Holdings: Handbook of nonlinear partial differential equations

Handbook of nonlinear partial differential equations:

Saved in:

Bibliographic Details
Main Author:	Poljanin, Andrej D. 1951- (Author)
Format:	Book
Language:	English
Published:	Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2004
Subjects:	Mécanique non linéaire - Mathématiques Équations différentielles non linéaires - Solutions numériques Mathematik Differential equations, Nonlinear > Numerical solutions Nonlinear mechanics > Mathematics Numerische Mathematik Nichtlineare partielle Differentialgleichung Nichtlineare Mechanik
Online Access:	Book review (E-STREAMS) Inhaltsverzeichnis
Physical Description:	XX, 814 S. Ill., graph. Darst.
ISBN:	1584883553

Staff View

MARC


LEADER	00000nam a2200000zc 4500
001	BV021518442
003	DE-604
005	20061121
007	t
008	060320s2004 xxuad\|\| \|\|\|\| 00\|\|\| eng d
010			\|a 2003058473
020			\|a 1584883553 \|c acidfree paper \|9 1-58488-355-3
035			\|a (OCoLC)52766184
035			\|a (DE-599)BVBBV021518442
040			\|a DE-604 \|b ger \|e aacr
041	0		\|a eng
044			\|a xxu \|c US
049			\|a DE-29T \|a DE-20 \|a DE-824 \|a DE-634 \|a DE-83 \|a DE-11
050		0	\|a QA372
082	0		\|a 515/.355 \|2 22
084			\|a QH 150 \|0 (DE-625)141534: \|2 rvk
084			\|a SK 540 \|0 (DE-625)143245: \|2 rvk
084			\|a 35-00 \|2 msc
084			\|a 35F20 \|2 msc
084			\|a 35G20 \|2 msc
100	1		\|a Poljanin, Andrej D. \|d 1951- \|e Verfasser \|0 (DE-588)128391251 \|4 aut
245	1	0	\|a Handbook of nonlinear partial differential equations \|c Andrei D. Polyanin ; Valentin F. Zaitsev
246	1	3	\|a Nonlinear partial differential equations
264		1	\|a Boca Raton, Fla. [u.a.] \|b Chapman & Hall/CRC \|c 2004
300			\|a XX, 814 S. \|b Ill., graph. Darst.
336			\|b txt \|2 rdacontent
337			\|b n \|2 rdamedia
338			\|b nc \|2 rdacarrier
650		4	\|a Mécanique non linéaire - Mathématiques
650		4	\|a Équations différentielles non linéaires - Solutions numériques
650		4	\|a Mathematik
650		4	\|a Differential equations, Nonlinear \|x Numerical solutions
650		4	\|a Nonlinear mechanics \|x Mathematics
650	0	7	\|a Numerische Mathematik \|0 (DE-588)4042805-9 \|2 gnd \|9 rswk-swf
650	0	7	\|a Nichtlineare partielle Differentialgleichung \|0 (DE-588)4128900-6 \|2 gnd \|9 rswk-swf
650	0	7	\|a Nichtlineare Mechanik \|0 (DE-588)4042095-4 \|2 gnd \|9 rswk-swf
689	0	0	\|a Nichtlineare partielle Differentialgleichung \|0 (DE-588)4128900-6 \|D s
689	0		\|5 DE-604
689	1	0	\|a Nichtlineare Mechanik \|0 (DE-588)4042095-4 \|D s
689	1	1	\|a Numerische Mathematik \|0 (DE-588)4042805-9 \|D s
689	1		\|5 DE-604
700	1		\|a Zajcev, Valentin F. \|e Sonstige \|0 (DE-588)12839126X \|4 oth
856	4		\|u http://www.e-streams.com/es0710/es0710*9.html \|3 Book review (E-STREAMS)
856	4	2	\|m HEBIS Datenaustausch Darmstadt \|q application/pdf \|u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014734968&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA \|3 Inhaltsverzeichnis
999			\|a oai:aleph.bib-bvb.de:BVB01-014734968

Record in the Search Index

_version_	1804135258425131008
adam_text	HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ANDREI D. POLYANIN VALENTIN F. ZAITSEV CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON NEW YORK WASHINGTON, D.C. CONTENTS AUTHORS XV FOREWORD XVII SOME NOTATIONS AND REMARKS XIX 1. PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 1 1.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 1 1.1.1. EQUATIONS OF THE FORM §F = A^JG- +BW + CW 2 1 1.1.2. EQUATIONS OF THE FORM ^F =A-^R + B 0 + BIW + B 2 W 2 + B3W 3 2 1.1.3. EQUATIONS OF THE FORM §F = A^F + F(W) 6 1.1.4. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 9 1.1.5. EQUATIONS OF THE FORM FF = AF^ + /O)FF + G(W) 9 1.1.6. EQUATIONS OF THE FORM §F =A\|^F + F(X,T,W)^ + G(X,T,W) 13 1.1.7. EQUATIONS OF THE FORM ^F =A^R+B(^) 2 + F(X,T,W) 15 1.1.8. EQUATIONS OF THE FORM ^? =A^ + F(X,T,W,%) 17 1.1.9. EQUATIONS OF THE FORM %JR =AW K ^-+ F(X,T,W,^) 18 1.1.10. EQUATIONS OF THE FORM ^ = A-§^ (W M FJ) 25 1.1.11. EQUATIONS OF THE FORM \|F = A JR(* M F!) +BWK 32 1.1.12. EQUATIONS OF THE FORM ^ = A-^ W M ^) +BW + AW KI + C 2 W K2 + C 3 W K3 .... 37 1.1.13. EQUATIONS OF THE FORM F^ = ;\|/[/ »§§] +9(W) 40 1.1.14. EQUATIONSOFTHEFONN^ = [F(W)^]+G(X,T,W,^) 42 1.1.15. OTHER EQUATIONS 46 1.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 52 1.2.1. EQUATIONS OF THE FORM ^ = A^+B O + B X E XW + B 2 E 2XW 52 1.2.2. EQUATIONS OF THE FORM \|F = A- (E XW %) + F(W) 53 1.2.3. EQUATIONS OF THE FORM §F = [/(,)&] +G ( W ) 5 6 1.2.4. OTHER EQUATIONS EXPLICITLY INDEPENDENT OF X AND T 58 1.2.5. EQUATIONS EXPLICITLY DEPENDENT ON X AND/OR T 61 1.3. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 62 1.3.1. EQUATIONS INVOLVING HYPERBOLIC COSINE 62 1.3.2. EQUATIONS INVOLVING HYPERBOLIC SINE 63 1.3.3. EQUATIONS INVOLVING HYPERBOLIC TANGENT 63 1.3.4. EQUATIONS INVOLVING HYPERBOLIC COTANGENT 64 1.4. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 64 1.4.1. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 64 1.4.2. OTHER EQUATIONS 66 1.5. EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES 68 1.5.1. EQUATIONS INVOLVING COSINE 68 1.5.2. EQUATIONS INVOLVING SINE 69 1.5.3. EQUATIONS INVOLVING TANGENT 70 1.5.4. EQUATIONS INVOLVING COTANGENT 70 1.5.5. EQUATIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS 71 VI CONTENTS 1.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 71 1.6.1. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 71 1.6.2. EQUATIONS OF THE FORM §F = AF^F + F(X, I)\|F + G(X, T,W) 75 1.6.3. EQUATIONS OF THE FORM ^-= A^R + F(X,T,W)^ + G(X,T,W) 78 1.6.4. EQUATIONSOFTHEFORM FF =A^+B(^F + F(X,T,W) 81 1.6.5. EQUATIONS OF THE FORM FF=A\|^F + B(^F + F(X,T,W)J%- +G(X,T,W) 83 1.6.6. EQUATIONS OF THE FORM §F=A^ + F(X, T,W)(^) 2 + G(X,T,W)^+ H(X,T,W) 84 1.6.7. EQUATIONS OF THE FORM ^ =A^R+F(X T T,W T ^) 88 1.6.8. EQUATIONS OFTHE FORM FF = F(X, T)F^- + G(X, T, W, F^) 89 1.6.9. EQUATIONS OFTHE FORM FF = AW^F + F(X, T, W)^ + G(X, T,W) 92 1.6.10. EQUATIONS OF THE FORM FF = (AW + B)^- + F(X, T,W)(^) 2 + G(X, T,W)^ + H(X, T,W) 94 1.6.11. EQUATIONS OFTHE FORM FF = AW M ^ + FIX, T)^ + G(X, T,W) 97 1.6.12. EQUATIONS OFTHE FORM ^=A-§^ (W%) + F(X, T)$ + G(X, T,W) 98 1.6.13. EQUATIONS OF THE FORM $. = A-J^(W M ^) + F(X,T)^ + G(X,T,W) 100 1.6.14. EQUATIONS OF THE FORM \|F = A-^(E XW ^)+F(X,T,W) 102 1.6.15. EQUATIONS OFTHE FORM %F = ^ [F(W)^]+G(X, T,W,^) 10 4 1.6.16. EQUATIONS OFTHE FORM §F = F(X,W)^ ILL 1.6.17. EQUATIONS OF THE FORM ^ = F(X,T,W)^ + G(X,T,W, %) 113 1.6.18. EQUATIONS OF THE FORM ^F = F(X,W, ^ ) ^ T + G(X,T,W, F^) 120 1.6.19. NONLINEAR EQUATIONS OFTHE THERMAL (DIFFUSION) BOUNDARY LAYER 124 1.7. NONLINEAR SCHRODINGER EQUATIONS AND RELATED EQUATIONS 125 1.7.1. EQUATIONS OF THE FORM I -^ + \|^F- + F( W )W = 0 INVOLVING ARBITRARY PARAMETERS 125 1.7.2. EQUATIONS OF THE FORM I^- + -^ -§^ (X N F^) + F( W )W = 0 INVOLVING ARBITRARY PARAMETERS 128 1.7.3. