Arbitrary Functions3.4.1. Equations of the Form

2wt2

= a 2w

x2+ f (x, t,w)


t2= a

2wx2

+ f(x, t,w, w

x

)


= f (x) 2wx2

+ g(x, t,w, w

x

)


t2= f (w) 2w

x2+ g

(x, t,w, w

x

)


= f (x,w) 2wx2

+ g(x, t,w, w

x

)

Page vii



= f (t,w) 2wx2

+ g(x, t,w, w

x

)

3.4.7. Other Equations Linear in the Highest Derivatives

3.5. Equations of the Form 2wxy

= F(x, y,w, w

x,

wy

)

3.5.1. Equations Involving Arbitrary Parameters of the Form 2wxy

= f (w)3.5.2. Other Equations Involving Arbitrary Parameters3.5.3. Equations Involving Arbitrary Functions

4. Hyperbolic Equations with Two or Three Space Variables4.1. Equations with Two Space Variables Involving PowerLaw Nonlinearities


t2=

x

[f (x) w

x

]+

y

[g(y) w

y

]+ awp


t2= a

x

(wn w

x

)+ b

y

(wk w

y

)


t2=

x

[f (w) w

x

]+

y

[g(w) w

y

]

4.1.4. Other Equations4.2. Equations with Two Space Variables Involving Exponential Nonlinearities


t2=

x

[f (x) w

x

]+

y

[g(y) w

y

]+ aew


t2= a

x

(ew w

x

)+ b

y

(ew w

y

)

4.2.3. Other Equations4.3. Nonlinear Telegraph Equations with Two Space Variables

4.3.1. Equations Involving PowerLaw Nonlinearities4.3.2. Equations Involving Exponential Nonlinearities

4.4. Equations with Two Space Variables Involving Arbitrary Functions4.4.1. Equations of the Form

2wt2

=x

[f (x) w

x

]+

y

[g(y) w

y

]+ h(w)


t2=

x

[f (w) w

x

]+

y

[g(w) w

y

]+ h(w)

4.4.3. Other Equations4.5. Equations with Three Space Variables Involving Arbitrary Parameters


=x

[f (x) w

x

]+

y

[g(y) w

y

]+

z

[h(z) w

z

]+ awp


=x

[f (x) w

x

]+

y

[g(y) w

y

]+

z

[h(z) w

z

]+ aew


= a x

(wn w

x

)+ b

y

(wm w

y

)+ c

z

(wk w

z

)+ swp


= a x

(e1w w

x

)+ b

y

(e2w w

y

)+ c

z

(e3w w

z

)+

sew

4.6. Equations with Three Space Variables Involving Arbitrary Functions4.6.1. Equations of the Form

2wt2

=x

[f1(x) wx

]+

y

[f2(y) wy

]+

z

[f3(z) wz

]+g(w)


t2=

x

[f1(w) wx

]+

y

[f2(w) wy

]+

z

[f3(w) wz

]+g(w)


5. Elliptic Equations with Two Space Variables5.1. Equations with PowerLaw Nonlinearities

5.1.1. Equations of the Form 2wx2

+ 2w

y2= aw + bwn + cw2n1


+ 2w

y2= f (x, y,w)


+ a 2w

y2= F

(x, y,w, w

x,

wy

)

5.1.4. Equations of the Form x

[f1(x, y) wx

]+

y

[f2(x, y) wy

]= g(w)


[f1(w) wx

]+

y

[f2(w) wy

]= g(w)

5.1.6. Other Equations Involving Arbitrary Parameters

Page viii


5.2. Equations with Exponential Nonlinearities5.2.1. Equations of the Form 2w

x2+

2wy2

= a + bew + cew


+ 2w

y2= f (x, y,w)


[f1(x, y) wx

]+

y

[f2(x, y) wy

]= g(w)


[f1(w) wx

]+

y

[f2(w) wy

]= g(w)

5.2.5. Other Equations Involving Arbitrary Parameters5.3. Equations Involving Other Nonlinearities

5.3.1. Equations with Hyperbolic Nonlinearities5.3.2. Equations with Logarithmic Nonlinearities5.3.3. Equations with Trigonometric Nonlinearities

5.4. Equations Involving Arbitrary Functions5.4.1. Equations of the Form 2w

x2+

2wy2

= F (x, y,w)5.4.2. Equations of the Form a 2w

x2+ b

2wy2

= F(x, y,w, w

x,

wy

)

