HANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS

Transcription

1 HANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS Andrei D. Polyanin Chapman & Hall/CRC Taylor & Francis Group Boca Raton London New York Singapore

2 Foreword Basic Notation and Remarks Author iii v vii Introduction. Some Definitions, Formulas, Methods, and Solutions Classification of Second-Order Partial Differential Equations Equations with Two Independent Variables Equations with Many Independent Variables Basic Problems of Mathematical Physics Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems First, Second, Third, and Mixed Boundary Value Problems Properties and Particular Solutions of Linear Equations Homogeneous Linear Equations Nonhomogeneous Linear Equations Separation of Variables Method General Description of the Separation of Variables Method Solution of Boundary Value Problems for Parabolic and Hyperbolic Equations Integral Transforms Method Main Integral Transforms Laplace Transform and Its Application in Mathematical Physics Fourier Transform and Its Application in Mathematical Physics Representation of the Solution of the Cauchy Problem via the Fundamental Solution Cauchy Problem for Parabolic Equations Cauchy Problem for Hyperbolic Equations Nonhomogeneous Boundary Value Problems with One Space Variable. Representation of Solutions via the Green's Function Problems for Parabolic Equations Problems for Hyperbolic Equations Nonhomogeneous Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green's Function Problems for Parabolic Equations Problems for Hyperbolic Equations Problems for Elliptic Equations Comparison of the Solution Structures for Boundary Value Problems for * Equations of Various Types Construction of the Green's Functions. General Formulas and Relations Green's Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains Green's Functions Admitting Incomplete Separation of Variables Construction of Green's Functions via Fundamental Solutions 35

3 x Duhamel's Principles in Nonstationary Problems Problems for Homogeneous Linear Equations Problems for Nonhomogeneous Linear Equations Transformations Simplifying Initial and Boundary Conditions Transformations That Lead to Homogeneous Boundary Conditions Transformations That Lead to Homogeneous Initial and Boundary Conditions Parabolic Equations with One Space Variable Constant Coefficient Equations HeatEquation ^ = a EquationoftheForm^f =af^ + $(z,t) Equation of the Form ^ =a^r + bw + ^(x,t) Equation of the Form ^ = a ^ + &f^ + $(x,t) Equation of the Form f = a%$:+ b% +av + $(x,t) Heat Equation with Axial or Central Symmetry and Related Equations Equation of the Form &$ =a( p L + ^) EquationoftheForm^f=a( ^ + ^)+$(M) EquationoftheForm^=o(^ + J^) EquationoftheForm^f=a(^ + ff)+$(m) Equation of the Form ^ = ^ + ^ ^ Equation of the Form ^ = ^ +^2 f + $(;M) Equations Containing Power Functions and Arbitrary Parameters Equations of the Form ^ = af^- + f(x, t)w Equations of the Form ^ = a ^ + /(x,0 j Equations of the Form ^ =a%$- +f(x,t)& +g(x,t)w + h(x,t) Equations of the Form ^ = (ax + 6)0 + f(x, t)&± + g(x, t)w Equations of the Form ^ = (ax 2 + bx + c)&$- + f(x, t)^+g(x,t)w Equations of the Form ^ = f(x)%$-+ g(x, t)ff + h(x, t)w Equations of the Form f = f(x,t)^t + g(x,t)%±+ h(x,t)w Liquid-Film Mass Transfer Equation (1 - y 1 )^ = a ^ Equations of the Form f(x,y)% +g(x,y)&z = %$ + h(x,y).* Equations Containing Exponential Functions and Arbitrary Parameters Equations of the Form ^f = a ^ + f(x, t)w EquationsoftheForm^ = a^ + /(x,t) j Equations of the Form ff = a 7&r + /<* Olj+ 30M)w Equations of theiform f = ax n % + f(x,t)% + g(xj)w Equations of the Form ff = ae' 3x^- + f(x f t)^+g(x,t)w Other Equations Equations Containing Hyperbolic Functions and Arbitrary Parameters Equations Containing a Hyperbolic Cosine Equations Containing a Hyperbolic Sine Equations Containing a Hyperbolic Tangent Equations Containing a Hyperbolic Cotangent 120