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 130 1.7.4. EQUATIONS WITH CUBIC NONLINEARITIES INVOLVING ARBITRARY FUNCTIONS 131 1.7.5. EQUATIONS OF GENERAL FORM INVOLVING ARBITRARY FUNCTIONS OF A SINGLE ARGUMENT 134 1.7.6. EQUATIONS OF GENERAL FORM INVOLVING ARBITRARY FUNCTIONS OF TWO ARGUMENTS .. 137 2. PARABOLIC EQUATIONS WITH TWO OR MORE SPACE VARIABLES ... 141 2.1. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 141 2.1.1. EQUATIONSOFTHEFORM^ = [/(X)\|F]+^[ 5 (T/)^]+A^ 141 2.1.2. EQUATIONS OF THE FORM §F =A-§^{W N ^)+B-§^{W K ^) 142 2.1.3. EQUATIONS OFTHE FORM §F = [/(0F?] + ^ LD(W)^] + H(W) 147 2.1.4. OTHER EQUATIONS^ 149 2.2. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 154 2.2.1. EQUATIONS OF THE FORM ^F = [F(X)^] + ^;[G(Y)^}+AE XW 154 2.2.2. EQUATIONS OF THE FORM ^= A ^(E^^)+ B-^ (E XW ^) + F(W) 155 2.3. OTHER EQUATIONS WITH TWO SPACE VARIABLES INVOLVING ARBITRARY PARAMETERS 157 2.3.1. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 157 2.3.2. EQUATIONS WITH TRIGONOMETRICAL NONLINEARITIES 158 CONTENTS VII 2.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 159 2.4.1. HEAT AND MASS TRANSFER EQUATIONS IN QUIESCENT OR MOVING MEDIA WITH CHEMICAL REACTIONS 159 2.4.2. EQUATIONS OFTHE FORM §F = [F(X)$] + % [G(Y)%] + H(W) 161 2.4.3. EQUATIONS OFTHE FORM ^ = -^ [/C^I^F] + IR~ .9( W }LT~ + H(T, W ) 162 2.4.4. OTHER EQUATIONS LINEAR IN THE HIGHEST DERIVATIVES 165 2.4.5. NONLINEAR DIFFUSION BOUNDARY LAYER EQUATIONS 168 2.5. EQUATIONS WITH THREE OR MORE SPACE VARIABLES 169 2.5.1. EQUATIONS OF MASS TRANSFER IN QUIESCENT OR MOVING MEDIA WITH CHEMICAL REACTIONS 169 2.5.2. HEAT EQUATIONS WITH POWER-LAW OR EXPONENTIAL TEMPERATURE-DEPENDENT THERMAL DIFFUSIVITY 173 2.5.3. EQUATIONS OF HEAT AND MASS TRANSFER IN ANISOTROPIC MEDIA 174 2.5.4. OTHER EQUATIONS WITH THREE SPACE VARIABLES 177 2.5.5. EQUATIONS WITH N SPACE VARIABLES 179 2.6. NONLINEAR SCHRODINGER EQUATIONS 186 2.6.1. TWO-DIMENSIONAL EQUATIONS 186 2.6.2. THREE AND N-DIMENSIONAL EQUATIONS 189 3. HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE 191 3.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 191 3.1.1. EQUATIONS OFTHE FORM ^§- = ^§- +AW + BW N + CW 2N ~ L 191 3.1.2. EQUATIONS OFTHE FORM ^ = AF^F + F(X, T,W) 193 3.1.3. EQUATIONS OFTHE FORM & = A#F + FIX, T, W, &) 196 ^ OV- OX A J OX ! 3.1.4. EQUATIONS OF THE FORM ^ = F(X)^$-+ G(X,T,W, F^) 198 3.1.5. EQUATIONS OF THE FORM O^F = AW N ^JF + F(X, W) 202 3.1.6. EQUATIONS OFTHE FORM ^ = A-§^ (W N ^)+F(W) 20 4 3.1.7. OTHER EQUATIONS . * 209 3.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 213 3.2.1. EQUATIONS OFTHE FORM ^ = A ^ + BE? W + CE^ W 213 3.2.2. EQUATIONS OF THE FORM ^ = A ^ + F(X,T,W) 215 3.2.3. EQUATIONS OF THE FORM ^-= F(X)^ + G(X,T,W, \|F) 217 3.2.4. OTHER EQUATIONS 222 3.3. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 225 3.3.1. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 225 3.3.2. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 226 3.3.3. SINE-GORDON EQUATION AND OTHER EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES . 227 3.3.4. EQUATIONS OFTHE FORM ^ + A\|F = -^ [/(W)FS 230 3.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 234 3.4.1. EQUATIONS OFTHE FORM ^ = A^T + F(X, T,W) 234 3.4.2. EQUATIONS OFTHE FORM FE = A^F + FIX, T, W, ^) 239 X OT OX ** OX / 3.4.3. EQUATIONS OFTHE FORM ^ = F(.X)^T +G(X, T, W, F^F) 245 3.4.4. EQUATIONS OFTHE FORM 4#- = F(W)^R + Q(X, T, W, ^-) 250 - 1 OT OX J V OX / 3.4.5. EQUATIONS OFTHE FORM 4^ = F(X, W)^R + G{X, T, W, ^F) 257 VIII CONTENTS 3.4.6. EQUATIONS OFTHE FORM F^ = F(T, W)^ +G(X,T,W,^) 259 3.4.7. OTHER EQUATIONS LINEAR IN THE HIGHEST DERIVATIVES 260 3.5. EQUATIONS OFTHE FORM ^ = F(X,Y,W, FJ, F^) 266 3.5.1. EQUATIONS INVOLVING ARBITRARY PARAMETERS OFTHE FORM -§^§- = F(W) 266 3.5.2. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 270 3.5.3. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 271 4. HYPERBOLIC EQUATIONS WITH TWO OR THREE SPACE VARIABLES 275 4.1. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 275 4.1.1. EQUATIONS OFTHE FORM ^ = [/(Z)F^] + ^ [G(Y)%] +A* P 275 4.1.2. EQUATIONS OF THE FORM #$. = A&(W N %) + B^ (W K ^) 277 4.1.3. EQUATIONSOFTHEFORM^ = ^[/(^)\|^]+^[^)^] 285 4.1.4. OTHER EQUATIONS 290 4.2. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 292 4.2.1. EQUATIONS OFTHE FORM F^ = [/()F^] + \| (V)^] + AE XW 292 4.2.2. EQUATIONS OF THE FORM ^ = A^(E^ FF)+ B^ {E XW ^) 294 4.2.3. OTHER EQUATIONS 299 4.3. NONLINEAR TELEGRAPH EQUATIONS WITH TWO SPACE VARIABLES 299 4.3.1. EQUATIONS INVOLVING POWER-LAW NONLINEARITIES 299 4.3.2. EQUATIONS INVOLVING EXPONENTIAL NONLINEARITIES 303 4.4. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING ARBITRARY FUNCTIONS 305 4.4.1. EQUATIONSOFTHEFORM^ = ^[/(^]+^[ 5 (Y)^]+/I(I T ;) 305 4.4.2. EQUATIONSOFTHEFORM^ = ^[/( U ;)\|^]+^[ 5 ( W ;)^]+/I( W ;) 308 4.4.3. OTHER EQUATIONS 313 4.5. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING ARBITRARY PARAMETERS 317 4.5.1. EQUATIONS OF THE FORM ^ = [F( X )^] + ^[G( Y) ^] + - +AW R 317 4.5.2. EQUATIONS OFTHE FORM ^ = & [/(Z)F^] + [G(Y)%] + & [H(Z)%] + AE XW 318 4.5.3. EQ UATIONSOFTHEFORM^ = A^K\|^) + 6^(^^) +C ^(^^) +S ^ 320 4.5.4. EQUATIONS OF THE FORM ^ = A ( EX ^^)+B-^{E X W %)+C(E X W ^) + SEP W 327 4.6. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING ARBITRARY FUNCTIONS 334 4.6.1. EQUATIONS OF THE FORM ^ = [/I()FE] +^ [H(V)^] +F Z [M)TH] + 9(W) 334 4.6.2. EQUATIONSOFTHEFORM^ = ^[/ 1 (^)\|^]+^[/ 2 ( W ;)^]+^[/3(;)^]+ 5 (;) 338 4.6.3. OTHER EQUATIONS 344 5. ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 347 5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 347 5.1.1. EQUATIONS OFTHE FORM FT + & = AW + BW N + CW 2N ~ L 347 5.1.2. EQUATIONS OFTHE FORM 0 + ^ = F(X,Y,W) 349 5.1.3. EQUATIONS OF THE FORM 0 + A§F = F(X,Y,W,^,^) 350 5.1.4. EQUATIONS OF THE FORM ^-[F L ( X ,Y)^]+^[F 1 ( X ,Y)^}= G(W) 351 5.1.5. EQUATIONS OF THE FORM J-[F,(W)^]+-^[H(W)^]=G(W) 353 5.1.6. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 358 CONTENTS IX 5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 364 5.2.1. EQUATIONS OFTHE FORM ^R + §R = A + BE 3 * + CE^ W 364 5.2.2. EQUATIONS OFTHE FORM \|^X + \|^F = F(X, Y,W) 367 5.2.3. EQUATIONSOFTHEFORM^[/ 1 ( A; ,Y)^]+^[/ 2 ( A :, 2 /)\|^]=^(^) 367 5.2.4. EQUATIONS OF THE FORM -FL [H(W)^} + -^ [F 2 (W)%] = G(W) 370 5.2.5. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 373 5.3. EQUATIONS INVOLVING OTHER NONLINEARITIES 376 5.3.1. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 376 5.3.2. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 377 5.3.3. EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES 380 5.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 382 5.4.1. EQUATIONS OFTHE FORM \|^ + ^ = F(X, Y,W) 382 5.4.2. EQUATIONS OFTHE FORM A ^F + BF^-= F(X,Y,W,^,^) 387 5.4.3. HEAT AND MASS TRANSFER EQUATIONS OF THE FORM -^[F(X)^]+-^[G(Y)^]=H(W) 391 5.4.4. EQUATIONS OFTHE FORM -^ [F(X, Y, W)^]+^ [G(X, Y, W)^} = H(X, Y,W) .... 393 5.4.5. OTHER EQUATIONS 399 6. ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES 405 6.1. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 405 6.1.1. EQUATIONS OFTHE FORM [/(X)^] + [G(Y)%] + -§~ Z [H(Z)^ =AW* 405 6.1.2. EQUATIONS OFTHE FORM & [F(W)%] + % [ & )%] + [G(W)%?] =0 408 6.2. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 413 6.2.1. EQUATIONS OF THE FORM &[F(X)%]+% ; [G(V)%JL]+&[H(Z)%]=AE^ .... 413 6.2.2. EQUATIONSOFTHEFORMA 1 ^(E A -\|^)+A 2 ^(E A ^^)+A3^(E A ^^)=6 E ^ 416 6.3. THREE-DIMENSIONAL EQUATIONS INVOLVING ARBITRARY FUNCTIONS 420 6.3.1. HEAT AND MASS TRANSFER EQUATIONS OF THE FORM ^ [/I(A;)FJ] + ^ [MY)$] + 420 6.3.2. HEAT AND MASS TRANSFER EQUATIONS WITH COMPLICATING FACTORS 423 6.3.3. OTHER EQUATIONS 426 6.4. EQUATIONS WITH N INDEPENDENT VARIABLES 428 6.4.1. EQUATIONS OF THE FORM -^ [F 1 ( XL )J-]+.. .+^_ [/(») \| ] =G(X U ...,X N ,W) 428 6.4.2. OTHER EQUATIONS ? ? 430 7. EQUATIONS INVOLVING MIXED DERIVATIVES AND SOME OTHER EQUATIONS 433 7.1. EQUATIONS LINEAR IN THE MIXED DERIVATIVE 433 7.1.1. CALOGERO EQUATION 433 7.1.2. KHOKHLOV-ZABOLOTSKAYA EQUATION 435 7.1.3. EQUATION OF UNSTEADY TRANSONIC GAS FLOWS 440 ,- 7.1.4. E Q UATIONSOFTHEFORM^^-^^ = F(X, ? /,^,^) 443 7.1.5. OTHER EQUATIONS WITH TWO INDEPENDENT VARIABLES 445 7.1.6. OTHER EQUATIONS WITH THREE INDEPENDENT VARIABLES 448 7.2. EQUATIONS QUADRATIC IN THE HIGHEST DERIVATIVES 449 7.2.1. EQUATIONS OFTHE FORM F^ \|^F = F(X, Y) 449 7.2.2. MONGE-AMPEREEQUATION(^) 2 -^0=F(^Y) 451 7.2.3. EQUATIONSOFTHEFORM(^) 2 -^^=F(A ; , ?/ , U ;,^,\|^) 461 X CONTENTS 7.2.4. EQUATIONS OF THE FORM {^F = F(X, 2/)FF \|^ + G(X, Y) 465 7.2.5. OTHER EQUATIONS 469 7.3. BELLMAN TYPE EQUATIONS AND RELATED EQUATIONS 472 7.3.1. EQUATIONS WITH QUADRATIC NONLINEARITIES 472 7.3.2. EQUATIONS WITH POWER-LAW NONLINEARITIES 475 8. SECOND-ORDER EQUATIONS OF GENERAL FORM 479 8.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 479 8.1.1. EQUATIONS OFTHE FORM FF = F(W, F^, FF) 479 8.1.2. EQUATIONS OFTHE FORM §F = F(T,W, F^, &$) 486 8.1.3. EQUATIONS OFTHE FORM FF = F(X, W, §^, \|^) 490 8.1.4. EQUATIONS OFTHE FORM FF = F(X, T, W, F^, ^F) 494 8.1.5. EQUATIONS OF THE FORM F (A;, T,W, FF -FJ, FS^F) =0 499 8.1.6. EQUATIONS WITH THREE INDEPENDENT VARIABLES 500 8.2. EQUATIONS INVOLVING TWO OR MORE SECOND DERIVATIVES 501 8.2.1. EQUATIONS OFTHE FORM ^$- = F(W, F^, F^) 501 8.2.2. EQUATIONS OF THE FORM ^ = F(X,T,W,^,^-,^R) 505 8.2.3. EQUATIONS LINEAR IN THE MIXED DERIVATIVE 508 8.2.4. EQUATIONS WITH TWO INDEPENDENT VARIABLES, NONLINEAR IN TWO OR MORE HIGHEST DERIVATIVES 509 8.2.5. EQUATIONS WITH N INDEPENDENT VARIABLES 512 9. THIRD-ORDER EQUATIONS 515 9.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 515 9.1.1. KORTEWEG-DE VRIES EQUATION FF + AF^F + BW^ =0 515 9.1.2. CYLINDRICAL, SPHERICAL, AND MODIFIED KORTEWEG-DE VRIES EQUATIONS 521 9.1.3. GENERALIZED KORTEWEG-DE VRIES EQUATION FF + A^F + /(W)FJ =0 524 9.1.4. EQUATIONS REDUCIBLE TO THE KORTEWEG-DE VRIES EQUATION 526 9.1.5. EQUATIONS OFTHE FORM FF + A^F + F(W, §^) = 0 529 9.1.6. EQUATIONS OFTHE FORM FF + A^F + F(X, T, W, F^) = 0 530 9.1.7. BURGERS-KORTEWEG-DE VRIES EQUATION AND OTHER EQUATIONS 532 9.2. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 536 9.2.1. EQUATIONS WITH QUADRATIC NONLINEARITIES 536 9.2.2. OTHER EQUATIONS 539 9.3. HYDRODYNAMIC BOUNDARY LAYER EQUATIONS 540 9.3.1. STEADY HYDRODYNAMIC BOUNDARY LAYER EQUATIONS FOR A NEWTONIAN FLUID 540 9.3.2. STEADY BOUNDARY LAYER EQUATIONS FOR NON-NEWTONIAN FLUIDS 547 9.3.3. UNSTEADY BOUNDARY LAYER EQUATIONS FOR A NEWTONIAN FLUID 553 9.3.4. UNSTEADY BOUNDARY LAYER EQUATIONS FOR NON-NEWTONIAN FLUIDS 564 9.3.5. RELATED EQUATIONS 568 9.4. EQUATIONS OF MOTION OF IDEAL FLUID (EULER EQUATIONS) 570 9.4.1. STATIONARY EQUATIONS 570 9.4.2. NONSTATIONARY EQUATIONS 574 9.5. OTHER THIRD-ORDER NONLINEAR EQUATIONS 580 9.5.1. EQUATIONS INVOLVING SECOND-ORDER MIXED DERIVATIVES 580 9.5.2. EQUATIONS INVOLVING THIRD-ORDER MIXED DERIVATIVES 583 9.5.3. EQUATIONS INVOLVING ^ AND ^ 587 CONTENTS XI 10. FOURTH-ORDER EQUATIONS 589 10.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 589 10.1.1. EQUATIONS OF THE FORM F =A\|F+F(I,T ( ;,F) 589 10.1.2. OTHER EQUATIONS 593 10.2. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 595 10.2.1. BOUSSINESQ EQUATION AND ITS MODIFICATIONS 595 10.2.2. EQUATIONS WITH QUADRATIC NONLINEARITIES 600 10.2.3. OTHER EQUATIONS 603 10.3. EQUATIONS INVOLVING MIXED DERIVATIVES 605 10.3.1. KADOMTSEV-PETVIASHVILI EQUATION 605 10.3.2. STATIONARY HYDRODYNAMIC EQUATIONS (NAVIER-STOKES EQUATIONS) 607 10.3.3. NONSTATIONARY HYDRODYNAMIC EQUATIONS (NAVIER-STOKES EQUATIONS) 616 10.3.4. OTHER EQUATIONS 628 11. EQUATIONS OF HIGHER ORDERS 631 11.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T AND LINEAR IN THE HIGHEST DERIVATIVE .... 631 11.1.1. FIFTH-ORDER EQUATIONS 631 11.1.2. EQUATIONS OF THE FORM &$* =A% + F(X,T,W) 633 11.1.3. EQUATIONS OFTHE FORM FF = AF^F + /(W)F^ 635 11.1.4. EQUATIONS OF THE FORM FF = AF^F + F(X,T,W)^ + G(X,T,W) 637 11.1.5. EQUATIONS OF THE FORM FF = A§^R +F(X,T,W, F^) 640 11.1.6. EQUATIONS OF THE FORM FF =A% +F(X,T,W, F^---,FJIF) 645 11.1.7. EQUATIONS OFTHE FORM FF = AW^- + F(X, T, W)\|F + G(X, T,W) 64 7 11.1.8. OTHER EQUATIONS 648 11.2. GENERAL FORM EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 651 11.2.1. EQUATIONS OFTHE FORM §F = F(W, §^,..., FJ) 651 11.2.2. EQUATIONS OFTHE FORM §F = F(T, W, %,..., -GF) 656 11.2.3. EQUATIONS OF THE FORM %?= F(X,W, %,..., %) 659 11.2.4. EQUATIONS OF THE FORM =F(X,T, W, %,..., -F) 662 11.3. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 666 11.3.1. EQUATIONS OFTHE FORM ^ = AF^ + F(X,T, W) 666 11.3.2. EQUATIONS OFTHE FORM \| ^ = A^-+F(X,T,W, F^-) 667 11.3.3. EQUATIONS OF THE FORM ^-= A^-+ F(X,T,W, FJ, * * , IJSF) 671 11.3.4. EQUATIONS OF THE FORM ^ = AW^-+ F(X,T,W)^+G(X,T,W) 673 11.3.5. EQUATIONS OF THE FORM ^=F(X,T, W, F^,-* ) 675 11.4. OTHER EQUATIONS 676 11.4.1 . EQUATIONS INVOLVING MIXED DERIVATIVES 676 11.4.2. EQUATIONS INVOLVING F^ AND F^F 680 SUPPLEMENTS. EXACT METHODS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS .... 