5.4.3. Heat and Mass Transfer Equations of the Form x

[f (x) w

x

]+

y

[g(y) w

y

]=h(w)


[f (x, y,w) w

x

]+

y

[g(x, y,w) w

y

]= h(x, y,w)


6. Elliptic Equations with Three or More Space Variables6.1. Equations with Three Space Variables Involving PowerLaw Nonlinearities


[f (x) w

x

]+

y

[g(y) w

y

]+

z

[h(z) w

z

]= awp


[f (w) w

x

]+

y

[g(w) w

y

]+

z

[g(w) w

z

]= 0

6.2. Equations with Three Space Variables Involving Exponential Nonlinearities6.2.1. Equations of the Form

x

[f (x) w

x

]+

y

[g(y) w

y

]+

z

[h(z) w

z

]= aew

6.2.2. Equations of the Form a1 x(e1w w

x

)+a2 y

(e2w w

y

)+a3 y

(e2w w

y

)= bew 416

6.3. ThreeDimensional Equations Involving Arbitrary Functions6.3.1. Heat and Mass Transfer Equations of the Form

x

[f1(x) wx

]+

y

[f2(y) wy

]+

z

[f3(z) wz

]= g(w)

6.3.2. Heat and Mass Transfer Equations with Complicating Factors6.3.3. Other Equations

6.4. Equations with n Independent Variables6.4.1. Equations of the Form

x1

[f1(x1) wx1

]+ +

xn

[fn(xn) wxn

]= g(x1, . . . ,xn,w)


7. Equations Involving Mixed Derivatives and Some Other Equations7.1. Equations Linear in the Mixed Derivative

7.1.1. Calogero Equation7.1.2. KhokhlovZabolotskaya Equation7.1.3. Equation of Unsteady Transonic Gas Flows7.1.4. Equations of the Form w

y2wxy

wx

2wy2

= F(x, y, w

x,

wy

)

7.1.5. Other Equations with Two Independent Variables7.1.6. Other Equations with Three Independent Variables

7.2. Equations Quadratic in the Highest Derivatives7.2.1. Equations of the Form

2wx2

2wy2

= F (x, y)7.2.2. MongeAmpere equation

(2wxy

)2

2wx2

2wy2

= F (x, y)7.2.3. Equations of the Form

(2wxy

)2

2wx2

2wy2

= F(x, y,w, w

x,

wy

)

Page ix


7.2.4. Equations of the Form(

2wxy

)2= f (x, y) 2w

x22wy2

+ g(x, y)7.2.5. Other Equations

7.3. Bellman Type Equations and Related Equations7.3.1. Equations with Quadratic Nonlinearities7.3.2. Equations with PowerLaw Nonlinearities

8. SecondOrder Equations of General Form8.1. Equations Involving the First Derivative in t


= F(w, w

x,

2wx2

)


= F(t,w, w

x,

2wx2

)


= F(x,w, w

x,

2wx2

)


= F(x, t,w, w

x,

2wx2

)

8.1.5. Equations of the Form F(x, t,w, w

t,

wx

,2wx2

)= 0

8.1.6. Equations with Three Independent Variables8.2. Equations Involving Two or More Second Derivatives


t2= F

(w, w

x,

2wx2

)


t2= F

(x, t,w, w

x,

wt

,2wx2

)

8.2.3. Equations Linear in the Mixed Derivative8.2.4. Equations with Two Independent Variables, Nonlinear in Two or More Highest

Derivatives8.2.5. Equations with n Independent Variables

9. ThirdOrder Equations9.1. Equations Involving the First Derivative in t

9.1.1. Kortewegde Vries Equation wt

+ a 3w

x3+ bw w

x= 0

9.1.2. Cylindrical, Spherical, and Modied Kortewegde Vries Equations9.1.3. Generalized Kortewegde Vries Equation w

t+ a

3wx3

+ f (w) wx

= 09.1.4. Equations Reducible to the Kortewegde Vries Equation9.1.5. Equations of the Form w

t+ a

3wx3

+ f(w, w

x

)= 0


+ a 3w

x3+ F

(x, t,w, w

x

)= 0

9.1.7. BurgersKortewegde Vries Equation and Other Equations9.2. Equations Involving the Second Derivative in t

9.2.1. Equations with Quadratic Nonlinearities9.2.2. Other Equations

9.3. Hydrodynamic Boundary Layer Equations9.3.1. Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid9.3.2. Steady Boundary Layer Equations for NonNewtonian Fluids9.3.3. Unsteady Boundary Layer Equations for a Newtonian Fluid9.3.4. Unsteady Boundary Layer Equations for NonNewtonian Fluids9.3.5. Related Equations