4 xi 1.6. Equations Containing Logarithmic Functions and Arbitrary Parameters Equations of the Form f = af^f + f(x, t)^+g(x,t)w Equations of the Form ^ = ax k %$ + f(x,t)% + g(x,t)w Equations Containing Trigonometric Functions and Arbitrary Parameters Equations Containing a Cosine Equations Containing a Sine Equations Containing a Tangent Equations Containing a Cotangent Equations Containing Arbitrary Functions Equations of the Form ^ = a^- + f(x,t)w Equations of the Form f = a^r + f(x,t)^ Equations of the Form ^ = a ^ + f(x, t)& + g(x, t)w Equations of the Form f- = ax n & + f(x,t)%± + g(x,t)w Equations of the Form f = ae 0x^ + f(x,t)% + g(x,t)w Equations of the Form ^ = f(x)%$:+g(x,t)% + h(x,t)w Equations of the Form f = f(t)^$-+g(x, *)f^ + h(x, t)w Equations of the Form ^ = f(x,t)^pr + g(x,t)^ + h(x,t)w Equations of the Form s(x) f = -^ [p( )fj] -q(x)w + $(x, t) Equations of Special Form Equations of the Diffusion (Thermal) Boundary Layer One-Dimensional Schrodinger Equation i%^ = ^ ^ + U(x)w Parabolic Equations with Two Space Variables Heat Equation ^ = aa 2 w Boundary Value Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems Heat Equation with a Source ^f = aa 2 w + $(z, y, t) Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems Other Equations Equations Containing Arbitrary Parameters Equations Containing Arbitrary Functions Parabolic Equations with Three or More Space Variables Heat Equation f = aa 3 w Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Heat Equation with Source ^ = aa 3 w + $(x;y, z,t) Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Other Equations with Three Space Variables Equations Containing Arbitrary Parameters Equations Containing Arbitrary Functions Equations of the Form p(x, y, z)^- =div[a(x, y, z)vw]-q(x, y, z)w+$(x, y, z, t) 266

5 xlt Equations with n Space Variables 268» Equations of the Form f = aa n w + $(x\ x n,t) Other Equations Containing Arbitrary Parameters Equations Containing Arbitrary Functions Hyperbolic Equations with One Space Variable Constant Coefficient Equations WaveEquation ^ =a Equations of the Form ^ = a 2 f^f + $(x, t) Equation of the Form ^ =a 2 f^ -bw + $(x,t) Equation of the Form ^=o 2 0-6^+ $(x,t) Equation of the Form ^ = a 1 &$ + b% +cw + $(x,t) Wave Equation with Axial or Central Symmetry Equations of the Form ^ = a 2 ( ^ + f) EquationoftheForm^=a 2 (^ + ^)+#(r,t) EquationoftheForm^ = a 2 (^ +!f) EquationoftheForm^=a 2 (^f+ f^)+$(r,t) EquationoftneForm ^=a 2 (^f+ ±^)-&w + $(r,t) EquationoftheForm^ = a 2 (^ + ^)-6«; + $(r, Equations Containing Power Functions and Arbitrary Parameters Equations of the Form ^- = (ax + b) ^f + c^ + kw + $(x,t) Equations of the Form &$- = (ax 2 + 6)0 + cx^ + kw + $(z, t) Other Equations Equations Containing the First Time Derivative Equations of the Form ^ + fc f =a 2 $-+ b%%+cw + $(x,t) Equations of the Form ^-+ k^-= f(x)^r + g(x)^+ h(x)w + ^>(x,t) Other Equations Equations Containing Arbitrary Functions Equations of the Form s(x) ^r =-^ [pte)f^] - q(x)w + $(z, t) Equations of the Form ^ + a(t)^ = b(t){- ^ [l*x)^] - q(x)w} + $(x, t) Other Equations Hyperbolic Equations with Two Space Variables Wave Equation ^ = a 2 A 2 it> Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems Nonhomogeneous Wa've Equation ^- = a 2 Aiw + $(x, y,t) Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems Equations of the Form ^- = a 2 Aiw -bw + $(#, y,i) Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems r 371