683 S.I. CLASSIFICATION OF SECOND-ORDER SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES 683 5.1.1. TYPES OF EQUATIONS. CHARACTERISTIC EQUATION 683 5.1.2. CANONICAL FORM OF PARABOLIC EQUATIONS 683 5.1.3. CANONICAL FORM OF HYPERBOLIC EQUATIONS 684 5.1.4. CANONICAL FORM OF ELLIPTIC EQUATIONS 684 XII CONTENTS 5.2. TRANSFORMATIONS OF EQUATIONS OF MATHEMATICAL PHYSICS 685 5.2.1. POINT TRANSFORMATIONS 685 5.2.2. HODOGRAPH TRANSFORMATION 686 5.2.3. CONTACT TRANSFORMATIONS. LEGENDRE AND EULER TRANSFORMATIONS 688 5.2.4. BACKHAND TRANSFORMATIONS. DIFFERENTIAL SUBSTITUTIONS 690 5.3. TRAVELING-WAVE SOLUTIONS AND SELF-SIMILAR SOLUTIONS. SIMILARITY METHODS 693 5.3.1. PRELIMINARY REMARKS 693 5.3.2. TRAVELING-WAVE SOLUTIONS. INVARIANCE OF EQUATIONS UNDER TRANSLATIONS 694 5.3.3. SELF-SIMILAR SOLUTIONS. INVARIANCE OF EQUATIONS UNDER SCALING TRANSFORMATIONS 695 5.3.4. EXPONENTIAL SELF-SIMILAR SOLUTIONS. EQUATIONS INVARIANT UNDER COMBINED TRANSLATION AND SCALING 696 5.4. METHOD OF GENERALIZED SEPARATION OF VARIABLES 698 5.4.1. INTRODUCTION 698 5.4.2. STRUCTURE OF GENERALIZED SEPARABLE SOLUTIONS 700 5.4.3. SOLUTION OF FUNCTIONAL-DIFFERENTIAL EQUATIONS BY DIFFERENTIATION 701 5.4.4. SOLUTION OF FUNCTIONAL-DIFFERENTIAL EQUATIONS BY SPLITTING 705 5.4.5. SIMPLIFIED SCHEME FOR CONSTRUCTING GENERALIZED SEPARABLE SOLUTIONS 709 5.4.6. TITOV-GALAKTIONOV METHOD 710 5.5. METHOD OF FUNCTIONAL SEPARATION OF VARIABLES 713 5.5.1. STRUCTURE OF FUNCTIONAL SEPARABLE SOLUTIONS 713 5.5.2. SPECIAL FUNCTIONAL SEPARABLE SOLUTIONS 713 5.5.3. DIFFERENTIATION METHOD 718 5.5.4. SPLITTING METHOD. REDUCTION TO A FUNCTIONAL EQUATION WITH TWO VARIABLES 721 5.5.5. SOLUTIONS OF SOME NONLINEAR FUNCTIONAL EQUATIONS AND THEIR APPLICATIONS .... 723 5.6. GENERALIZED SIMILARITY REDUCTIONS OF NONLINEAR EQUATIONS 728 5.6.1. CLARKSON-KRUSKAL DIRECT METHOD: A SPECIAL FORM FOR SIMILARITY REDUCTION .... 728 5.6.2. CLARKSON-KRUSKAL DIRECT METHOD: THE GENERAL FORM FOR SIMILARITY REDUCTION .. 731 5.6.3. SOME MODIFICATIONS AND GENERALIZATIONS 732 5.7. GROUP ANALYSIS METHODS 735 5.7.1. CLASSICAL METHOD FOR SYMMETRY REDUCTIONS 735 5.7.2. NONCLASSICAL METHOD FOR SYMMETRY REDUCTIONS 744 5.8. DIFFERENTIAL CONSTRAINTS METHOD 747 5.8.1. DESCRIPTION OFTHE METHOD 747 5.8.2. FIRST-ORDER DIFFERENTIAL CONSTRAINTS 749 5.8.3. SECOND- AND HIGHER-ORDER DIFFERENTIAL CONSTRAINTS 754 5.8.4. CONNECTION BETWEEN THE DIFFERENTIAL CONSTRAINTS METHOD AND OTHER METHODS .. 756 5.9. PAINLEVE TEST FOR NONLINEAR EQUATIONS OF MATHEMATICAL PHYSICS 758 5.9.1. MOVABLE SINGULARITIES OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 758 5.9.2. SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH A MOVABLE POLE. DESCRIPTION OF THE METHOD 760 5.9.3. EXAMPLES OF THE^PAINLEVE TEST APPLICATIONS 761 5.10. INVERSE SCATTERING METHOD 764 5.10.1. LAX PAIR METHOD 764 5.10.2. METHOD BASED ON THE COMPATIBILITY CONDITION FOR TWO LINEAR EQUATIONS .... 766 5.10.3. METHOD BASED ON LINEAR INTEGRAL EQUATIONS 767 S.I 1. CONSERVATION LAWS 769 S.I 1.1. BASIC DEFINITIONS AND EXAMPLES 769 S.I 1.2. EQUATIONS ADMITTING VARIATIONAL FORMULATION. NOETHERIAN SYMMETRIES 770 CONTENTS XIU S.12. HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS 772 5.12.1. CONSERVATION LAWS. SOME EXAMPLES 772 5.12.2. CAUCHY PROBLEM, RIEMANN PROBLEM, AND INITIAL-BOUNDARY VALUE PROBLEM ... 773 5.12.3. CHARACTERISTIC LINES. HYPERBOLIC SYSTEMS. RIEMANN INVARIANTS 773 5.12.4. SELF-SIMILAR CONTINUOUS SOLUTIONS. RAREFACTION WAVES 777 5.12.5. SHOCKWAVES. RANKINE-HUGONIOT JUMP CONDITIONS 779 5.12.6. EVOLUTIONARY SHOCKS. LAX CONDITION (VARIOUS FORMULATIONS) 780 5.12.7. SOLUTIONS FOR THE RIEMANN PROBLEM 782 5.12.8. INITIAL-BOUNDARY VALUE PROBLEMS OF SPECIAL FORM 786 5.12.9. EXAMPLES OF NONSTRICT HYPERBOLIC SYSTEMS 786 REFERENCES 791 INDEX 809
adam_txt	HANDBOOK OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS ANDREI D. POLYANIN VALENTIN F. ZAITSEV CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON NEW YORK WASHINGTON, D.C. CONTENTS AUTHORS XV FOREWORD XVII SOME NOTATIONS AND REMARKS XIX 1. PARABOLIC EQUATIONS WITH ONE SPACE VARIABLE 1 1.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 1 1.1.1. EQUATIONS OF THE FORM §F = A^JG- +BW + CW 2 1 1.1.2. EQUATIONS OF THE FORM ^F =A-^R + B 0 + BIW + B 2 W 2 + B3W 3 2 1.1.3. EQUATIONS OF THE FORM §F = A^F + F(W) 6 1.1.4. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 9 1.1.5. EQUATIONS OF THE FORM FF = AF^ + /O)FF + G(W) 9 1.1.6. EQUATIONS OF THE FORM §F =A\|^F + F(X,T,W)^ + G(X,T,W) 13 1.1.7. EQUATIONS OF THE FORM ^F =A^R+B(^) 2 + F(X,T,W) 15 1.1.8. EQUATIONS OF THE FORM ^? =A^ + F(X,T,W,%) 17 1.1.9. EQUATIONS OF THE FORM %JR =AW K ^-+ F(X,T,W,^) 18 1.1.10. EQUATIONS OF THE FORM ^ = A-§^ (W M FJ) 25 1.1.11. EQUATIONS OF THE FORM \|F = A JR(* M F!) +BWK 32 1.1.12. EQUATIONS OF THE FORM ^ = A-^ \W M ^) +BW + AW KI + C 2 W K2 + C 3 W K3 . 37 1.1.13. EQUATIONS OF THE FORM F^ = ;\|/[/ »§§] +9(W) 40 1.1.14. EQUATIONSOFTHEFONN^ = [F(W)^]+G(X,T,W,^) 42 1.1.15. OTHER EQUATIONS 46 1.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 52 1.2.1. EQUATIONS OF THE FORM ^ = A^+B O + B X E XW + B 2 E 2XW 52 1.2.2. EQUATIONS OF THE FORM \|F = A- (E XW %) + F(W) 53 1.2.3. EQUATIONS OF THE FORM §F = [/(,)&] +G ( W ) 5 6 1.2.4. OTHER EQUATIONS EXPLICITLY INDEPENDENT OF X AND T 58 1.2.5. EQUATIONS EXPLICITLY DEPENDENT ON X AND/OR T 61 1.3. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 62 1.3.1. EQUATIONS INVOLVING HYPERBOLIC COSINE 62 1.3.2. EQUATIONS INVOLVING HYPERBOLIC SINE 63 1.3.3. EQUATIONS INVOLVING HYPERBOLIC TANGENT 63 1.3.4. EQUATIONS INVOLVING HYPERBOLIC COTANGENT 64 1.4. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 64 '' 1.4.1. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 64 1.4.2. OTHER EQUATIONS 66 1.5. EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES 68 1.5.1. EQUATIONS INVOLVING COSINE 68 1.5.2. EQUATIONS INVOLVING SINE 69 1.5.3. EQUATIONS INVOLVING TANGENT 70 1.5.4. EQUATIONS INVOLVING COTANGENT 70 1.5.5. EQUATIONS INVOLVING INVERSE TRIGONOMETRIC FUNCTIONS 71 VI CONTENTS 1.6. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 71 1.6.1. EQUATIONS OF THE FORM §F = A^F + F(X, T,W) 71 1.6.2. EQUATIONS OF THE FORM §F = AF^F + F(X, I)\|F + G(X, T,W) 75 1.6.3. EQUATIONS OF THE FORM ^-= A^R + F(X,T,W)^ + G(X,T,W) 78 1.6.4. EQUATIONSOFTHEFORM FF =A^+B(^F + F(X,T,W) 81 1.6.5. EQUATIONS OF THE FORM FF=A\|^F + B(^F + F(X,T,W)J%- +G(X,T,W) 83 1.6.6. EQUATIONS OF THE FORM §F=A^ + F(X, T,W)(^) 2 + G(X,T,W)^+ H(X,T,W) 84 1.6.7. EQUATIONS OF THE FORM ^ =A^R+F(X T T,W T ^) 88 1.6.8. EQUATIONS OFTHE FORM FF = F(X, T)F^- + G(X, T, W, F^) 89 1.6.9. EQUATIONS OFTHE FORM FF = AW^F + F(X, T, W)^ + G(X, T,W) 92 1.6.10. EQUATIONS OF THE FORM FF = (AW + B)^- + F(X, T,W)(^) 2 + G(X, T,W)^ + H(X, T,W) 94 1.6.11. EQUATIONS OFTHE FORM FF = AW M ^ + FIX, T)^ + G(X, T,W) 97 1.6.12. EQUATIONS OFTHE FORM ^=A-§^ (W%) + F(X, T)$ + G(X, T,W) 98 1.6.13. EQUATIONS OF THE FORM $. = A-J^(W M ^) + F(X,T)^ + G(X,T,W) 100 1.6.14. EQUATIONS OF THE FORM \|F = A-^(E XW ^)+F(X,T,W) 102 1.6.15. EQUATIONS OFTHE FORM %F = ^ [F(W)^]+G(X, T,W,^) 10 4 1.6.16. EQUATIONS OFTHE FORM §F = F(X,W)^ ILL 1.6.17. EQUATIONS OF THE FORM ^ = F(X,T,W)^ + G(X,T,W, %) 113 1.6.18. EQUATIONS OF THE FORM ^F = F(X,W, ^ ) ^ T + G(X,T,W, F^) 120 1.6.19. NONLINEAR EQUATIONS OFTHE THERMAL (DIFFUSION) BOUNDARY LAYER 124 1.7. NONLINEAR SCHRODINGER EQUATIONS AND RELATED EQUATIONS 125 1.7.1. EQUATIONS OF THE FORM I -^ + \|^F- + F(\W\)W = 0 INVOLVING ARBITRARY PARAMETERS 125 1.7.2. EQUATIONS OF THE FORM I^- + -^ -§^ (X N F^) + F(\W\)W = 0 INVOLVING ARBITRARY PARAMETERS 128 1.7.3. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 130 1.7.4. EQUATIONS WITH CUBIC NONLINEARITIES INVOLVING ARBITRARY FUNCTIONS 131 1.7.5. EQUATIONS OF GENERAL FORM INVOLVING ARBITRARY FUNCTIONS OF A SINGLE ARGUMENT 134 1.7.6. EQUATIONS OF GENERAL FORM INVOLVING ARBITRARY FUNCTIONS OF TWO ARGUMENTS . 137 2. PARABOLIC EQUATIONS WITH TWO OR MORE SPACE VARIABLES . 141 2.1. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 141 2.1.1. EQUATIONSOFTHEFORM^ = [/(X)\|F]+^[ 5 (T/)^]+A^ 141 2.1.2. EQUATIONS OF THE FORM §F =A-§^{W N ^)+B-§^{W K ^) 142 2.1.3. EQUATIONS OFTHE FORM §F = [/(0F?] + ^ LD(W)^] + H(W) 147 2.1.4. OTHER EQUATIONS^ 149 2.2. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 154 2.2.1. EQUATIONS OF THE FORM ^F = [F(X)^] + ^;[G(Y)^}+AE XW 154 2.2.2. EQUATIONS OF THE FORM ^= A ^(E^^)+ B-^ (E XW ^) + F(W) 155 2.3. OTHER EQUATIONS WITH TWO SPACE VARIABLES INVOLVING ARBITRARY PARAMETERS 157 2.3.1. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 157 2.3.2. EQUATIONS WITH TRIGONOMETRICAL NONLINEARITIES 158 CONTENTS VII 2.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 159 2.4.1. HEAT AND MASS TRANSFER EQUATIONS IN QUIESCENT OR MOVING MEDIA WITH CHEMICAL REACTIONS 159 2.4.2. EQUATIONS OFTHE FORM §F = [F(X)$] + % [G(Y)%] + H(W) 161 2.4.3. EQUATIONS OFTHE FORM ^ = -^ [/C^I^F] + IR~ \.9( W }LT~\ + H(T, W ) 162 2.4.4. OTHER EQUATIONS LINEAR IN THE HIGHEST DERIVATIVES 165 2.4.5. NONLINEAR DIFFUSION BOUNDARY LAYER EQUATIONS 168 2.5. EQUATIONS WITH THREE OR MORE SPACE VARIABLES 169 2.5.1. EQUATIONS OF MASS TRANSFER IN QUIESCENT OR MOVING MEDIA WITH CHEMICAL REACTIONS 169 2.5.2. HEAT EQUATIONS WITH POWER-LAW OR EXPONENTIAL TEMPERATURE-DEPENDENT THERMAL DIFFUSIVITY 173 2.5.3. EQUATIONS OF HEAT AND MASS TRANSFER IN ANISOTROPIC MEDIA 174 2.5.4. OTHER EQUATIONS WITH THREE SPACE VARIABLES 177 2.5.5. EQUATIONS WITH N SPACE VARIABLES 179 2.6. NONLINEAR SCHRODINGER EQUATIONS 186 2.6.1. TWO-DIMENSIONAL EQUATIONS 186 2.6.2. THREE AND N-DIMENSIONAL EQUATIONS 189 3. HYPERBOLIC EQUATIONS WITH ONE SPACE VARIABLE 191 3.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 191 3.1.1. EQUATIONS OFTHE FORM ^§- = ^§- +AW + BW N + CW 2N ~ L 191 3.1.2. EQUATIONS OFTHE FORM ^ = AF^F + F(X, T,W) 193 3.1.3. EQUATIONS OFTHE FORM & = A#F + FIX, T, W, &) 196 ^ OV- OX A J \ OX ! 3.1.4. EQUATIONS OF THE FORM ^ = F(X)^$-+ G(X,T,W, F^) 198 3.1.5. EQUATIONS OF THE FORM O^F = AW N ^JF + F(X, W) 202 3.1.6. EQUATIONS OFTHE FORM ^ = A-§^ (W N ^)+F(W) 20 4 3.1.7. OTHER EQUATIONS . * 209 3.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 213 3.2.1. EQUATIONS OFTHE FORM ^ = A ^ + BE? W + CE^ W 213 3.2.2. EQUATIONS OF THE FORM ^ = A ^ + F(X,T,W) 215 3.2.3. EQUATIONS OF THE FORM ^-= F(X)^ + G(X,T,W, \|F) 217 3.2.4. OTHER EQUATIONS 222 3.3. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 225 3.3.1. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 225 3.3.2. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 226 3.3.3. SINE-GORDON EQUATION AND OTHER EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES . 227 3.3.4. EQUATIONS OFTHE FORM ^ + A\|F = -^ [/(W)FS 230 3.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 234 3.4.1. EQUATIONS OFTHE FORM ^ = A^T + F(X, T,W) 234 3.4.2. EQUATIONS OFTHE FORM FE = A^F + FIX, T, W, ^) 239 X OT OX ** \ OX / 3.4.3. EQUATIONS OFTHE FORM ^ = F(.X)^T +G(X, T, W, F^F) 245 3.4.4. EQUATIONS OFTHE FORM 4#- = F(W)^R + Q(X, T, W, ^-) 250 - 1 OT " OX J V OX / 3.4.5. EQUATIONS OFTHE FORM 4^ = F(X, W)^R + G{X, T, W, ^F) 257 VIII CONTENTS 3.4.6. EQUATIONS OFTHE FORM F^ = F(T, W)^ +G(X,T,W,^) 259 3.4.7. OTHER EQUATIONS LINEAR IN THE HIGHEST DERIVATIVES 260 3.5. EQUATIONS OFTHE FORM ^ = F(X,Y,W, FJ, F^) 266 3.5.1. EQUATIONS INVOLVING ARBITRARY PARAMETERS OFTHE FORM -§^§- = F(W) 266 3.5.2. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 270 3.5.3. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 271 4. HYPERBOLIC EQUATIONS WITH TWO OR THREE SPACE VARIABLES 275 4.1. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 275 4.1.1. EQUATIONS OFTHE FORM ^ = [/(Z)F^] + ^ [G(Y)%] +A* P 275 4.1.2. EQUATIONS OF THE FORM #$. = A&(W N %) + B^ (W K ^) 277 4.1.3. EQUATIONSOFTHEFORM^ = ^[/(^)\|^]+^[^)^] 285 4.1.4. OTHER EQUATIONS 290 4.2. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 292 4.2.1. EQUATIONS OFTHE FORM F^ = [/()F^] + \| (V)^] + AE XW 292 4.2.2. EQUATIONS OF THE FORM ^ = A^(E^ FF)+ B^ {E XW ^) 294 4.2.3. OTHER EQUATIONS 299 4.3. NONLINEAR TELEGRAPH EQUATIONS WITH TWO SPACE VARIABLES 299 4.3.1. EQUATIONS INVOLVING POWER-LAW NONLINEARITIES 299 4.3.2. EQUATIONS INVOLVING EXPONENTIAL NONLINEARITIES 303 4.4. EQUATIONS WITH TWO SPACE VARIABLES INVOLVING ARBITRARY FUNCTIONS 305 4.4.1. EQUATIONSOFTHEFORM^ = ^[/(^]+^[ 5 (Y)^]+/I(I T ;) 305 4.4.2. EQUATIONSOFTHEFORM^ = ^[/( U ;)\|^]+^[ 5 ( W ;)^]+/I( W ;) 308 4.4.3. OTHER EQUATIONS 313 4.5. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING ARBITRARY PARAMETERS 317 4.5.1. EQUATIONS OF THE FORM ^ = [F( X )^] + ^[G( Y) ^] + - +AW R 317 4.5.2. EQUATIONS OFTHE FORM ^ = & [/(Z)F^] + [G(Y)%] + & [H(Z)%] + AE XW 318 4.5.3. EQ UATIONSOFTHEFORM^ = A^K\|^) + 6^(^^) +C ^(^^) +S ^ 320 4.5.4. EQUATIONS OF THE FORM ^ = A ( EX ^^)+B-^{E X W %)+C(E X W ^) + SEP W 327 4.6. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING ARBITRARY FUNCTIONS 334 4.6.1. EQUATIONS OF THE FORM ^ = [/I()FE] +^ [H(V)^] +F Z [M)TH] + 9(W) 334 4.6.2. EQUATIONSOFTHEFORM^ = ^[/ 1 (^)\|^]+^[/ 2 ( W ;)^]+^[/3(;)^]+ 5 (;) 338 4.6.3. OTHER EQUATIONS 344 5. ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 347 5.1. EQUATIONS WITH POWER-LAW NONLINEARITIES 347 5.1.1. EQUATIONS OFTHE FORM FT + & = AW + BW N + CW 2N ~ L 347 5.1.2. EQUATIONS OFTHE FORM 0 + ^ = F(X,Y,W) 349 5.1.3. EQUATIONS OF THE FORM 0 + A§F = F(X,Y,W,^,^) 350 5.1.4. EQUATIONS OF THE FORM ^-[F L ( X ,Y)^]+^[F 1 ( X ,Y)^}= G(W) 351 5.1.5. EQUATIONS OF THE FORM J-[F,(W)^]+-^[H(W)^]=G(W) 353 5.1.6. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 358 CONTENTS IX 5.2. EQUATIONS WITH EXPONENTIAL NONLINEARITIES 364 5.2.1. EQUATIONS OFTHE FORM ^R + §R = A + BE 3 * + CE^ W 364 5.2.2. EQUATIONS OFTHE FORM \|^X + \|^F = F(X, Y,W) 367 5.2.3. EQUATIONSOFTHEFORM^[/ 1 ( A; ,Y)^]+^[/ 2 ( A :, 2 /)\|^]=^(^) 367 5.2.4. EQUATIONS OF THE FORM -FL [H(W)^} + -^ [F 2 (W)%] = G(W) 370 5.2.5. OTHER EQUATIONS INVOLVING ARBITRARY PARAMETERS 373 5.3. EQUATIONS INVOLVING OTHER NONLINEARITIES 376 5.3.1. EQUATIONS WITH HYPERBOLIC NONLINEARITIES 376 5.3.2. EQUATIONS WITH LOGARITHMIC NONLINEARITIES 377 5.3.3. EQUATIONS WITH TRIGONOMETRIC NONLINEARITIES 380 5.4. EQUATIONS INVOLVING ARBITRARY FUNCTIONS 382 5.4.1. EQUATIONS OFTHE FORM \|^ + ^ = F(X, Y,W) 382 5.4.2. EQUATIONS OFTHE FORM A ^F + BF^-= F(X,Y,W,^,^) 387 5.4.3. HEAT AND MASS TRANSFER EQUATIONS OF THE FORM -^[F(X)^]+-^[G(Y)^]=H(W) 391 5.4.4. EQUATIONS OFTHE FORM -^ [F(X, Y, W)^]+^ [G(X, Y, W)^} = H(X, Y,W) . 393 5.4.5. OTHER EQUATIONS 399 6. ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES 405 6.1. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING POWER-LAW NONLINEARITIES 405 6.1.1. EQUATIONS OFTHE FORM [/(X)^] + [G(Y)%] + -§~ Z [H(Z)^\ =AW* 405 6.1.2. EQUATIONS OFTHE FORM & [F(W)%] + % [ & )%] + [G(W)%?] =0 408 6.2. EQUATIONS WITH THREE SPACE VARIABLES INVOLVING EXPONENTIAL NONLINEARITIES 413 6.2.1. EQUATIONS OF THE FORM &[F(X)%]+% ; [G(V)%JL]+&[H(Z)%]=AE^ . 413 6.2.2. EQUATIONSOFTHEFORMA 1 ^(E A -\|^)+A 2 ^(E A ^^)+A3^(E A ^^)=6 E ^ 416 6.3. THREE-DIMENSIONAL EQUATIONS INVOLVING ARBITRARY FUNCTIONS 420 6.3.1. HEAT AND MASS TRANSFER EQUATIONS OF THE FORM ^ [/I(A;)FJ] + ^ [MY)$] + 420 6.3.2. HEAT AND MASS TRANSFER EQUATIONS WITH COMPLICATING FACTORS 423 6.3.3. OTHER EQUATIONS 426 6.4. EQUATIONS WITH N INDEPENDENT VARIABLES 428 6.4.1. EQUATIONS OF THE FORM -^ [F 1 ( XL )J-]+. .+^_ [/(») \| ] =G(X U .,X N ,W) 428 6.4.2. OTHER EQUATIONS ? ? 430 7. EQUATIONS INVOLVING MIXED DERIVATIVES AND SOME OTHER EQUATIONS 433 7.1. EQUATIONS LINEAR IN THE MIXED DERIVATIVE 433 7.1.1. CALOGERO EQUATION 433 7.1.2. KHOKHLOV-ZABOLOTSKAYA EQUATION 435 7.1.3. EQUATION OF UNSTEADY TRANSONIC GAS FLOWS 440 ,- 7.1.4. E Q UATIONSOFTHEFORM^^-^^ = F(X, ? /,^,^) 443 7.1.5. OTHER EQUATIONS WITH TWO INDEPENDENT VARIABLES 445 7.1.6. OTHER EQUATIONS WITH THREE INDEPENDENT VARIABLES 448 7.2. EQUATIONS QUADRATIC IN THE HIGHEST DERIVATIVES 449 7.2.1. EQUATIONS OFTHE FORM F^ \|^F = F(X, Y) 449 7.2.2. MONGE-AMPEREEQUATION(^) 2 -^0=F(^Y) 451 7.2.3. EQUATIONSOFTHEFORM(^) 2 -^^=F(A ; , ?/ , U ;,^,\|^) 461 X CONTENTS 7.2.4. EQUATIONS OF THE FORM {^F = F(X, 2/)FF \|^ + G(X, Y) 465 7.2.5. OTHER EQUATIONS 469 7.3. BELLMAN TYPE EQUATIONS AND RELATED EQUATIONS 472 7.3.1. EQUATIONS WITH QUADRATIC NONLINEARITIES 472 7.3.2. EQUATIONS WITH POWER-LAW NONLINEARITIES 475 8. SECOND-ORDER EQUATIONS OF GENERAL FORM 479 8.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 479 8.1.1. EQUATIONS OFTHE FORM FF = F(W, F^, FF) 479 8.1.2. EQUATIONS OFTHE FORM §F = F(T,W, F^, &$) 486 8.1.3. EQUATIONS OFTHE FORM FF = F(X, W, §^, \|^) 490 8.1.4. EQUATIONS OFTHE FORM FF = F(X, T, W, F^, ^F) 494 8.1.5. EQUATIONS OF THE FORM F (A;, T,W, FF'-FJ, FS^F) =0 499 8.1.6. EQUATIONS WITH THREE INDEPENDENT VARIABLES 500 8.2. EQUATIONS INVOLVING TWO OR MORE SECOND DERIVATIVES 501 8.2.1. EQUATIONS OFTHE FORM ^$- = F(W, F^, F^) 501 8.2.2. EQUATIONS OF THE FORM ^ = F(X,T,W,^,^-,^R) 505 8.2.3. EQUATIONS LINEAR IN THE MIXED DERIVATIVE 508 8.2.4. EQUATIONS WITH TWO INDEPENDENT VARIABLES, NONLINEAR IN TWO OR MORE HIGHEST DERIVATIVES 509 8.2.5. EQUATIONS WITH N INDEPENDENT VARIABLES 512 9. THIRD-ORDER EQUATIONS 515 9.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 515 9.1.1. KORTEWEG-DE VRIES EQUATION FF + AF^F + BW^ =0 515 9.1.2. CYLINDRICAL, SPHERICAL, AND MODIFIED KORTEWEG-DE VRIES EQUATIONS 521 9.1.3. GENERALIZED KORTEWEG-DE VRIES EQUATION FF + A^F + /(W)FJ =0 524 9.1.4. EQUATIONS REDUCIBLE TO THE KORTEWEG-DE VRIES EQUATION 526 9.1.5. EQUATIONS OFTHE FORM FF + A^F + F(W, §^) = 0 529 9.1.6. EQUATIONS OFTHE FORM FF + A^F + F(X, T, W, F^) = 0 530 9.1.7. BURGERS-KORTEWEG-DE VRIES EQUATION AND OTHER EQUATIONS 532 9.2. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 536 9.2.1. EQUATIONS WITH QUADRATIC NONLINEARITIES 536 9.2.2. OTHER EQUATIONS 539 9.3. HYDRODYNAMIC BOUNDARY LAYER EQUATIONS 540 9.3.1. STEADY HYDRODYNAMIC BOUNDARY LAYER EQUATIONS FOR A NEWTONIAN FLUID 540 9.3.2. STEADY BOUNDARY LAYER EQUATIONS FOR NON-NEWTONIAN FLUIDS 547 9.3.3. UNSTEADY BOUNDARY LAYER EQUATIONS FOR A NEWTONIAN FLUID 553 9.3.4. UNSTEADY BOUNDARY LAYER EQUATIONS FOR NON-NEWTONIAN FLUIDS 564 9.3.5. RELATED EQUATIONS 568 9.4. EQUATIONS OF MOTION OF IDEAL FLUID (EULER EQUATIONS) 570 9.4.1. STATIONARY EQUATIONS 570 9.4.2. NONSTATIONARY EQUATIONS 574 9.5. OTHER THIRD-ORDER NONLINEAR EQUATIONS 580 9.5.1. EQUATIONS INVOLVING SECOND-ORDER MIXED DERIVATIVES 580 9.5.2. EQUATIONS INVOLVING THIRD-ORDER MIXED DERIVATIVES 583 9.5.3. EQUATIONS INVOLVING ^ AND ^ 587 CONTENTS XI 10. FOURTH-ORDER EQUATIONS 589 10.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 589 10.1.1. EQUATIONS OF THE FORM F =A\|F+F(I,T ( ;,F) 589 10.1.2. OTHER EQUATIONS 593 10.2. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 595 10.2.1. BOUSSINESQ EQUATION AND ITS MODIFICATIONS 595 10.2.2. EQUATIONS WITH QUADRATIC NONLINEARITIES 600 10.2.3. OTHER EQUATIONS 603 10.3. EQUATIONS INVOLVING MIXED DERIVATIVES 605 10.3.1. KADOMTSEV-PETVIASHVILI EQUATION 605 10.3.2. STATIONARY HYDRODYNAMIC EQUATIONS (NAVIER-STOKES EQUATIONS) 607 10.3.3. NONSTATIONARY HYDRODYNAMIC EQUATIONS (NAVIER-STOKES EQUATIONS) 616 10.3.4. OTHER EQUATIONS 628 11. EQUATIONS OF HIGHER ORDERS 631 11.1. EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T AND LINEAR IN THE HIGHEST DERIVATIVE . 631 11.1.1. FIFTH-ORDER EQUATIONS 631 11.1.2. EQUATIONS OF THE FORM &$* =A% + F(X,T,W) 633 11.1.3. EQUATIONS OFTHE FORM FF = AF^F + /(W)F^ 635 11.1.4. EQUATIONS OF THE FORM FF = AF^F + F(X,T,W)^ + G(X,T,W) 637 11.1.5. EQUATIONS OF THE FORM FF = A§^R +F(X,T,W, F^) 640 11.1.6. EQUATIONS OF THE FORM FF =A% +F(X,T,W, F^---,FJIF) 645 11.1.7. EQUATIONS OFTHE FORM FF = AW^- + F(X, T, W)\|F + G(X, T,W) 64 7 11.1.8. OTHER EQUATIONS 648 11.2. GENERAL FORM EQUATIONS INVOLVING THE FIRST DERIVATIVE IN T 651 11.2.1. EQUATIONS OFTHE FORM §F = F(W, §^,., FJ) 651 11.2.2. EQUATIONS OFTHE FORM §F = F(T, W, %,., -GF) 656 11.2.3. EQUATIONS OF THE FORM %?