9.4. Equations of Motion of Ideal Fluid (Euler Equations)9.4.1. Stationary Equations9.4.2. Nonstationary Equations

9.5. Other ThirdOrder Nonlinear Equations9.5.1. Equations Involving SecondOrder Mixed Derivatives9.5.2. Equations Involving ThirdOrder Mixed Derivatives9.5.3. Equations Involving 3w

x3and

3wy3

Page x


10. FourthOrder Equations10.1. Equations Involving the First Derivative in t


= a 4w

x4+ F

(x, t,w, w

x

)

10.1.2. Other Equations10.2. Equations Involving the Second Derivative in t

10.2.1. Boussinesq Equation and Its Modications10.2.2. Equations with Quadratic Nonlinearities10.2.3. Other Equations

10.3. Equations Involving Mixed Derivatives10.3.1. KadomtsevPetviashvili Equation10.3.2. Stationary Hydrodynamic Equations (NavierStokes Equations)10.3.3. Nonstationary Hydrodynamic Equations (NavierStokes equations)10.3.4. Other Equations

11. Equations of Higher Orders11.1. Equations Involving the First Derivative in t and Linear in the Highest Derivative

11.1.1. FifthOrder Equations11.1.2. Equations of the Form w

t= a

nwxn

+ f (x, t,w)11.1.3. Equations of the Form w

t= a

nwxn

+ f (w) wx


= a nw

xn+ f (x, t,w) w

x+ g(x, t,w)


= a nw

xn+ F

(x, t,w, w

x

)


= a nw

xn+ F

(x, t,w, w

x, . . . ,

n1wxn1

)


= aw nw

xn+ f (x, t,w) w

x+ g(x, t,w)

11.1.8. Other Equations11.2. General Form Equations Involving the First Derivative in t


= F(w, w

x, . . . ,

nwxn

)


= F(t,w, w

x, . . . ,

nwxn

)


= F(x,w, w

x, . . . ,

nwxn

)


= F(x, t,w, w

x, . . . ,

nwxn

)

11.3. Equations Involving the Second Derivative in t11.3.1. Equations of the Form

2wt2

= a nw

xn+ f (x, t,w)


t2= a

nwxn

+ F(x, t,w, w

x

)


t2= a

nwxn

+ F(x, t,w, w

x, . . . ,

n1wxn1

)


= aw nw

xn+ f (x, t,w) w

x+ g(x, t,w)


= F(x, t,w, w

x, . . . ,

nwxn

)

11.4. Other Equations11.4.1. Equations Involving Mixed Derivatives11.4.2. Equations Involving

nwxn

and mwym

Supplements. Exact Methods for Solving Nonlinear Partial Differential EquationsS.1. Classication of SecondOrder Semilinear Partial Differential Equations in Two

Independent VariablesS.1.1. Types of Equations. Characteristic EquationS.1.2. Canonical Form of Parabolic EquationsS.1.3. Canonical Form of Hyperbolic EquationsS.1.4. Canonical Form of Elliptic Equations

Page xi


S.2. Transformations of Equations of Mathematical PhysicsS.2.1. Point TransformationsS.2.2. Hodograph TransformationS.2.3. Contact Transformations. Legendre and Euler TransformationsS.2.4. Backlund Transformations. Differential Substitutions

S.3. TravelingWave Solutions and SelfSimilar Solutions. Similarity MethodsS.3.1. Preliminary RemarksS.3.2. TravelingWave Solutions. Invariance of Equations Under TranslationsS.3.3. SelfSimilar Solutions. Invariance of Equations Under Scaling TransformationsS.3.4. Exponential SelfSimilar Solutions. Equations Invariant Under Combined

Translation and ScalingS.4. Method of Generalized Separation of Variables

S.4.1. IntroductionS.4.2. Structure of Generalized Separable SolutionsS.4.3. Solution of FunctionalDifferential Equations by DifferentiationS.4.4. Solution of FunctionalDifferential Equations by SplittingS.4.5. Simplied Scheme for Constructing Generalized Separable SolutionsS.4.6. TitovGalaktionov Method

S.5. Method of Functional Separation of VariablesS.5.1. Structure of Functional Separable SolutionsS.5.2. Special Functional Separable SolutionsS.5.3. Differentiation MethodS.5.4. Splitting Method. Reduction to a Functional Equation with Two VariablesS.5.5. Solutions of Some Nonlinear Functional Equations and Their Applications