6 xiii 5.4. Telegraph Equation ^ + Jfe^jf- = a 2 A 2 w -bw + $(x, y,t) Problems in Cartesian Coordinates Problems in Polar Coordinates Axisymmetric Problems Other Equations with Two Space Variables Hyperbolic Equations with Three or More Space Variables Wave Equation ^ = a 2 A 3 w Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Nonhomogeneous Wave Equation ^- = a 1 A 3 w + $(x, y, z, t) Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Equations of the Form ^$- = a 2 A 3 iu - bw + $(x, y, z, t) Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Telegraph Equation ^ + *^f = a 2 A 3 w-bw + $(x,y,z,t) Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Other Equations with Three Space Variables Equations Containing Arbitrary Parameters Equation of the Form p(x, y, z)-^- = div [a(x, y, z) Vw] -q(x, y, z)w+$(x, y, z, t) Equations with n Space Variables Wave Equation j = a 2 A n w Nonhomogeneous Wave Equation ^ = a 2 A n w + $(xi,..., x n, t) Equations of the Form ^ = a 2 A n w -bw + $(xi,...,x n,t) Equations Containing the First Time Derivative Elliptic Equations with Two Space Variables Laplace Equation A2W = Problems in Cartesian Coordinate System Problems in Polar Coordinate System Other Coordinate Systems. Conformal Mappings Method Poisson Equation A 2 w = -$(x) Preliminary Remarks. Solution Structure 478 ' Problems in Cartesian Coordinate System Problems in Polar Coordinate System Arbitrary Shape Domain. Conformal Mappings Method Helmholtz Equation A 2 w + Xw = -$(x) General Remarks, Results, and Formulas Problems in Cartesian Coordinate System Problems in Polar Coordinate System Other.Orthogonal Coordinate Systems. Elliptic Domain 508

7 xiv ' 7.4. Other Equations Stationary Schrodinger Equation A 2 w = f(x, y)w Convective Heat and Mass Transfer Equations Equations of Heat and Mass Transfer in Anisotropic Media Other Equations Arising in Applications Equations of the Form a(x)%$-+ fi + b(x)^+c(x)w = -$(x,y) Elliptic Equations with Three or More Space Variables Laplace Equation A 3 w = Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Other Orthogonal Curvilinear Systems of Coordinates Poisson Equation A 3 w + $(x) = Preliminary Remarks. Solution Structure Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Helmholtz Equation A 3 w + Xw = -$(x) General Remarks, Results, and Formulas Problems in Cartesian Coordinates Problems in Cylindrical Coordinates Problems in Spherical Coordinates Other Orthogonal Curvilinear Coordinates Other Equations with Three Space Variables Equations Containing Arbitrary Functions Equations of the Form div [a(x, y, z)vw] - q(x, y, z)w = -$(x, y, z) Equations with n Space Variables Laplace Equation A n w = Other Equations Higher-Order Partial Differential Equations Third-Order Partial Differential Equations Fourth-Order One-Dimensional Nonstationary Equations Equations of the Form f- + a 2 f^ = $(z,t) Equations of the Form ^ +a 2 fjr = Equations of the Form ^r+ a 2 0 = $(x,*) Equations of the Form ^ + a 2 ^- + kw = $(x,t) Other Equations Two-Dimensional Nonstationary Fourth-Order Equations EquationsofmeForm f+ a 2 (0 + f^)=$(x,y,*) Two-Dimensional Equations of the Form ^ + a 2 AAw = Three- and n-dimensional Equations of the Form ^ + a 2 AAu> = Equations of the Form ^ + a 2 AAw + kw = $(x, y,t) Equations of the Form ^ + o 2 (&$-+ ft)+kw = $(x,y,t) Fourth-Order Stationary Equations Biharmonic Equation AAiu = Equationsof the Form AAiu = $(x,y) 625

