**EXACT SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS **

First-Order Quasilinear Equations

Equations with Two Independent Variables Containing Arbitrary Parameters

Equations with Two Independent Variables Containing Arbitrary Functions

Other Quasilinear Equations

**First-Order Equations with Two Independent Variables Quadratic in Derivatives **

Equations Containing Arbitrary Parameters

Equations Containing Arbitrary Functions

**First-Order Nonlinear Equations with Two Independent Variables of General Form **

Nonlinear Equations Containing Arbitrary Parameters

Equations Containing Arbitrary Functions of Independent Variables

Equations Containing Arbitrary Functions of Derivatives

**First-Order Nonlinear Equations with Three or More Independent Variables**

Nonlinear Equations with Three Variables Quadratic in Derivatives

Other Nonlinear Equations with Three Variables Containing Parameters

Nonlinear Equations with Three Variables Containing Arbitrary Functions

Nonlinear Equations with Four Independent Variables

Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Parameters

Nonlinear Equations with Arbitrary Number of Variables Containing Arbitrary Functions

**Second-Order Parabolic Equations with One Space Variable **

Equations with Power Law Nonlinearities

Equations with Exponential Nonlinearities

Equations with Hyperbolic Nonlinearities

Equations with Logarithmic Nonlinearities

Equations with Trigonometric Nonlinearities

Equations Involving Arbitrary Functions

Nonlinear Schrödinger Equations and Related Equations

**Second-Order Parabolic Equations with Two or More Space Variables **

Equations with Two Space Variables Involving Power Law Nonlinearities

Equations with Two Space Variables Involving Exponential Nonlinearities

Other Equations with Two Space Variables Involving Arbitrary Parameters

Equations Involving Arbitrary Functions

Equations with Three or More Space Variables

Nonlinear Schrödinger Equations

**Second-Order Hyperbolic Equations with One Space Variable **

Equations with Power Law Nonlinearities

Equations with Exponential Nonlinearities

Other Equations Involving Arbitrary Parameters

Equations Involving Arbitrary Functions

Equations of the Form

**Second-Order Hyperbolic Equations with Two or More Space Variables**

Equations with Two Space Variables Involving Power Law Nonlinearities

Equations with Two Space Variables Involving Exponential Nonlinearities

Nonlinear Telegraph Equations with Two Space Variables

Equations with Two Space Variables Involving Arbitrary Functions

Equations with Three Space Variables Involving Arbitrary Parameters

Equations with Three or More Space Variables Involving Arbitrary Functions

**Second-Order Elliptic Equations with Two Space Variables **

Equations with Power Law Nonlinearities

Equations with Exponential Nonlinearities

Equations Involving Other Nonlinearities

Equations Involving Arbitrary Functions

**Second-Order Elliptic Equations with Three or More Space Variables **

Equations with Three Space Variables Involving Power Law Nonlinearities

Equations with Three Space Variables Involving Exponential Nonlinearities

Three-Dimensional Equations Involving Arbitrary Functions

Equations with *n* Independent Variables

**Second-Order Equations Involving Mixed Derivatives and Some Other Equations **

Equations Linear in the Mixed Derivative

Equations Quadratic in the Highest Derivatives

Bellman-Type Equations and Related Equations

**Second-Order Equations of General Form **

Equations Involving the First Derivative in *t*

Equations Involving Two or More Second Derivatives

**Third-Order Equations **

Equations Involving the First Derivative in *t*

Equations Involving the Second Derivative in *t *

Hydrodynamic Boundary Layer Equations

Equations of Motion of Ideal Fluid (Euler Equations)

Other Third-Order Nonlinear Equations

**Fourth-Order Equations **

Equations Involving the First Derivative in *t*

Equations Involving the Second Derivative in *t*

Equations Involving Mixed Derivatives

**Equations of Higher Orders **

Equations Involving the First Derivative in *t *and Linear in the Highest Derivative

General Form Equations Involving the First Derivative in *t*

Equations Involving the Second Derivative in *t*

Other Equations

**Systems of Two First-Order Partial Differential Equations **

Systems of the Form

Other Systems of Two Equations

**Systems of Two Parabolic** **Equations **

Systems of the Form

Other Systems of Two Parabolic Equations

**Systems of Two Second-Order Klein–Gordon Type Hyperbolic Equations **

Systems of the Form

**Systems of Two Elliptic Equations **

Systems of the Form

Other Systems of Two Second-Order Elliptic Equations

Von Kármán Equations (Fourth-Order Elliptic Equations)

**First-Order Hydrodynamic and Other Systems Involving Three or More Equations **

Equations of Motion of Ideal Fluid (Euler Equations)

Adiabatic Gas Flow

Systems Describing Fluid Flows in the Atmosphere, Seas, and Oceans

Chromatography Equations

Other Hydrodynamic-Type Systems

Ideal Plasticity with the von Mises Yield Criterion

**Navier–Stokes and Related Equations **

Navier–Stokes Equations

Solutions with One Nonzero Component of the Fluid Velocity

Solutions with Two Nonzero Components of the Fluid Velocity

Solutions with Three Nonzero Fluid Velocity Components Dependent on Two Space Variables

Solutions with Three Nonzero Fluid Velocity Components Dependent on Three Space Variables

Convective Fluid Motions

Boundary Layer Equations (Prandtl Equations)

**Systems of General Form **

Nonlinear Systems of Two Equations Involving the First Derivatives with Respect to *t*

Nonlinear Systems of Two Equations Involving the Second Derivatives with Respect to *t *

Other Nonlinear Systems of Two Equations

Nonlinear Systems of Many Equations Involving the First Derivatives with Respect to *t *