S.6. Generalized Similarity Reductions of Nonlinear EquationsS.6.1. ClarksonKruskal Direct Method: a Special Form for Similarity ReductionS.6.2. ClarksonKruskal Direct Method: the General Form for Similarity ReductionS.6.3. Some Modications and Generalizations

S.7. Group Analysis MethodsS.7.1. Classical Method for Symmetry ReductionsS.7.2. Nonclassical Method for Symmetry Reductions

S.8. Differential Constraints MethodS.8.1. Description of the MethodS.8.2. FirstOrder Differential ConstraintsS.8.3. Second and HigherOrder Differential ConstraintsS.8.4. Connection Between the Differential Constraints Method and Other Methods

S.9. Painleve Test for Nonlinear Equations of Mathematical PhysicsS.9.1. Movable Singularities of Solutions of Ordinary Differential EquationsS.9.2. Solutions of Partial Differential Equations with a Movable Pole. Description of

the MethodS.9.3. Examples of the Painleve Test Applications

S.10. Inverse Scattering MethodS.10.1. Lax Pair MethodS.10.2. Method Based on the Compatibility Condition for Two Linear EquationsS.10.3. Method Based on Linear Integral Equations

S.11. Conservation LawsS.11.1. Basic Denitions and ExamplesS.11.2. Equations Admitting Variational Formulation. Noetherian Symmetries

Page xii


S.12. Hyperbolic Systems of Quasilinear EquationsS.12.1. Conservation Laws. Some ExamplesS.12.2. Cauchy Problem, Riemann Problem, and InitialBoundary Value ProblemS.12.3. Characteristic Lines. Hyperbolic Systems. Riemann InvariantsS.12.4. SelfSimilar Continuous Solutions. Rarefaction WavesS.12.5. Shock Waves. RankineHugoniot Jump ConditionsS.12.6. Evolutionary Shocks. Lax Condition (Various Formulations)S.12.7. Solutions for the Riemann ProblemS.12.8. InitialBoundary Value Problems of Special FormS.12.9. Examples of Nonstrict Hyperbolic Systems

References

Page xiii


AUTHORS

Andrei D. Polyanin, Ph.D., D.Sc., is a noted scientist of broadinterests, who works in various areas of mathematics, mechanics,and chemical engineering sciences.

A. D. Polyanin graduated from the Department of Mechanicsand Mathematics of the Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at theInstitute for Problems in Mechanics of the Russian (former USSR)Academy of Sciences. Since 1975, A. D. Polyanin has been amember of the staff of the Institute for Problems in Mechanics ofthe Russian Academy of Sciences. He is a member of the RussianNational Committee on Theoretical and Applied Mechanics.

Professor Polyanin is an author of 33 books in English, Russian,German, and Bulgarian, as well as over 120 research papers andthree patents. He has written a number of fundamental handbooks,including A. D. Polyanin and V. F. Zaitsev, Handbook of ExactSolutions for Ordinary Differential Equations, CRC Press, 1995and 2002; A. D. Polyanin and A. V. Manzhirov, Handbook of Inte

gral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equationsfor Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, andA. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002;and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Mathematical Physics Equations,Fizmatlit, 2002. Professor Polyanin is Editor of the book series Differential and Integral Equations and Their Applications, Taylor & Francis, London, and Physical and Mathematical ReferenceLiterature, Fizmatlit, Moscow.

In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciencesfor his research in mechanics. In 2001, he received an award from the Ministry of Education of theRussian Federation.Address: Institute for Problems in Mechanics, RAS, 101 Vernadsky Avenue, Building 1, 119526 Moscow, RussiaE-mail: [email protected]

Valentin F. Zaitsev, Ph.D., D.Sc., is a noted scientist in the eldsof ordinary differential equations, mathematical physics, and nonlinear mechanics.

V. F. Zaitsev graduated from the Radio Electronics Faculty ofthe Leningrad Polytechnical Institute (now SaintPetersburg Technical University) in 1969 and received his Ph.D. degree in 1983 atthe Leningrad State University. His Ph.D. thesis was devoted to thegroup approach to the study of some classes of ordinary differentialequations. In 1992, Professor Zaitsev received his Doctor of Sciences degree; his D.Sc. thesis was dedicated to the discretegroupanalysis of ordinary differential equations.

