Holdings: Handbook of first order partial differential equations

Handbook of first order partial differential equations:

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Bibliographic Details
Main Authors:	Poljanin, Andrej D. 1951- (Author), Zajcev, Valentin F. (Author), Moussiaux, Alain (Author)
Format:	Book
Language:	English
Published:	London [u.a.] Taylor & Francis 2002
Edition:	1. publ.
Series:	Differential and integral equations and their applications 1
Subjects:	Partielle Differentialgleichung
Online Access:	Inhaltsverzeichnis
Physical Description:	XIV, 500 S. graph. Darst.
ISBN:	041527267X

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_version_	1804129270780395521
adam_text	HANDBOOK OF FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS A.D. POLYANIN INSTITUTE FOR PROBLEMS IN MECHANICS RUSSIAN ACADEMY OF SCIENCES MOSCOW, RUSSIA V.F. ZAITSEV RUSSIAN STATE PEDAGOGICAL UNIVERSITY ST PETERSBURG, RUSSIA AND A. MOUSSIAUX UNIVERSITY DE NAMUR NAMUR BELGIUM LONDON AND NEW YORK CONTENTS PREFACE XV AUTHORS XVI ANNOTATION XV SOME NOTATION AND REMARKS XVIII PART I. LINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES 1 1. EQUATIONS CONTAINING ONE DERIVATIVE 3 2. LINEAR EQUATIONS OF THE FORM F(X,Y)^ + G(X,Y)^ =0 5 2.1. PRELIMINARY REMARKS 5 2.1.1. SOLUTION METHOD 5 2.1.2. CAUCHY PROBLEM (INITIAL VALUE PROBLEM) 6 2.1.3. EXAMPLES 6 2.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 7 2.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X AND Y 7 2.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X AND Y 9 2.2.3. COEFFICIENTS OF EQUATIONS CONTAIN INTEGER POWERS OF X AND Y 12 2.2.4. COEFFICIENTS OF EQUATIONS CONTAIN FRACTIONAL POWERS 13 2.2.5. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF X AND Y 14 2.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 24 2.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 24 2.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 25 2.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 30 2.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 30 2.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 30 2.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 31 2.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 32 2.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 33 2.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 34 2.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 34 2.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 34 2.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 37 2.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 37 2.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 39 2.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 40 2.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 42 2.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 43 2.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 45 2.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 45 2.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 46 2.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 48 2.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 49 CONTENTS 2.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF A; 51 2.8.1. EQUATIONS CONTAIN ARBITRARY AND POWER-LAW FUNCTIONS 51 2.8.2. EQUATIONS CONTAIN ARBITRARY AND EXPONENTIAL FUNCTIONS 53 2.8.3. EQUATIONS CONTAIN ARBITRARY AND HYPERBOLIC FUNCTIONS 54 2.8.4. EQUATIONS CONTAIN ARBITRARY AND LOGARITHMIC FUNCTIONS 55 2.8.5. EQUATIONS CONTAIN ARBITRARY AND TRIGONOMETRIC FUNCTIONS 55 2.8.6. EQUATIONS CONTAIN ARBITRARY FUNCTIONS AND THEIR DERIVATIVES 56 2.9. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DIFFERENT ARGUMENTS 58 2.9.1. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X AND ARBITRARY FUNCTIONS OF Y 58 2.9.2. EQUATIONS CONTAIN ONE ARBITRARY FUNCTION OF COMPLICATED ARGUMENT 59 2.9.3. EQUATIONS CONTAIN SEVERAL ARBITRARY FUNCTIONS 61 3. LINEAR EQUATIONS OF THE FORM F(X, Y)^ + G(.X, Y)^ = H(X, Y) 65 3.1. PRELIMINARY REMARKS 65 3.1.1. SOLUTION METHODS 65 3.1.2. CAUCHY PROBLEM 66 3.1.3. EXAMPLES 67 3.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 68 3.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X AND Y 68 3.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X AND Y 69 3.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWER-LAW FUNCTIONS 70 3.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF AND Y 70 3.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 72 3.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 72 3.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 73 3.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 74 3.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 74 3.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 75 3.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 76 3.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 76 3.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 77 3.