8 XV Equations of the Form A An; - Xw = $(x, y) Equations of the Form f^- + f^ = $(z,y) Equations of the Form f^r + fpr +fcw = $(x,y) Stokes Equation (Axisymmetric Flows of Viscous Fluids) Higher-Order Linear Equations with Constant Coefficients Fundamental Solutions. Cauchy Problem Elliptic Equations Hyperbolic Equations Regular Equations. Number of Initial Conditions in the Cauchy Problem Some Special-Type Equations Higher-Order Linear Equations with Variable Coefficients Equations Containing the First Time Derivative Equations Containing the Second Time Derivative Nonstationary Problems with Many Space Variables Some Special-Type Equations 651 Supplement A. Special Functions and Their Properties 655 A.I. Some Symbols and Coefficients 655 A.1.1. Factorials 655 A.I.2. Binomial Coefficients 655 A.1.3. Pochhammer Symbol 656 A.1.4. Bernoulli Numbers 656 A.2. Error Functions and Exponential Integral 656 A.2.1. Error Function and Complementary Error Function 656 A.2.2. Exponential Integral 656 A.2.3. Logarithmic Integral 657 A.3. Sine Integral and Cosine Integral. Fresnel Integrals 657 A.3.1. Sine Integral 657 A.3.2. Cosine Integral 658 A.3.3. Fresnel Integrals 658 A.4. Gamma and Beta Functions 659 A.4.1. Gamma Function 659 A.4.2. Beta Function 660 A.5. Incomplete Gamma and Beta Functions 660 A.5.1. Incomplete Gamma Function 660 A.5.2. Incomplete Beta Function 661 A.6. Bessel Functions 661 A.6.1. Definitions and Basic Formulas 661 A.6.2. Integral Representations and Asymptotic Expansions 663 y A.6.3. Zeros and Orthogonality Properties of Bessel Functions 664 A.6.4. Hankel Functions (Bessel Functions of the Third Kind) 665 A.7. Modified Bessel Functions 666 A.7.1. Definitions. Basic Formulas 666 A.7.2. Integral Representations and Asymptotic Expansions 667 A.8. Airy Functions 668 A.8.1. Definition and Basic Formulas 668 A.8.2. PowetSeries and Asymptotic Expansions 668

9 xvi A.9. Degenerate Hypergeometric Functions 669 A.9.1. Definitions and Basic Formulas 669 A.9.2. Integral Representations and Asymptotic Expansions 671 A.10. Hypergeometric Functions 672 A Definition and Some Formulas 672 A Basic Properties and Integral Representations 672 A.11. Whittaker Functions 672 A.12. Legendre Polynomials and Legendre Functions 674 A Definitions. Basic Formulas 674 A Zeros of Legendre Polynomials and the Generating Function 674 A Associated Legendre Functions 674 A.13. Parabolic Cylinder Functions 675 A Definitions. Basic Formulas 675 A Integral Representations and Asymptotic Expansions 675 A.14. Mathieu Functions 675 A Definitions and Basic Formulas 675 A.15. Modified Mathieu Functions 677 A.16. Orthogonal Polynomials 677 A Laguerre Polynomials and Generalized Laguerre Polynomials 678 A Chebyshev Polynomials and Functions 679 A Hermite Polynomial 679 A Jacobi Polynomials 680 Supplement B. Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics 681 B.I. Introduction 681 B.I. 1. Preliminary Remarks 681 B.I.2. Simple Cases of Variable Separation in Nonlinear Equations 682 B.1.3. Examples of Nontrivial Variable Separation in Nonlinear Equations 683 B.2. Methods of Generalized Separation of Variables 685 B.2.1. Structure of Generalized Separable Solutions 685 B.2.2. Solution of Functional Differential Equations by Differentiation 686 B.2.3. Solution of Functional Differential Equations by Splitting 688 B.2.4. Simplified Scheme for Constructing Exact Solutions of Equations with Quadratic Nonlinearities 691 B.3. Methods of Functional Separation of Variables * B.3.1. Structure of Functional Separable Solutions 693 B.3.2. Special Functional Separable Solutions 693 B.3.3. Differentiation Method 696 B.3.4. Splitting Method. Reduction to a Functional Equation with Two Variables B.3.5. Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations 701 B.4. First-Order Nonlinear Equations 706 B.4.1. Preliminary Remarks 706 B.4.2. Individual Equations 706 B.5. Second-Order Nonlinear Equations 709 B.5.1. Parabolic Equations 709 B.5.2. Hyperbolic Equations 721 B.5.3. Elliptic Equations 727 B.5.4. Equations Containing Mixed Derivatives 733

10 xvn B.5.5. General Form Equations 736 B.6. Third-Order Nonlinear Equations 739 B.6.1. Stationary Hydrodynamic Boundary Layer Equations 739 B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations 741 B.7. Fourth-Order Nonlinear Equations 749 B.7.1. Stationary Hydrodynamic Equations (Navier-Stokes Equations) 749 B.7.2. Nonstationary Hydrodynamic Equations 752 B.8. Higher-Order Nonlinear Equations 757 B.8.1. Equations of the Form f = F(x,t,w, f^ -gf) 757 B.8.2. EquationsoftheForm^ = F(z,t,w,f^,..., ) 761 B.8.3. Other Equations 765 References 769 Index 777