In 19711996, V. F. Zaitsev worked in the Research Institute forComputational Mathematics and Control Processes of the St. Petersburg State University. Since 1996, Professor Zaitsev has beena member of the staff of the Russian State Pedagogical University(St. Petersburg).

Professor Zaitsev has made important contributions to newmethods in the theory of ordinary and partial differential equations. He is an author of morethan 130 scientic publications, including 18 books and one patent.Address: Russian State Pedagogical University, 48 Naberezhnaya reki Moiki, 191186 Saint-Petersburg, RussiaE-mail: valentin [email protected]

Page xv


FOREWORD

Nonlinear partial differential equations are encountered in various elds of mathematics,physics,chemistry, and biology, and numerous applications.

Exact (closedform) solutions of differential equations play an important role in the properunderstanding of qualitative features of many phenomena and processes in various areas of naturalscience. Exact solutions of nonlinear equations graphically demonstrate and allow unravelingthe mechanisms of many complex nonlinear phenomena such as spatial localization of transferprocesses, multiplicity or absence steady states under various conditions, existence of peakingregimes and many others. Furthermore, simple solutions are often used in teaching many coursesas specic examples illustrating basic tenets of a theory that admit mathematical formulation.

Even those special exact solutions that do not have a clear physical meaning can be used astest problems to verify the consistency and estimate errors of various numerical, asymptotic, andapproximate analytical methods. Exact solutions can serve as a basis for perfecting and testingcomputer algebra software packages for solving differential equations. It is signicant that manyequations of physics, chemistry, and biology contain empirical parameters or empirical functions.Exact solutions allow researchers to design and run experiments, by creating appropriate naturalconditions, to determine these parameters or functions.

This book contains more than 1600 nonlinear mathematical physics equations and nonlinearpartial differential equations and their solutions. A large number of new exact solutions to nonlinearequations are described. Equations of parabolic, hyperbolic, elliptic, mixed, and general types arediscussed. Second, third, fourth, and higherorder nonlinear equations are considered. The bookpresents exact solutions to equations of heat and mass transfer, wave theory, nonlinear mechanics,hydrodynamics, gas dynamics, plasticity theory, nonlinear acoustics, combustion theory, nonlinearoptics, theoretical physics, differential geometry, control theory, chemical engineering sciences,biology, and other elds.

Special attention is paid to generalform equations that depend on arbitrary functions; exactsolutions of such equations are of principal value for testing numerical and approximate methods.Almost all other equations contain one or more arbitrary parameters (in fact, this book deals withwhole families of partial differential equations), which can be xed by the reader at will. In total,the handbook contains signicantly more nonlinear PDEs and exact solutions than any other bookcurrently available.

The supplement of the book presents exact analytical methods for solving nonlinear mathematical physics equations. When selecting the material, the authors have given a pronounced preferenceto practical aspects of the matter; that is, to methods that allow effectively constructing exactsolutions. Apart from the classical methods, the book also describes widerange methods that havebeen greatly developed over the last decade (the nonclassical and direct methods for symmetryreductions, the differential constraints method, the method of generalized separation of variables,and others). For the readers better understanding of the methods, numerous examples of solving specic differential equations and systems of differential equations are given throughout thebook.

For the convenience of a wide audience with different mathematical backgrounds, the authorstried to do their best, wherever possible, to avoid special terminology. Therefore, some of themethods are outlined in a schematic and somewhat simplied manner, with necessary referencesmade to books where these methods are considered in more detail. Many sections were written sothat they could be read independently from each other. This allows the reader to quickly get to theheart of the matter.

The handbook consists of chapters, sections, and subsections. Equations and formulas arenumbered separately in each subsection. The equations within subsections are arranged in increasingorder of complexity. The extensive table of contents provides rapid access to the desired equations.

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Separate parts of the book may be used by lecturers of universities and colleges for practicalcourses and lectures on nonlinear mathematical physics equations for graduate and postgraduatestudents. Furthermore, the books may be used as a database of test problems for numerical andapproximate methods for solving nonlinear partial differential equations.

We would like to express our deep gratitude to Alexei Zhurov for fruitful discussions andvaluable remarks.

The authors hope that this book will be helpful for a wide range of scientists, university teachers,engineers, and students engaged in the elds of mathematics, physics, mechanics, control, chemistry,and engineering sciences.

