Székelyhidi L. Ordinary and partial differential equations for the beginner. - Singapore: World Scientific, 2016. - xiv, 239 p. - Bibliogr.: p.231-232. - Ind.: p.233-239. - ISBN 978-981-4723-98-5 Шифр: (И/В16-S99) 02  Место хранения:   02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents

1 ORDINARY DIFFERENTIAL EQUATIONS .............................. 1 1.1 Basic concepts and terminology .......................... 1 1.2 Problems ................................................ 4 1.3 Auxiliary results from functional analysis .............. 5 1.4 Problems ................................................ 7 1.5 Approximate solutions. Peano's theorem .................. 8 1.6 Problems ............................................... 13 1.7 Existence and uniqueness ............................... 13 1.8 Problems ............................................... 18 1.9 Parametric differential equations ...................... 19 1.10 Problems ............................................... 26 1.11 Characteristic function ................................ 26 1.12 Problems ............................................... 28 2 ELEMENTARY SOLUTION METHODS ................................. 29 2.1 Separable differential equations ....................... 29 2.2 Problems ............................................... 31 2.3 Differential equations of homogeneous degree ........... 32 2.4 Problems . ........................................... 35 2.5 First order linear differential equations .............. 36 2.6 Problems ............................................... 38 2.7 Bernoulli equations .................................... 39 2.8 Problems ............................................... 39 2.9 Riccati equations ...................................... 40 2.1 Problems ............................................... 40 2.11 Exact differential equations ........................... 41 2.12 Problems ............................................... 45 2.13 Incomplete differential equations ...................... 46 2.14 Problems ............................................... 50 2.15 Implicit differential equations ........................ 50 2.16 Problems ............................................... 52 2.17 Lagrange and Clairut equations ......................... 52 2.18 Problems ............................................... 54 3 LINEAR DIFFERENTIAL EQUATIONS ............................... 55 3.1 Integrals of linear differential equations ............. 55 3.2 Problems ............................................... 59 3.3 Linear differential equations with constant coefficients ........................................... 59 3.4 Problems ............................................... 61 3.5 Computation of the exponential matrix .................. 63 3.6 Problems ............................................... 66 4 FUNCTIONAL DEPENDENCE, INDEPENDENCE ......................... 69 4.1 Functional independence ................................ 69 4.2 Functional expressibility .............................. 72 4.3 First integrals ........................................ 74 4.4 Problems ............................................... 76 5 HIGHER ORDER DIFFERENTIAL EQUATIONS ......................... 77 5.1 A reduction principle .................................. 77 5.2 Problems ............................................... 79 5.3 Intermediate integrals ................................. 79 5.4 Problems ............................................... 83 5.5 Higher order linear differential equations ............. 83 5.6 Problems ............................................... 85 5.7 Linear differential equations with constant coefficients ........................................... 86 5.8 Problems ............................................... 87 5.9 Decreasing the order of linear homogeneous equations ... 88 5.10 Problems ............................................... 91 5.11 Euler differential equations ........................... 91 5.12 Problems ............................................... 92 5.13 Exponential polynomials ................................ 92 5.14 Problems ............................................... 95 5.15 Boundary value problems ................................ 95 5.16 Problems .............................................. 101 5.17 Power series solutions ................................ 101 5.18 Problems .............................................. 106 5.19 The Laplace transform ................................. 106 5.20 Problems .............................................. 109 5.21 The Fourier transform of exponential polynomials ...... 110 5.22 Problems .............................................. 118 6 FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS ................. 119 6.1 Homogeneous linear partial differential equations ..... 119 6.2 Problems .............................................. 123 6.3 Quasilinear partial differential equations ............ 124 6.4 Problems .............................................. 127 7 THEORY OF CHARACTERISTICS .................................. 129 7.1 First order partial differential equations ............ 129 7.2 Problems .............................................. 131 7.3 Cauchy problem for first order equations .............. 132 7.4 Problems .............................................. 139 7.5 Special Cauchy problem for first order partial differential equation ................................. 139 7.6 Problems .............................................. 