Handbook of mathematics for engineers and scienteists part 2 doc

CONTENTS vii
4.6. LineandPlaneinSpace 124
4.6.1. PlaneinSpace 124
4.6.2. LineinSpace 131
4.6.3. MutualArrangementofPoints,Lines,andPlanes 135
4.7. Quadric Surfaces (Quadrics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.7.1. Quadrics (Canonical Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
4.7.2. Quadrics(GeneralTheory) 148
ReferencesforChapter4 153
5. Algebra 155
5.1. Polynomials and Algebraic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1.1. Polynomials and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.1.2. LinearandQuadraticEquations 157
5.1.3. CubicEquations 158
5.1.4. Fourth-DegreeEquation 159
5.1.5. Algebraic Equations of Arbitrary Degree and Their Properties . . . . . . . . . . . . . . 161
5.2. MatricesandDeterminants 167
5.2.1. Matrices 167
5.2.2. Determinants 175
5.2.3. EquivalentMatrices.Eigenvalues 180
5.3. Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.3.1. Concept of a Linear Space. Its Basis and Dimension . . . . . . . . . . . . . . . . . . . . . . 187
5.3.2. Subspaces of Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
5.3.3. Coordinate Transformations Corresponding to Basis Transformations in a Linear
Space 191
5.4. Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.4.1. RealEuclideanSpace 192
5.4.2. Complex Euclidean Space (Unitary Space) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.4.3. Banach Spaces and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.5. SystemsofLinearAlgebraicEquations 197
5.5.1. Consistency Condition for a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

5.5.2. Finding Solutions of a System of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 198
5.6. LinearOperators 204
5.6.1. NotionofaLinearOperator.ItsProperties 204
5.6.2. LinearOperatorsinMatrixForm 208
5.6.3. EigenvectorsandEigenvaluesofLinearOperators 209
5.7. Bilinear and Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.7.1. Linear and Sesquilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
5.7.2. Bilinear Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.7.3. QuadraticForms 216
5.7.4. Bilinear and Quadratic Forms in Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . 219
5.7.5. Second-Order Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
5.8. SomeFactsfromGroupTheory 225
5.8.1. Groups and Their Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.8.2. Transformation Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
5.8.3. GroupRepresentations 230
ReferencesforChapter5 233
viii CONTENTS
6. Limits and Derivatives 235
6.1. BasicConceptsofMathematicalAnalysis 235
6.1.1. NumberSets.FunctionsofRealVariable 235
6.1.2. LimitofaSequence 237
6.1.3. LimitofaFunction.Asymptotes 240
6.1.4. Infinitely Small and InfinitelyLargeFunctions 242
6.1.5. Continuous Functions. Discontinuities of the First and the Second Kind . . . . . . . 243
6.1.6. ConvexandConcaveFunctions 245
6.1.7. Functions of Bounded Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
6.1.8. ConvergenceofFunctions 249
6.2. DifferentialCalculusforFunctionsofaSingleVariable 250
6.2.1. Derivative and Differential, Their Geometrical and Physical Meaning . . . . . . . . . 250
6.2.2. TableofDerivativesandDifferentiationRules 252