Andrei D. PolyaninValentin F. Zaitsev

Page xviii


SOME NOTATIONS AND REMARKS

Latin CharactersC1,C2, . . . are arbitrary constants;

r, , z cylindrical coordinates, r =x2 + y2 and x = r cos, y = r sin;

r, , spherical coordinates,r =x2 +y2 +z2 andx=r sin cos, y = sin sin, z =r cos ;

t time (t 0);w unknown function (dependent variable);

x, y, z space (Cartesian) coordinates;x1, . . . ,xn Cartesian coordinates in ndimensional space.

Greek Characters Laplace operator; in twodimensional case, =

2

x2+

2

y2;

in threedimensional case, = 2

x2+

2

y2+

2

z2; in ndimensional case, =

n

k=1

2

x2k

;

biharmonic operator; in twodimensional case, = 4

x4+ 2

4

x2y2+

4

y4.

Brief Notation for Derivatives

wx =w

x, wt =

w

t, wxx =

2w

x2, wxt =

2w

xt, wtt =

2w

t2, . . . (partial derivatives);

f x =df

dx, f xx =

d2f

dx2, f xxx =

d3f

dx3, f xxxx =

d4f

dx4, f (n)x =

dnf

dxn(derivatives for f = f (x)).

Brief Notation for Differential Operators

x =

x, y =

y, t =

t, w =

w(differential operators in x, y, t, and w);

Dx =

x+ wx

w+ wxx

wx+ wxt

wt+ (total differential operator in x);

Dt =

t+ wt

w+ wxt

wx+ wtt

wt+ (total differential operator in t).

In the last two relations, w is assumed to be dependent on x and t, w = w(x, t).

Remarks1. The book presents solutions of the following types:

(a) expressible in terms of elementary functions explicitly, implicitly, or parametrically;(b) expressible in terms of elementary functions and integrals of elementary functions;(c) expressible in terms of elementary functions, functions involved in the equation (if the equation

contains arbitrary functions), and integrals of the equation functions and/or other elementaryfunctions;

(d) expressible in terms of ordinary differential equations or nite systems of ordinary differentialequations;

(e) expressible in terms of solutions to nonlinear equations that can be reduced to linear partialdifferential equations or linear integral equations.

Page xix


2. The book also deals with solutions described by equations with fewer new variables thanthose in the original equations. An expression that solves an equations in three independent variablesand is determined by an equation in two independent variables will be called a twodimensionalsolution.

3. As a rule, the book does not present simple solutions that depend on only one of the variablesinvolved in the original equation.

4. Equations are numbered separately in each subsection. When referencing a particular equation, we use a notation like 3.1.2.5, which implies equation 5 from Subsection 3.1.2.

5. If a formula or a solution contains an expression like f (x)a 2

, it is often not stated that theassumption a 2 is implied.

6. Though incomplete, very simple and graphical classication of solutions by their appearanceis used in the book. For equations in two independent variables, x and t, and one unknown, w, thesolution name and structure are as follows (x and t in the solutions below can be swapped):

No. Solution name Solution structure1 Travelingwave solution w = F (z), z = x + t, 02 Additive separable solution w = (x) + (t)3 Multiplicative separable solution w = (x)(t)4 Selfsimilar solution w = tF (z), z = xt5 Generalized selfsimilar solution w = (t)F (z), z = x(t)6 Generalized travelingwave solution w = F (z), z = (t)x + (t)7 Generalized separable solution w = 1(x)1(t) + + n(x)n(t)8 Functional separable solution w = F (z), z = 1(x)1(t) + + n(x)n(t)

For uniformity of presentation, we also use this term in the cases where the variable t plays the role of a spatial coordinate. Sometimes, a solution of the form w = tF (z), z = xt , where x = x+C1 and t = t+C2 , will also be called a self-similarsolution.

7. The present book does not consider rstorder nonlinear partial differential equations. Forthese equations, see Kamke (1965), Rhee, Aris, and Amundson (1986, 1989), and Polyanin, Zaitsev,and Moussiaux (2002).

8. ODE and PDE are conventional abbreviations for ordinary differential equation and partialdifferential equation, respectively.

This symbol indicates references to literature sources whenever:(a) at least one of the solutions was obtained in the cited source (even though the solution contained

correctable misprints in signs or coefcients);(b) the cited source provides further information on the equation in question and their solutions.

Page xx


Handbook Of Nonlinear Partial Differential EquationsContentsAuthorsForewordSome Notations And RemarksLatin CharactersGreek CharactersBrief Notation For DerivativesBrief Notation For Differential OperatorsRemarks

all oall of equationf equation

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