144 7.7 Complete integral ..................................... 144 7.8 Problems .............................................. 150 8 HIGHER ORDER PARTIAL DIFFERENTIAL EQUATIONS ................ 151 8.1 Special Cauchy problems for higher order partial differential equations ................................ 151 8.2 Theorems of Kovalevskaya and Holmgren ................. 153 8.3 Linear partial differential operators ................. 154 8.4 Problems .............................................. 158 8.5 Exponential polynomial solutions of partial differential equations ................................ 158 8.6 Problems .............................................. 163 9 SECOND ORDER QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS .... 165 9.1 Second order partial differential equations with linear principal part ................................. 165 9.2 Problems .............................................. 166 9.3 Linear transformation, normal form .................... 167 9.4 Problems .............................................. 168 9.5 Reduced normal form ................................... 169 9.6 Problems .............................................. 172 9.7 Normal form in two variables .......................... 173 9.8 Hyperbolic equations .................................. 174 9.9 Problems .............................................. 176 9.10 Parabolic equations ................................... 177 9.11 Problems .............................................. 178 9.12 Elliptic equations .................................... 179 9.13 Problems .............................................. 180 10 SPECIAL PROBLEMS IN TWO VARIABLES .......................... 185 10.1 Goursat problem for hyperbolic equations .............. 185 10.2 Problems .............................................. 188 10.3 Cauchy problem for hyperbolic equations ............... 189 10.4 Problems .............................................. 192 10.5 Mixed problem for the wave equation ................... 192 10.6 Problems .............................................. 195 10.7 Fourier method ........................................ 195 10.8 Problems .............................................. 197 11 TABLE OF LAPLACE TRANSFORMS ................................ 199 12 ANSWERS TO SELECTED PROBLEMS ............................... 203 1.2 Basic concepts and terminology ........................ 203 1.10 Parametric differential equations ..................... 203 1.12 Characteristic function ............................... 203 2.2 Separable differential equations ...................... 204 2.4 Differential equations of homogeneous degree .......... 204 2.6 First order linear differential equations ............. 205 2.8 Bernoulli equations ................................... 205 2.10 Riccati equations ..................................... 206 2.12 Exact differential equations .......................... 206 2.14 Incomplete differential equations ..................... 207 2.16 Implicit differential equations ....................... 208 2.18 Lagrange and Clairut equations ........................ 210 3.2 Integrals of linear differential equations ............ 211 3.4 Linear differential equations with constant coefficients .......................................... 211 3.6 Computation of the exponential matrix ................. 213 4.4 First integrals ....................................... 214 5.4 Intermediate integrals ................................ 214 5.6 Higher order linear differential equations ............ 215 5.8 Linear differential equations with constant coefficients .......................................... 215 5.10 Decreasing the order of linear homogeneous equations .. 216 5.12 Euler differential equations .......................... 216 5.14 Exponential polynomials ............................... 216 5.16 Boundary value problems ............................... 216 5.20 Power series solutions ................................ 217 5.22 The Laplace transform ................................. 218 5.24 The Fourier transform of exponential polynomials ...... 218 6.2 Homogeneous linear partial differential equations ..... 219 6.4 Quasilinear partial differential equations ............ 219 7.2 First order partial differential equations ............ 220 7.4 Cauchy problem for first order equations .............. 221 7.6 Special Cauchy problem for first order partial differential equation ................................. 221 7.8 Complete integral ..................................... 221 8.6 Exponential polynomial solutions of partial differential equations ................................ 222 9.2 Second order partial differential equations with linear principal part ................................. 222 9.4 Linear transformation, normal form .................... 223 9.6 Reduced normal form ................................... 224 9.9 Hyperbolic equations .................................. 225 9.11 Parabolic equations ................................... 225 9.13 Elliptic equations .................................... 225 10.2 Goursat problem for hyperbolic equations .............. 229 10.4 Cauchy problem for hyperbolic equations ............... 229 10.6 Mixed problem for the wave equation ................... 229 10.8 Fourier method ........................................ 229 Bibliography .................................................. 231 Index ......................................................... 233