6.2.3. Theorems about Differentiable Functions. L’Hospital Rule . . . . . . . . . . . . . . . . . 254
6.2.4. Higher-Order Derivatives and Differentials. Taylor’s Formula . . . . . . . . . . . . . . . 255
6.2.5. Extremal Points. Points of Inflection 257
6.2.6. Qualitative Analysis of Functions and Construction of Graphs . . . . . . . . . . . . . . 259
6.2.7. Approximate Solution of Equations (Root-Finding Algorithms for Continuous
Functions) 260
6.3. FunctionsofSeveralVariables.PartialDerivatives 263
6.3.1. PointSets.Functions.LimitsandContinuity 263
6.3.2. DifferentiationofFunctionsofSeveralVariables 264
6.3.3. Directional Derivative. Gradient. GeometricalApplications 267
6.3.4. ExtremalPointsofFunctionsofSeveralVariables 269
6.3.5. DifferentialOperatorsoftheFieldTheory 272
ReferencesforChapter6 272
7. Integrals 273
7.1. IndefiniteIntegral 273
7.1.1. Antiderivative. IndefiniteIntegralandItsProperties 273
7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral. Integration
Examples 274
7.1.3. IntegrationofRationalFunctions 276
7.1.4. Integration of Irrational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
7.1.5. Integration of Exponential and Trigonometric Functions . . . . . . . . . . . . . . . . . . . 281
7.1.6. Integration of Polynomials Multiplied by Elementary Functions . . . . . . . . . . . . . 283
7.2. DefiniteIntegral 286
7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical Meaning of the
DefiniteIntegral 286
7.2.2. Properties of DefiniteIntegralsandUsefulFormulas 287
7.2.3. General Reduction Formulas for the Evaluation of Integrals . . . . . . . . . . . . . . . . 289
7.2.4. General Asymptotic Formulas for the Calculation of Integrals . . . . . . . . . . . . . . . 290
7.2.5. Mean Value Theorems. Properties of Integrals in Terms of Inequalities.
ArithmeticMeanandGeometricMeanofFunctions 295

7.2.6. Geometric and Physical Applications of the DefiniteIntegral 299
7.2.7. Improper Integrals with InfiniteIntegrationLimit 301
7.2.8. General Reduction Formulas for the Calculation of Improper Integrals . . . . . . . . 304
7.2.9. General Asymptotic Formulas for the Calculation of Improper Integrals . . . . . . . 307
7.2.10. Improper Integrals of Unbounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
7.2.11. Cauchy-Type Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
CONTENTS ix
7.2.12. Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
7.2.13. SquareIntegrableFunctions 314
7.2.14. Approximate (Numerical) Methods for Computation of Definite Integrals . . . . 315
7.3. Double and Triple Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3.1. Definition and Properties of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 317
7.3.2. Computation of the Double Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
7.3.3. Geometric and Physical Applications of the Double Integral . . . . . . . . . . . . . . . . 323
7.3.4. Definition and Properties of the Triple Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
7.3.5. Computation of the Triple Integral. Some Applications. Iterated Integrals and
AsymptoticFormulas 325
7.4. LineandSurfaceIntegrals 329
7.4.1. LineIntegraloftheFirstKind 329
7.4.2. LineIntegraloftheSecondKind 330
7.4.3. SurfaceIntegraloftheFirstKind 332
7.4.4. SurfaceIntegraloftheSecondKind 333
7.4.5. IntegralFormulasofVectorCalculus 334
ReferencesforChapter7 335
8. Series 337
8.1. Numerical Series and Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s Criterion . . . . . . . 337
8.1.2. Convergence Criteria for Series with Positive (Nonnegative) Terms . . . . . . . . . . 338
8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and Conditional
Convergence 341

8.1.4. Multiplication of Series. Some Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
8.1.5. Summation Methods. Convergence Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 344
8.1.6. Infinite Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
8.2. FunctionalSeries 348
8.2.1. Pointwise and Uniform Convergence of Functional Series . . . . . . . . . . . . . . . . . . 348
8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly Convergent
Series 349
8.3. PowerSeries 350
8.3.1. Radius of Convergence of Power Series. Properties of Power Series . . . . . . . . . . 350
8.3.2. TaylorandMaclaurinPowerSeries 352
8.3.3. Operations with Power Series. Summation Formulas for Power Series . . . . . . . . 354
8.4. FourierSeries 357
8.4.1. Representation of 2π-Periodic Functions by Fourier Series. Main Results . . . . . 357
8.4.2. Fourier Expansions of Periodic, Nonperiodic, Odd, and Even Functions . . . . . . . 359
8.4.3. Criteria of Uniform and Mean-Square Convergence of Fourier Series . . . . . . . . . 361
8.4.4. Summation Formulas for Trigonometric Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
8.5. AsymptoticSeries 363
8.5.1. Asymptotic Series of Poincar
´
e Type. Formulas for the Coefficients 363
8.5.2. OperationswithAsymptoticSeries 364
ReferencesforChapter8 366
9. Differential Geometry 367
9.1. TheoryofCurves 367
9.1.1. PlaneCurves 367
9.1.2. SpaceCurves 379
x CONTENTS
9.2. Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.1. Elementary Notions in Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
9.2.2. CurvatureofCurvesonSurface 392

