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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
References for Chapter 14 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 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. Point Transformations: Overview and Examples .. .. ... .. .. .. .. .. .. .. .. 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.
Similarity Method .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 667
15.3.1. Preliminary Remarks .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 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, and Their Solutions . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 674
15.3.5. Generalized Self-Similar Solutions . . . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 677
15.4. Exact Solutions with Simple Separation of Variables .. .. .. .. ... .. .. .. .. .. .. .. .. 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. Method of Generalized Separation of Variables .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 681
15.5.1. Structure of Generalized Separable Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 681
15.5.2. Simplified Scheme for Constructing Solutions Based on Presetting One System
of Coordinate Functions .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. .. .. .. 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–Galaktionov Method .. .. .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . . . . . . . 693
15.6. Method of Functional Separation of Variables . . . . . . . . . . . . . ... .. .. .. .. .. .. .. .. 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