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 77 3.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 77 3.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 78 3.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 79 3.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 79 3.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 79 3.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 80 3.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 80 3.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 81 3.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 82 3.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 82 3.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 83 3.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 83 3.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 84 3.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 85 3.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 85 3.8.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF A; AND ARBITRARY FUNCTIONS OF Y 87 3.8.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF COMPLICATED ARGUMENTS 88 3.8.4. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 89 CONTENTS II 4. LINEAR EQUATIONS OF THE FORM F(X, Y)-§F + G(X, Y)^ = H(X, Y)W 91 4.1. PRELIMINARY REMARKS 91 4.1.1. SOLUTION METHODS 91 4.1.2. EXAMPLES 92 4.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 93 4.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X AND Y 93 4.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X AND Y 94 4.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWER-LAW FUNCTIONS 95 4.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF A; AND Y 96 4.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 97 4.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 97 4.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 98 4.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 99 4.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 99 4.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 100 4.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 100 4.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 101 4.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 102 4.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 102 4.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 102 4.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 103 4.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 104 4.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 104 4.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 104 4.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 105 4.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 106 4.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 106 4.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 107 4.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 107 4.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 108 4.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 108 4.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 109 4.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 109 4.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 109 4.8.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF A; AND ARBITRARY FUNCTIONS OF Y 112 4.8.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF COMPLICATED ARGUMENTS 113 4.8.4. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 114 5. LINEAR EQUATIONS OF THE FORM F(X,Y)^ + G(X,Y)^ = HI(X,Y)W + HO(X,Y) 115 5.1. PRELIMINARY REMARKS 115 5.1.1. SOLUTION METHODS 115 5.1.2. EXAMPLES 116 5.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 117 5.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X AND Y 117 5.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X AND Y 118 5.2.3. COEFFICIENTS OF EQUATIONS CONTAIN SQUARE ROOTS 119 5.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF A; AND Y 120 VIII CONTENTS 5.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 122 5.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 122 5.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 123 5.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 125 5.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 125 5.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 125 5.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 126 5.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 126 5.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 127 5.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 127 5.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 127 5.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 128 5.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 129 5.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 129 5.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 130 5.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 130 5.