9.2.3. IntrinsicGeometryofSurface 395
ReferencesforChapter9 397
10. Functions of Complex Variable 399
10.1. BasicNotions 399
10.1.1. ComplexNumbers.FunctionsofComplexVariable 399
10.1.2. FunctionsofComplexVariable 401
10.2. MainApplications 419
10.2.1. ConformalMappings 419
10.2.2. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
ReferencesforChapter10 433
11. Integral Transforms 435
11.1. GeneralFormofIntegralTransforms.SomeFormulas 435
11.1.1. IntegralTransformsandInversionFormulas 435
11.1.2. Residues.JordanLemma 435
11.2. LaplaceTransform 436
11.2.1. LaplaceTransformandtheInverseLaplaceTransform 436
11.2.2. Main Properties of the Laplace Transform. Inversion Formulas for Some
Functions 437
11.2.3. Limit Theorems. Representation of Inverse Transforms as Convergent Series
andAsymptoticExpansions 440
11.3. Mellin Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.1. Mellin Transform and the Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 441
11.3.2. Main Properties of the Mellin Transform. Relation Among the Mellin,
Laplace, and Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442
11.4. VariousFormsoftheFourierTransform 443
11.4.1. FourierTransformandtheInverseFourierTransform 443
11.4.2. FourierCosineandSineTransforms 445
11.5. OtherIntegralTransforms 446
11.5.1. Integral Transforms Whose Kernels Contain Bessel Functions and Modified
BesselFunctions 446

11.5.2. Summary Table of Integral Transforms. Areas of Application of Integral
Transforms 448
ReferencesforChapter11 451
12. Ordinary Differential Equations 453
12.1. First-OrderDifferentialEquations 453
12.1.1. General Concepts. The Cauchy Problem. Uniqueness and Existence Theorems 453
12.1.2. Equations Solved for the Derivative. Simplest Techniques of Integration . . . . 456
12.1.3. ExactDifferentialEquations.IntegratingFactor 458
12.1.4. Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460
12.1.5. AbelEquationsoftheFirstKind 462
12.1.6. Abel Equations of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
12.1.7. EquationsNotSolvedfortheDerivative 465
12.1.8. ContactTransformations 468
12.1.9. Approximate Analytic Methods for Solution of Equations . . . . . . . . . . . . . . . . 469
12.1.10. NumericalIntegrationofDifferentialEquations 471
CONTENTS xi
12.2. Second-Order Linear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
12.2.1. Formulas for the General Solution. Some Transformations . . . . . . . . . . . . . . . 472
12.2.2. Representation of Solutions as a Series in the Independent Variable . . . . . . . . 475
12.2.3. AsymptoticSolutions 477
12.2.4. Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
12.2.5. EigenvalueProblems 482
12.2.6. TheoremsonEstimatesandZerosofSolutions 487
12.3. Second-Order Nonlinear Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.1. Form of the General Solution. Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . 488
12.3.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489
12.3.3. Methods of Regular Series Expansions with Respect to the Independent
Variable 492
12.3.4. Movable Singularities of Solutions of Ordinary Differential Equations.
Painlev