= F(X,W, %,., %) 659 11.2.4. EQUATIONS OF THE FORM =F(X,T, W, %,., -F) 662 11.3. EQUATIONS INVOLVING THE SECOND DERIVATIVE IN T 666 11.3.1. EQUATIONS OFTHE FORM ^ = AF^ + F(X,T, W) 666 11.3.2. EQUATIONS OFTHE FORM \| ^ = A^-+F(X,T,W, F^-) 667 11.3.3. EQUATIONS OF THE FORM ^-= A^-+ F(X,T,W, FJ, * * , "IJSF) 671 11.3.4. EQUATIONS OF THE FORM ^ = AW^-+ F(X,T,W)^+G(X,T,W) 673 11.3.5. EQUATIONS OF THE FORM ^=F(X,T, W, F^,-* ) 675 11.4. OTHER EQUATIONS 676 11.4.1'. EQUATIONS INVOLVING MIXED DERIVATIVES 676 11.4.2. EQUATIONS INVOLVING F^ AND F^F 680 SUPPLEMENTS. EXACT METHODS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS . 683 S.I. CLASSIFICATION OF SECOND-ORDER SEMILINEAR PARTIAL DIFFERENTIAL EQUATIONS IN TWO INDEPENDENT VARIABLES 683 5.1.1. TYPES OF EQUATIONS. CHARACTERISTIC EQUATION 683 5.1.2. CANONICAL FORM OF PARABOLIC EQUATIONS 683 5.1.3. CANONICAL FORM OF HYPERBOLIC EQUATIONS 684 5.1.4. CANONICAL FORM OF ELLIPTIC EQUATIONS 684 XII CONTENTS 5.2. TRANSFORMATIONS OF EQUATIONS OF MATHEMATICAL PHYSICS 685 5.2.1. POINT TRANSFORMATIONS 685 5.2.2. HODOGRAPH TRANSFORMATION 686 5.2.3. CONTACT TRANSFORMATIONS. LEGENDRE AND EULER TRANSFORMATIONS 688 5.2.4. BACKHAND TRANSFORMATIONS. DIFFERENTIAL SUBSTITUTIONS 690 5.3. TRAVELING-WAVE SOLUTIONS AND SELF-SIMILAR SOLUTIONS. SIMILARITY METHODS 693 5.3.1. PRELIMINARY REMARKS 693 5.3.2. TRAVELING-WAVE SOLUTIONS. INVARIANCE OF EQUATIONS UNDER TRANSLATIONS 694 5.3.3. SELF-SIMILAR SOLUTIONS. INVARIANCE OF EQUATIONS UNDER SCALING TRANSFORMATIONS 695 5.3.4. EXPONENTIAL SELF-SIMILAR SOLUTIONS. EQUATIONS INVARIANT UNDER COMBINED TRANSLATION AND SCALING 696 5.4. METHOD OF GENERALIZED SEPARATION OF VARIABLES 698 5.4.1. INTRODUCTION 698 5.4.2. STRUCTURE OF GENERALIZED SEPARABLE SOLUTIONS 700 5.4.3. SOLUTION OF FUNCTIONAL-DIFFERENTIAL EQUATIONS BY DIFFERENTIATION 701 5.4.4. SOLUTION OF FUNCTIONAL-DIFFERENTIAL EQUATIONS BY SPLITTING 705 5.4.5. SIMPLIFIED SCHEME FOR CONSTRUCTING GENERALIZED SEPARABLE SOLUTIONS 709 5.4.6. TITOV-GALAKTIONOV METHOD 710 5.5. METHOD OF FUNCTIONAL SEPARATION OF VARIABLES 713 5.5.1. STRUCTURE OF FUNCTIONAL SEPARABLE SOLUTIONS 713 5.5.2. SPECIAL FUNCTIONAL SEPARABLE SOLUTIONS 713 5.5.3. DIFFERENTIATION METHOD 718 5.5.4. SPLITTING METHOD. REDUCTION TO A FUNCTIONAL EQUATION WITH TWO VARIABLES 721 5.5.5. SOLUTIONS OF SOME NONLINEAR FUNCTIONAL EQUATIONS AND THEIR APPLICATIONS . 723 5.6. GENERALIZED SIMILARITY REDUCTIONS OF NONLINEAR EQUATIONS 728 5.6.1. CLARKSON-KRUSKAL DIRECT METHOD: A SPECIAL FORM FOR SIMILARITY REDUCTION . 728 5.6.2. CLARKSON-KRUSKAL DIRECT METHOD: THE GENERAL FORM FOR SIMILARITY REDUCTION . 731 5.6.3. SOME MODIFICATIONS AND GENERALIZATIONS 732 5.7. GROUP ANALYSIS METHODS 735 5.7.1. CLASSICAL METHOD FOR SYMMETRY REDUCTIONS 735 5.7.2. NONCLASSICAL METHOD FOR SYMMETRY REDUCTIONS 744 5.8. DIFFERENTIAL CONSTRAINTS METHOD 747 5.8.1. DESCRIPTION OFTHE METHOD 747 5.8.2. FIRST-ORDER DIFFERENTIAL CONSTRAINTS 749 5.8.3. SECOND- AND HIGHER-ORDER DIFFERENTIAL CONSTRAINTS 754 5.8.4. CONNECTION BETWEEN THE DIFFERENTIAL CONSTRAINTS METHOD AND OTHER METHODS . 756 5.9. PAINLEVE TEST FOR NONLINEAR EQUATIONS OF MATHEMATICAL PHYSICS 758 5.9.1. MOVABLE SINGULARITIES OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 758 5.9.2. SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS WITH A MOVABLE POLE. DESCRIPTION OF THE METHOD 760 5.9.3. EXAMPLES OF THE^PAINLEVE TEST APPLICATIONS 761 5.10. INVERSE SCATTERING METHOD 764 5.10.1. LAX PAIR METHOD 764 5.10.2. METHOD BASED ON THE COMPATIBILITY CONDITION FOR TWO LINEAR EQUATIONS . 766 5.10.3. METHOD BASED ON LINEAR INTEGRAL EQUATIONS 767 S.I 1. CONSERVATION LAWS 769 S.I 1.1. BASIC DEFINITIONS AND EXAMPLES 769 S.I 1.2. EQUATIONS ADMITTING VARIATIONAL FORMULATION. NOETHERIAN SYMMETRIES 770 CONTENTS XIU S.12. HYPERBOLIC SYSTEMS OF QUASILINEAR EQUATIONS 772 5.12.1. CONSERVATION LAWS. SOME EXAMPLES 772 5.12.2. CAUCHY PROBLEM, RIEMANN PROBLEM, AND INITIAL-BOUNDARY VALUE PROBLEM . 773 5.12.3. CHARACTERISTIC LINES. HYPERBOLIC SYSTEMS. RIEMANN INVARIANTS 773 5.12.4. SELF-SIMILAR CONTINUOUS SOLUTIONS. RAREFACTION WAVES 777 5.12.5. SHOCKWAVES. RANKINE-HUGONIOT JUMP CONDITIONS 779 5.12.6. EVOLUTIONARY SHOCKS. LAX CONDITION (VARIOUS FORMULATIONS) 780 5.12.7. SOLUTIONS FOR THE RIEMANN PROBLEM 782 5.12.8. INITIAL-BOUNDARY VALUE PROBLEMS OF SPECIAL FORM 786 5.12.9. EXAMPLES OF NONSTRICT HYPERBOLIC SYSTEMS 786 REFERENCES 791 INDEX 809
any_adam_object	1
any_adam_object_boolean	1
author	Poljanin, Andrej D. 1951-
author_GND	(DE-588)128391251 (DE-588)12839126X
author_facet	Poljanin, Andrej D. 1951-
author_role	aut
author_sort	Poljanin, Andrej D. 1951-
author_variant	a d p ad adp
building	Verbundindex
bvnumber	BV021518442
callnumber-first	Q - Science
callnumber-label	QA372
callnumber-raw	QA372
callnumber-search	QA372
callnumber-sort	QA 3372
callnumber-subject	QA - Mathematics
classification_rvk	QH 150 SK 540
ctrlnum	(OCoLC)52766184 (DE-599)BVBBV021518442
dewey-full	515/.355
dewey-hundreds	500 - Natural sciences and mathematics
dewey-ones	515 - Analysis
dewey-raw	515/.355
dewey-search	515/.355
dewey-sort	3515 3355
dewey-tens	510 - Mathematics
discipline	Mathematik Wirtschaftswissenschaften
discipline_str_mv	Mathematik Wirtschaftswissenschaften
format	Book
fullrecord	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>02375nam a2200565zc 4500</leader><controlfield tag="001">BV021518442</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">20061121 </controlfield><controlfield tag="007">t</controlfield><controlfield tag="008">060320s2004 xxuad\|\| \|\|\|\| 00\|\|\| eng d</controlfield><datafield tag="010" ind1=" " ind2=" "><subfield code="a">2003058473</subfield></datafield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">1584883553</subfield><subfield code="c">acidfree paper</subfield><subfield code="9">1-58488-355-3</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)52766184</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV021518442</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">aacr</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="044" ind1=" " ind2=" "><subfield code="a">xxu</subfield><subfield code="c">US</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-29T</subfield><subfield code="a">DE-20</subfield><subfield code="a">DE-824</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-83</subfield><subfield code="a">DE-11</subfield></datafield><datafield tag="050" ind1=" " ind2="0"><subfield code="a">QA372</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">515/.355</subfield><subfield code="2">22</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">QH 150</subfield><subfield code="0">(DE-625)141534:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">SK 540</subfield><subfield code="0">(DE-625)143245:</subfield><subfield code="2">rvk</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">35-00</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">35F20</subfield><subfield code="2">msc</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">35G20</subfield><subfield code="2">msc</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Poljanin, Andrej D.