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 131 5.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 132 5.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 133 5.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 133 5.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 133 5.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 134 5.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 134 5.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 135 5.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 135 5.8.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X AND ARBITRARY FUNCTIONS OF Y 137 5.8.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 138 PART II. LINEAR EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES 139 6. LINEAR EQUATIONS OF THEFORMF(X,Y,Z)^+G(X,Y,Z)^ + H(X,Y,Z)^ = 0 141 6.1. PRELIMINARY REMARKS 141 6.1.1. SOLUTION METHODS 141 6.1.2. CAUCHY PROBLEM (INITIAL VALUE PROBLEM) 142 6.1.3. EXAMPLES 142 6.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 143 6.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN A;, Y, AND Z 143 6.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X, Y, AND Z 147 6.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWERS OF X, Y, AND Z 150 6.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF X, Y, AND Z 151 6.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 154 6.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 154 6.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 155 6.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 157 6.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 157 6.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 158 6.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 159 6.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 159 6.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 160 CONTENTS 6.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 161 6.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 161 6.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 161 6.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 162 6.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 162 6.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 162 6.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 163 6.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 163 6.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 164 6.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 164 6.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 164 6.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 165 6.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 166 6.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 166 6.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 167 6.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 167 6.8.2. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF DIFFERENT VARIABLES . . 169 6.8.3. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 170 7. LINEAR EQUATIONS OF THE FORM /, F^ + F 2 G- + F 3 F^ = F 4 , F N = F N (X, Y, Z) 173 7.1. PRELIMINARY REMARKS 173 7.1.1. SOLUTION METHODS 173 7.1.2. EXAMPLES 174 7.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 175 7.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X, Y, AND Z 175 7.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X, Y, AND Z 176 7.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWERS IN X, Y, AND Z 177 7.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF X, Y, AND Z 178 7.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 180 7.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 180 7.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 181 7.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 182 * 7.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 182 7.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 183 7.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 184 7.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 185 7.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 186 7.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 186 7.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 186 7.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 187 7.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 187 7.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 187 7.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 188 7.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 189 7.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 190 7.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 190 7.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 191 7.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 191 7.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 192 CONTENTS 7.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 192 7.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 193 7.