´
eTranscendents 494
12.3.5. Perturbation Methods of Mechanics and Physics . . . . . . . . . . . . . . . . . . . . . . . 499
12.3.6. Galerkin Method and Its Modifications (Projection Methods) . . . . . . . . . . . . . 508
12.3.7. Iteration and Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
12.4. LinearEquationsofArbitraryOrder 514
12.4.1. Linear Equations with Constant Coefficients 514
12.4.2. Linear Equations with Variable Coefficients 518
12.4.3. AsymptoticSolutionsofLinearEquations 522
12.4.4. CollocationMethodandItsConvergence 523
12.5. NonlinearEquationsofArbitraryOrder 524
12.5.1. StructureoftheGeneralSolution.CauchyProblem 524
12.5.2. Equations Admitting Reduction of Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
12.6. LinearSystemsofOrdinaryDifferentialEquations 528
12.6.1. Systems of Linear Constant-CoefficientEquations 528
12.6.2. Systems of Linear Variable-CoefficientEquations 539
12.7. NonlinearSystemsofOrdinaryDifferentialEquations 542
12.7.1. Solutions and First Integrals. Uniqueness and Existence Theorems . . . . . . . . . 542
12.7.2. Integrable Combinations. Autonomous Systems of Equations . . . . . . . . . . . . . 545
12.7.3. Elements of Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
ReferencesforChapter12 550
13. First-Order Partial Differential Equations 553
13.1. Linear and Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
13.1.1. CharacteristicSystem.GeneralSolution 553
13.1.2. CauchyProblem.ExistenceandUniquenessTheorem 556
13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations . . 558
13.1.4. Quasilinear Equations of General Form. Generalized Solution, Jump
Condition, and Stability Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 567
13.2. NonlinearEquations 570
13.2.1. Solution Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 570

13.2.2. CauchyProblem.ExistenceandUniquenessTheorem 576
13.2.3. Generalized Viscosity Solutions and Their Applications . . . . . . . . . . . . . . . . . 579
ReferencesforChapter13 584
xii CONTENTS
14. Linear Partial Differential Equations 585
14.1. Classification of Second-Order Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 585
14.1.1. EquationswithTwoIndependentVariables 585
14.1.2. EquationswithManyIndependentVariables 589
14.2. BasicProblemsofMathematicalPhysics 590
14.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 590
14.2.2. First, Second, Third, and Mixed Boundary Value Problems . . . . . . . . . . . . . . . 593
14.3. Properties and Exact Solutions of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
14.3.1. Homogeneous Linear Equations and Their Particular Solutions . . . . . . . . . . . . 594
14.3.2. Nonhomogeneous Linear Equations and Their Particular Solutions . . . . . . . . . 598
14.3.3. General Solutions of Some Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . 600
14.4. Method of Separation of Variables (Fourier Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
14.4.1. Description of the Method of Separation of Variables. General Stage of
Solution 602
14.4.2. Problems for Parabolic Equations: Final Stage of Solution . . . . . . . . . . . . . . . 605
14.4.3. Problems for Hyperbolic Equations: Final Stage of Solution . . . . . . . . . . . . . . 607
14.4.4. Solution of Boundary Value Problems for Elliptic Equations . . . . . . . . . . . . . . 609
14.5. IntegralTransformsMethod 611
14.5.1. Laplace Transform and Its Application in Mathematical Physics . . . . . . . . . . . 611
14.5.2. Fourier Transform and Its Application in Mathematical Physics . . . . . . . . . . . 614
14.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution . . 615
14.6.1. CauchyProblemforParabolicEquations 615
14.6.2. CauchyProblemforHyperbolicEquations 617
14.7. Boundary Value Problems for Parabolic Equations with One Space Variable. Green’s
Function 618
14.7.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 618

14.7.2. Problems for Equation s(x)
∂w
∂t
=
∂
∂x

p(x)
∂w
∂x

–q(x)w + Φ(x, t) 620
14.8. Boundary Value Problems for Hyperbolic Equations with One Space Variable. Green’s
Function.GoursatProblem 623
14.8.1. Representation of Solutions via the Green’s Function . . . . . . . . . . . . . . . . . . . . 623
14.8.2. Problems for Equation s(x)
∂
2
w
∂t
2
=
∂
∂x

p(x)
∂w
∂x

–q(x)w + Φ(x, t) 624

14.8.3. Problems for Equation
∂
2
w
∂t
2
+ a(t)
∂w
∂t
= b(t)

∂
∂x

p(x)
∂w
∂x

– q(x)w

+ Φ(x, t) 626
14.8.4. Generalized Cauchy Problem with Initial Conditions Set Along a Curve . . . . . 627
14.8.5. Goursat Problem (a Problem with Initial Data of Characteristics) . . . . . . . . . . 629
14.9. Boundary Value Problems for Elliptic Equations with Two Space Variables . . . . . . . . . 631
14.9.1. Problems and the Green’s Functions for Equation
a(x)
∂
2
w
∂x