**EXACT METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS **

Methods for Solving First-Order Quasilinear **Equations **

Characteristic System. General Solution

Cauchy Problem. Existence and Uniqueness Theorem

Qualitative Features and Discontinuous Solutions of Quasilinear Equations

Quasilinear Equations of General Form

**Methods for Solving First-Order Nonlinear Equations **

Solution Methods

Cauchy Problem. Existence and Uniqueness Theorem

Generalized Viscosity Solutions and Their Applications

**Classification of Second-Order Nonlinear Equations **

Semilinear Equations in Two Independent Variables

Nonlinear Equations in Two Independent Variables

**Transformations of Equations of Mathematical Physics **

Point Transformations: Overview and Examples

Hodograph Transformations (Special Point Transformations)

Contact Transformations. Legendre and Euler Transformations

Differential Substitutions. Von Mises Transformation

Bäcklund Transformations. RF Pairs

Some Other Transformations

**Traveling-Wave Solutions and Self-Similar Solutions **

Preliminary Remarks

Traveling-Wave Solutions. Invariance of Equations under Translations

Self-Similar Solutions. Invariance of Equations under Scaling Transformations

**Elementary Theory of Using Invariants for Solving Equations **

Introduction. Symmetries. General Scheme of Using Invariants for Solving Mathematical Equations

Algebraic Equations and Systems of Equations

Ordinary Differential Equations

Partial Differential Equations

General Conclusions and Remarks

**Method of Generalized Separation of Variables **

Exact Solutions with Simple Separation of Variables

Structure of Generalized Separable Solutions

Simplified Scheme for Constructing Generalized Separable Solutions

Solution of Functional Differential Equations by Differentiation

Solution of Functional Differential Equations by Splitting

Titov–Galaktionov Method

**Method of Functional Separation of Variables **

Structure of Functional Separable Solutions. Solution by Reduction to Equations with Quadratic Nonlinearities

Special Functional Separable Solutions. Generalized Traveling-Wave Solutions

Differentiation Method

Splitting Method. Solutions of Some Nonlinear Functional Equations and Their Applications

**Direct Method of Symmetry Reductions of Nonlinear Equations **

Clarkson–Kruskal Direct Method

Some Modifications and Generalizations

**Classical Method of Symmetry Reductions **

One-Parameter Transformations and Their Local Properties

Symmetries of Nonlinear Second-Order Equations. Invariance Condition

Using Symmetries of Equations for Finding Exact Solutions. Invariant Solutions

Some Generalizations. Higher-Order Equations

Symmetries of Systems of Equations of Mathematical Physics

**Nonclassical Method of Symmetry Reductions **

General Description of the Method

Examples of Constructing Exact Solutions

**Method of Differential Constraints **

Preliminary Remarks. Method of Differential Constraints for Ordinary Differential Equations

Description of the Method for Partial Differential Equations

First-Order Differential Constraints for PDEs

Second-Order Differential Constraints for PDEs. Some Generalized

Connection between the Method of Differential Constraints and Other Methods

**Painlevé Test for Nonlinear Equations of Mathematical Physics **

Movable Singularities of Solutions of Ordinary Differential Equations

Solutions of Partial Differential Equations with a Movable Pole. Method Description

Performing the Painlevé Test and Truncated Expansions for Studying Some Nonlinear Equations

**Methods of the Inverse Scattering Problem (Soliton Theory) **

Method Based on Using Lax Pairs

Method Based on a Compatibility Condition for Systems of Linear Equations

Method Based on Linear Integral Equations

Solution of the Cauchy Problem by the Inverse Scattering Problem Method

**Conservation Laws **

Basic Definitions and Examples

Equations Admitting Variational Form. Noetherian Symmetries

**Nonlinear Systems of Partial Differential Equations **

Overdetermined Systems of Two Equations

Pfaffian Equations and Their Solutions. Connection with Overdetermined Systems

Systems of First-Order Equations Describing Convective Mass Transfer with Volume Reaction

First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type

Systems of Second-Order Equations of Reaction-Diffusion Type

**SYMBOLIC AND NUMERICAL SOLUTIONS OF NONLINEAR PDES WITH MAPLE, ***MATHEMATICA*, AND MATLAB

**Nonlinear Partial Differential Equations with Maple **

Introduction

Brief Introduction to Maple

Analytical Solutions and Their Visualizations

Analytical Solutions of Nonlinear Systems

Constructing Exact Solutions Using Symbolic Computation. What Can Go Wrong

Some Errors That People Commonly Do When Constructing Exact Solutions with the Use of Symbolic Computations

Numerical Solutions and Their Visualizations

Analytical-Numerical Solutions

**Nonlinear Partial Differential Equations with ***Mathematica *

Introduction

Brief Introduction to *Mathematica *

Analytical Solutions and Their Visualizations

Analytical Solutions of Nonlinear Systems

Numerical Solutions and Their Visualizations

Analytical-Numerical Solutions

**Nonlinear Partial Differential Equations with MATLAB **

Introduction

Brief Introduction to MATLAB

Numerical Solutions via Predefined Functions

Solving Cauchy Problems. Method of Characteristics

Constructing Finite-Difference Approximations

**SUPPLEMENTS**

**Painlevé Transcendents **

Preliminary Remarks. Singular Points of Solutions

First Painlevé Transcendent

Second Painlevé Transcendent

Third Painlevé Transcendent

Fourth Painlevé Transcendent

Fifth Painlevé Transcendent

Sixth Painlevé Transcendent

Examples of Solutions to Nonlinear Equations in Terms of Painlevé Transcendents

**Functional Equations **

Method of Differentiation in a Parameter

Method of Differentiation in Independent Variables

Method of Argument Elimination by Test Functions

Nonlinear Functional Equations Reducible to Bilinear Equations

**Bibliography **

**Index**