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 193 7.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 193 7.8.2. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF DIFFERENT VARIABLES .. 195 7.8.3. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 196 8. LINEAR EQUATIONS OF THE FORM FTFJ* + / 2 F^ + / 3 \|^ = FW, F N = F N (X,Y,Z) .... 199 8.1. PRELIMINARY REMARKS 199 8.1.1. SOLUTION METHODS 199 8.1.2. EXAMPLES 200 8.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 201 8.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X, Y, AND Z 201 8.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X, Y, AND Z 202 8.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWERS OF X, Y, AND Z 203 8.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF X, Y, AND Z 204 8.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 206 8.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 206 8.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 207 8.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 208 8.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 208 8.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 209 8.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 210 8.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 211 8.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 211 8.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 212 8.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 212 8.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 213 8.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 213 8.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 213 8.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 214 8.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 215 8.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 215 8.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 216 8.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 217 8.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 217 8.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 217 8.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 218 8.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 218 8.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 219 8.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 219 8.8.2. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF DIFFERENT VARIABLES .. 221 8.8.3. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 222 9. LINEAR EQUATIONS OF THE FORM/! F^ + /ZF^ + /SFR = F 4 W + F S , F N = F N (X,Y,Z) 225 9.1. PRELIMINARY REMARKS 225 9.1.1. SOLUTION METHODS 225 9.1.2. EXAMPLES 226 CONTENTS XI 9.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 226 9.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN X, Y, AND Z 226 9.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN X, Y, AND Z 228 9.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWERS OF X, Y, AND Z 228 9.2.4. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY POWERS OF X, Y, AND Z 229 9.3. EQUATIONS CONTAINING EXPONENTIAL FUNCTIONS 231 9.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 231 9.3.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL AND POWER-LAW FUNCTIONS 232 9.4. EQUATIONS CONTAINING HYPERBOLIC FUNCTIONS 233 9.4.1. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC SINE 233 9.4.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COSINE 234 9.4.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC TANGENT 235 9.4.4. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC COTANGENT 235 9.4.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT HYPERBOLIC FUNCTIONS 236 9.5. EQUATIONS CONTAINING LOGARITHMIC FUNCTIONS 237 9.5.1. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 237 9.5.2. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC AND POWER-LAW FUNCTIONS 237 9.6. EQUATIONS CONTAINING TRIGONOMETRIC FUNCTIONS 238 9.6.1. COEFFICIENTS OF EQUATIONS CONTAIN SINE 238 9.6.2. COEFFICIENTS OF EQUATIONS CONTAIN COSINE 239 9.6.3. COEFFICIENTS OF EQUATIONS CONTAIN TANGENT 240 9.6.4. COEFFICIENTS OF EQUATIONS CONTAIN COTANGENT 240 9.6.5. COEFFICIENTS OF EQUATIONS CONTAIN DIFFERENT TRIGONOMETRIC FUNCTIONS 241 9.7. EQUATIONS CONTAINING INVERSE TRIGONOMETRIC FUNCTIONS 242 9.7.1. COEFFICIENTS OF EQUATIONS CONTAIN ARCSINE 242 9.7.2. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOSINE 242 9.7.3. COEFFICIENTS OF EQUATIONS CONTAIN ARCTANGENT 243 9.7.4. COEFFICIENTS OF EQUATIONS CONTAIN ARCCOTANGENT 243 9.8. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 244 9.8.1. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF X 244 9.8.2. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF DIFFERENT VARIABLES .. 245 9.8.3. COEFFICIENTS OF EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 246 10. LINEAR EQUATIONS WITH FOUR OR MORE INDEPENDENT VARIABLES 249 10.1. PRELIMINARY REMARKS 249 10.1.1. LINEAR HOMOGENEOUS EQUATIONS 249 10.1.2. LINEAR NONHOMOGENEOUS EQUATIONS 250 10.2. SPECIFIC EQUATIONS 251 10.2.1. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 251 10.2.2. OTHER EQUATIONS CONTAINING ARBITRARY PARAMETERS 254 10.2.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 256 PART III. NONLINEAR EQUATIONS 259 11. QUASILINEAR EQUATIONS OF THE FORM F(X, Y)\|F- + G(X, Y)F^ = H(X, Y,W) 261 11.1. PRELIMINARY REMARKS 261 11.1.1. SOLUTION METHODS 261 11.1.2. EXAMPLES 262 XII CONTENTS 11.2. EQUATIONS CONTAINING ARBITRARY PARAMETERS 263 11.2.1. COEFFICIENTS OF EQUATIONS CONTAIN POWER-LAW FUNCTIONS 263 11.2.2. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 264 11.2.3. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC FUNCTIONS 266 11.2.4. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 267 11.2.5. COEFFICIENTS OF EQUATIONS CONTAIN TRIGONOMETRIC FUNCTIONS 268 11.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 268 11.3.1. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF ONE VARIABLE 268 11.3.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 271 12. QUASILINEAR EQUATIONS OF THE FORM F{X, Y,W)^ + G(X, Y,W)^- = H(X, Y,W) ... 273 12.1. PRELIMINARY REMARKS 273 12.1.1. SOLUTION METHODS 273 12.1.2. CAUCHY PROBLEM. EXISTENCE AND UNIQUENESS THEOREM 274 12.1.3. EQUATION ^ + F(W)^=0. QUALITATIVE FEATURES AND DISCONTINUOUS SOLUTIONS 276 12.1.4. GENERALIZED SOLUTIONS OF QUASILINEAR EQUATIONS 286 12.2. EQUATIONS CONTAINING POWER-LAW FUNCTIONS 290 12.2.1. COEFFICIENTS OF EQUATIONS ARE LINEAR IN W 290 12.2.2. COEFFICIENTS OF EQUATIONS ARE QUADRATIC IN W 293 12.2.3. COEFFICIENTS OF EQUATIONS CONTAIN OTHER POWERS OF W 295 12.3. OTHER EQUATIONS CONTAINING ARBITRARY PARAMETERS 297 12.3.1. COEFFICIENTS OF EQUATIONS CONTAIN EXPONENTIAL FUNCTIONS 297 12.3.2. COEFFICIENTS OF EQUATIONS CONTAIN HYPERBOLIC FUNCTIONS 299 12.3.3. COEFFICIENTS OF EQUATIONS CONTAIN LOGARITHMIC FUNCTIONS 302 12.3.4. COEFFICIENTS OF EQUATIONS CONTAIN TRIGONOMETRIC FUNCTIONS 304 12.4. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 306 12.4.1. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF INDEPENDENT VARIABLES 306 12A.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF THE UNKNOWN VARIABLE 310 12.4.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 314 13. EQUATIONS WITH TWO INDEPENDENT VARIABLES QUADRATIC IN DERIVATIVES 317 13.1. PRELIMINARY REMARKS 317 13.2. EQUATIONS CONTAINING ARBITRARY PARAMETERS 317 13.2.1. EQUATIONS OF THE FORM F^FJ- = F(X,Y,W) 317 13.2.2. EQUATIONS OF THE FORM /(X, Y, TU)\|F F^ + 0(Z, 2/, W)FF = H(X, Y, W) 319 13.2.3. EQUATIONS OF THE FORM F(X,Y,W)^^ + G(X,Y,W)^- + H{X,Y,W)^ = S(X, Y,W) 321 13.2.4. EQUATIONS OF THE FORM % + F(X,Y,W)(^) 2 = G(X,Y,W) 324 13.2.5. EQUATIONS OF THE FORM $ + F(X,Y,W) (FF) 2 + G(X,Y,W)%J-= H(X, Y, W) ... 331 13.2.6. EQUATIONS OF THEFORM F(X,Y,W)(^) 2 + G(X,Y,W)(^) 2 = H(X,Y,W) 335 13.2.7. EQUATIONSOFTHEFORM/(X,Y)(\|^) 2 + G(X,Y)\|F F^ = H(X,Y,W) 340 13.2.8. OTHER EQUATIONS 343 13.3. EQUATIONS CONTAINING ARBITRARY FUNCTIONS 347 13.3.1. EQUATIONS OF THE FORM FJF^R = /(Z,2/,W) 347 13.3.2. EQUATIONS OF THE FORM F( X ,Y)JGJ% + FF (X,Y)FF = H(X,Y,W) 349 13.3.3. EQUATIONS OF THE FORM F(X,Y)% + G(X,Y,W)(^) 2 = H(X,Y,W) 351 13.3.4. EQUATIONS OF THE FORM \|^+/(X, 2 /,U;)(\|F) 2 + FF (X,Y ) U;)\|^ = /I(X, 2/, W) ... 356 13.3.5. EQUATIONS OF THEFORM F(X,Y,W)(^-) 2 + G(X,Y,W)(^J-) 2 = H(X,Y,W) 359 CONTENTS XIII 13.3.6. EQUATIONS OF THE FORM (F^) 2 + F(X,Y,W)$- F^- = G(X,Y,W) 361 13.3.7. OTHER EQUATIONS 364 14. NONLINEAR EQUATIONS WITH TWO INDEPENDENT VARIABLES OF GENERAL FORM 367 14.1. PRELIMINARY REMARKS 367 14.1.1. SOLUTION METHODS 367 14.1.2. CAUCHY PROBLEM. EXISTENCE AND UNIQUENESS THEOREM 371 14.1.3. GENERALIZED VISCOSITY SOLUTIONS AND THEIR APPLICATIONS 374 14.2. EQUATIONS CONTAINING CUBIC NONLINEARITIES WITH RESPECT TO DERIVATIVES 378 14.2.1. EQUATIONS OF THE FORM §^-(F^) 2 = /(X,Y,IU) 378 14.2.2. EQUATIONS OF THE FORM F(X, Y, W)(^Y + G{X, Y, W)^ = H(X, Y, W) 379 14.2.3. EQUATIONS OF THE FORM FIX, Y, W){%%-) 3 + G(X, Y, W)(^) 2 = H{X,Y,W) 381 14.2.4. EQUATIONS OF THE FORM F(X,Y,W)(^Y + GIX,Y,W)%Z%F = H(X, Y, W) 382 14.2.5. OTHER EQUATIONS 383 14.3. NONLINEAR EQUATIONS CONTAINING ARBITRARY PARAMETERS 385 14.3.1. EQUATIONS CONTAIN THE FOURTH POWERS OF DERIVATIVES 385 14.3.2. EQUATIONS CONTAIN DERIVATIVES IN RADICANDS 387 14.3.3. EQUATIONS CONTAIN ARBITRARY POWERS OF DERIVATIVES 387 14.3.4. MORE COMPLICATED EQUATIONS 390 14.4. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF INDEPENDENT VARIABLES 392 14.4.1. EQUATIONS CONTAIN ONE ARBITRARY POWER OF DERIVATIVE 392 14.4.2. EQUATIONS CONTAIN TWO OR THREE ARBITRARY POWERS OF DERIVATIVES 395 14.5. EQUATIONS CONTAINING ARBITRARY FUNCTIONS OF DERIVATIVES 397 14.5.1. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF ONE VARIABLE 397 14.5.2. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 400 14.5.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF THREE VARIABLES 404 14.5.4. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF FOUR VARIABLES 406 15. NONLINEAR EQUATIONS WITH THREE OR MORE INDEPENDENT VARIABLES 407 15.1. PRELIMINARY REMARKS 407 15.1.1. QUASILINEAR EQUATIONS 407 15.1.2. NONLINEAR EQUATIONS 409 15.1.3. GENERALIZED VISCOSITY SOLUTIONS 416 15.2. QUASILINEAR EQUATIONS 418 15.2.1. EQUATIONS WITH THREE VARIABLES 418 15.2.2. EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES 422 15.3. NONLINEAR EQUATIONS WITH THREE VARIABLES QUADRATIC IN DERIVATIVES 424 15.3.1. EQUATIONS CONTAIN SQUARES OF ONE OR TWO DERIVATIVES 424 15.3.2. EQUATIONS CONTAIN SQUARES OF THREE DERIVATIVES 429 15.3.3. EQUATIONS CONTAIN PRODUCTS OF DERIVATIVES WITH RESPECT TO DIFFERENT VARIABLES 430 15.3.4. EQUATIONS CONTAIN SQUARES AND PRODUCTS OF DERIVATIVES 432 15.4. OTHER NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING PARAMETERS 432 15.4.1. EQUATIONS CUBIC IN DERIVATIVES 432 15.4.2. EQUATIONS CONTAIN ROOTS AND MODULI OF DERIVATIVES 433 15.4.3. EQUATIONS CONTAIN ARBITRARY POWERS OF DERIVATIVES 434 15.5. NONLINEAR EQUATIONS WITH THREE VARIABLES CONTAINING ARBITRARY FUNCTIONS 437 15.5.1. EQUATIONS QUADRATIC IN DERIVATIVES 437 15.5.2. EQUATIONS WITH POWER NONLINEARITY IN DERIVATIVES 443 15.5.3. EQUATIONS WITH ARBITRARY DEPENDENCE ON DERIVATIVES 445 15.5.4. NONLINEAR EQUATIONS OF GENERAL FORM 446 CONTENTS 15.6. NONLINEAR EQUATIONS WITH FOUR INDEPENDENT VARIABLES 450 15.6.1. EQUATIONS QUADRATIC IN DERIVATIVES 450 15.6.2. EQUATIONS CONTAIN POWER-LAW FUNCTIONS OF DERIVATIVES 452 15.7. NONLINEAR EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY PARAMETERS ? 454 15.7.1. EQUATIONS QUADRATIC IN DERIVATIVES 454 15.7.2. EQUATIONS WITH POWER-LAW NONLINEARITY IN DERIVATIVES 456 15.8. NONLINEAR EQUATIONS WITH ARBITRARY NUMBER OF VARIABLES CONTAINING ARBITRARY FUNCTIONS 457 15.8.1. EQUATIONS QUADRATIC IN DERIVATIVES 457 15.8.2. EQUATIONS WITH POWER-LAW NONLINEARITY IN DERIVATIVES 462 15.8.3. EQUATIONS CONTAIN ARBITRARY FUNCTIONS OF TWO VARIABLES 463 15.8.4. NONLINEAR EQUATIONS OF GENERAL FORM 464 SUPPLEMENT. SOLUTION OF DIFFERENTIAL EQUATIONS THROUGH THE CONVODE SOFTWARE 469 5.1. INTRODUCTION 469 5.1.1. PRELIMINARY REMARKS 469 5.1.2. REDUCE NOTATION USED IN CONVODE 469 5.1.3. HOW CONVODE SOLVES EQUATIONS 470 5.2. EXAMPLES OF SOLVING ORDINARY DIFFERENTIAL EQUATIONS 471 5.2.1. RICCATI EQUATION (EXAMPLE 1) 471 5.2.2. RICCATI EQUATION (EXAMPLE 2) 475 5.2.3. A NONLINEAR EQUATION QUADRATIC IN THE DERIVATIVE 479 5.3. EXAMPLES OF SOLVING PARTIAL DIFFERENTIAL EQUATIONS 481 5.3.1. A FIRST ORDER LINEAR EQUATION (EXAMPLE 1) 481 5.3.2. A FIRST ORDER LINEAR EQUATION (EXAMPLE 2) 483 5.3.3. A SECOND ORDER NONLINEAR EQUATION 486 5.4. HOW TO USE CONVODE 490 5.4.1. ARGUMENTS OF THE CONVODE PROCEDURE 490 5.4.2. GLOBAL VARIABLES 491 5.4.3. CONVODE VIA E-MAIL 492 REFERENCES 493 INDEX 497
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author	Poljanin, Andrej D. 1951- Zajcev, Valentin F. Moussiaux, Alain
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spelling	Poljanin, Andrej D. 1951- Verfasser (DE-588)128391251 aut Handbook of first order partial differential equations A. D. Polyanin, V. F. Zaitsev and A. Moussiaux 1. publ. London [u.a.] Taylor & Francis 2002 XIV, 500 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential and integral equations and their applications 1 Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Zajcev, Valentin F. Verfasser (DE-588)12839126X aut Moussiaux, Alain Verfasser aut Differential and integral equations and their applications 1 (DE-604)BV014412675 1 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009857283&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Poljanin, Andrej D. 1951- Zajcev, Valentin F. Moussiaux, Alain Handbook of first order partial differential equations Differential and integral equations and their applications Partielle Differentialgleichung (DE-588)4044779-0 gnd
subject_GND	(DE-588)4044779-0
title	Handbook of first order partial differential equations
title_auth	Handbook of first order partial differential equations
title_exact_search	Handbook of first order partial differential equations
title_full	Handbook of first order partial differential equations A. D. Polyanin, V. F. Zaitsev and A. Moussiaux
title_fullStr	Handbook of first order partial differential equations A. D. Polyanin, V. F. Zaitsev and A. Moussiaux
title_full_unstemmed	Handbook of first order partial differential equations A. D. Polyanin, V. F. Zaitsev and A. Moussiaux
title_short	Handbook of first order partial differential equations
title_sort	handbook of first order partial differential equations
topic	Partielle Differentialgleichung (DE-588)4044779-0 gnd
topic_facet	Partielle Differentialgleichung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009857283&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV014412675
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