2
+
∂
2
w
∂y
2
+ b(x)
∂w
∂x
+ c(x)w =–Φ(x, y) 631
14.9.2. Representation of Solutions to Boundary Value Problems via the Green’s
Functions 633
14.10. Boundary Value Problems with Many Space Variables. Representation of Solutions
viatheGreen’sFunction 634
14.10.1. ProblemsforParabolicEquations 634
14.10.2. Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636
14.10.3. Problems for Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
14.10.4. Comparison of the Solution Structures for Boundary Value Problems for
EquationsofVariousTypes 638
CONTENTS xiii
14.11. Construction of the Green’s Functions. General Formulas and Relations . . . . . . . . . . 639
14.11.1. Green’s Functions of Boundary Value Problems for Equations of Various
Types in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639
14.11.2. Green’s Functions Admitting Incomplete Separation of Variables . . . . . . . . 640
14.11.3. Construction of Green’s Functions via Fundamental Solutions . . . . . . . . . . 642
14.12. Duhamel’s Principles in Nonstationary Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
14.12.1. Problems for Homogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . . . . 646
14.12.2. Problems for Nonhomogeneous Linear Equations . . . . . . . . . . . . . . . . . . . . 648
14.13. Transformations Simplifying Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . 649

14.13.1. Transformations That Lead to Homogeneous Boundary Conditions . . . . . . 649
14.13.2. Transformations That Lead to Homogeneous Initial and Boundary
Conditions 650
ReferencesforChapter14 650
15. Nonlinear Partial Differential Equations 653
15.1. Classification of Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 653
15.1.1. Classification of Semilinear Equations in Two Independent Variables . . . . . . . 653
15.1.2. Classification of Nonlinear Equations in Two Independent Variables . . . . . . . . 653
15.2. Transformations of Equations of Mathematical Physics . . . . . . . . . . . . . . . . . . . . . . . . . 655
15.2.1. PointTransformations:OverviewandExamples 655
15.2.2. Hodograph Transformations (Special Point Transformations) . . . . . . . . . . . . . 657
15.2.3. Contact Transformations. Legendre and Euler Transformations . . . . . . . . . . . . 660
15.2.4. B
¨
acklund Transformations. Differential Substitutions . . . . . . . . . . . . . . . . . . . 663
15.2.5. Differential Substitutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666
15.3. Traveling-Wave Solutions, Self-Similar Solutions, and Some Other Simple Solutions.
SimilarityMethod 667
15.3.1. PreliminaryRemarks 667
15.3.2. Traveling-Wave Solutions. Invariance of Equations Under Translations . . . . . 667
15.3.3. Self-Similar Solutions. Invariance of Equations Under Scaling
Transformations 669
15.3.4. Equations Invariant Under Combinations of Translation and Scaling
Transformations,andTheirSolutions 674
15.3.5. GeneralizedSelf-SimilarSolutions 677
15.4. ExactSolutionswithSimpleSeparationofVariables 678
15.4.1. Multiplicative and Additive Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . 678
15.4.2. Simple Separation of Variables in Nonlinear Partial Differential Equations . . . 678
15.4.3. Complex Separation of Variables in Nonlinear Partial Differential Equations . 679
15.5. MethodofGeneralizedSeparationofVariables 681

15.5.1. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 681
15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System
ofCoordinateFunctions 683
15.5.3. Solution of Functional Differential Equations by Differentiation . . . . . . . . . . . 684
15.5.4. Solution of Functional-Differential Equations by Splitting . . . . . . . . . . . . . . . . 688
15.5.5. Titov–GalaktionovMethod 693
15.6. MethodofFunctionalSeparationofVariables 697
15.6.1. Structure of Functional Separable Solutions. Solution by Reduction to
Equations with Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
15.6.2. Special Functional Separable Solutions. Generalized Traveling-Wave
Solutions 697