</subfield><subfield code="d">1951-</subfield><subfield code="e">Verfasser</subfield><subfield code="0">(DE-588)128391251</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Handbook of nonlinear partial differential equations</subfield><subfield code="c">Andrei D. Polyanin ; Valentin F. Zaitsev</subfield></datafield><datafield tag="246" ind1="1" ind2="3"><subfield code="a">Nonlinear partial differential equations</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Boca Raton, Fla. [u.a.]</subfield><subfield code="b">Chapman & Hall/CRC</subfield><subfield code="c">2004</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XX, 814 S.</subfield><subfield code="b">Ill., graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Mécanique non linéaire - Mathématiques</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Équations différentielles non linéaires - Solutions numériques</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Mathematik</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Differential equations, Nonlinear</subfield><subfield code="x">Numerical solutions</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nonlinear mechanics</subfield><subfield code="x">Mathematics</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Numerische Mathematik</subfield><subfield code="0">(DE-588)4042805-9</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Nichtlineare partielle Differentialgleichung</subfield><subfield code="0">(DE-588)4128900-6</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="650" ind1="0" ind2="7"><subfield code="a">Nichtlineare Mechanik</subfield><subfield code="0">(DE-588)4042095-4</subfield><subfield code="2">gnd</subfield><subfield code="9">rswk-swf</subfield></datafield><datafield tag="689" ind1="0" ind2="0"><subfield code="a">Nichtlineare partielle Differentialgleichung</subfield><subfield code="0">(DE-588)4128900-6</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="0" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="689" ind1="1" ind2="0"><subfield code="a">Nichtlineare Mechanik</subfield><subfield code="0">(DE-588)4042095-4</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2="1"><subfield code="a">Numerische Mathematik</subfield><subfield code="0">(DE-588)4042805-9</subfield><subfield code="D">s</subfield></datafield><datafield tag="689" ind1="1" ind2=" "><subfield code="5">DE-604</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Zajcev, Valentin F.</subfield><subfield code="e">Sonstige</subfield><subfield code="0">(DE-588)12839126X</subfield><subfield code="4">oth</subfield></datafield><datafield tag="856" ind1="4" ind2=" "><subfield code="u">http://www.e-streams.com/es0710/es0710*9.html</subfield><subfield code="3">Book review (E-STREAMS)</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="m">HEBIS Datenaustausch Darmstadt</subfield><subfield code="q">application/pdf</subfield><subfield code="u">http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014734968&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA</subfield><subfield code="3">Inhaltsverzeichnis</subfield></datafield><datafield tag="999" ind1=" " ind2=" "><subfield code="a">oai:aleph.bib-bvb.de:BVB01-014734968</subfield></datafield></record></collection>
id	DE-604.BV021518442
illustrated	Illustrated
index_date	2024-07-02T14:21:40Z
indexdate	2024-07-09T20:37:39Z
institution	BVB
isbn	1584883553
language	English
lccn	2003058473
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-014734968
oclc_num	52766184
open_access_boolean
owner	DE-29T DE-20 DE-824 DE-634 DE-83 DE-11
owner_facet	DE-29T DE-20 DE-824 DE-634 DE-83 DE-11
physical	XX, 814 S. Ill., graph. Darst.
publishDate	2004
publishDateSearch	2004
publishDateSort	2004
publisher	Chapman & Hall/CRC
record_format	marc
spelling	Poljanin, Andrej D. 1951- Verfasser (DE-588)128391251 aut Handbook of nonlinear partial differential equations Andrei D. Polyanin ; Valentin F. Zaitsev Nonlinear partial differential equations Boca Raton, Fla. [u.a.] Chapman & Hall/CRC 2004 XX, 814 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mécanique non linéaire - Mathématiques Équations différentielles non linéaires - Solutions numériques Mathematik Differential equations, Nonlinear Numerical solutions Nonlinear mechanics Mathematics Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare Mechanik (DE-588)4042095-4 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s DE-604 Nichtlineare Mechanik (DE-588)4042095-4 s Numerische Mathematik (DE-588)4042805-9 s Zajcev, Valentin F. Sonstige (DE-588)12839126X oth http://www.e-streams.com/es0710/es0710*9.html Book review (E-STREAMS) HEBIS Datenaustausch Darmstadt application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014734968&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Poljanin, Andrej D. 1951- Handbook of nonlinear partial differential equations Mécanique non linéaire - Mathématiques Équations différentielles non linéaires - Solutions numériques Mathematik Differential equations, Nonlinear Numerical solutions Nonlinear mechanics Mathematics Numerische Mathematik (DE-588)4042805-9 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Nichtlineare Mechanik (DE-588)4042095-4 gnd
subject_GND	(DE-588)4042805-9 (DE-588)4128900-6 (DE-588)4042095-4
title	Handbook of nonlinear partial differential equations
title_alt	Nonlinear partial differential equations
title_auth	Handbook of nonlinear partial differential equations
title_exact_search	Handbook of nonlinear partial differential equations
title_exact_search_txtP	Handbook of nonlinear partial differential equations
title_full	Handbook of nonlinear partial differential equations Andrei D. Polyanin ; Valentin F. Zaitsev
title_fullStr	Handbook of nonlinear partial differential equations Andrei D. Polyanin ; Valentin F. Zaitsev
title_full_unstemmed	Handbook of nonlinear partial differential equations Andrei D. Polyanin ; Valentin F. Zaitsev
title_short	Handbook of nonlinear partial differential equations
title_sort	handbook of nonlinear partial differential equations
topic	Mécanique non linéaire - Mathématiques Équations différentielles non linéaires - Solutions numériques Mathematik Differential equations, Nonlinear Numerical solutions Nonlinear mechanics Mathematics Numerische Mathematik (DE-588)4042805-9 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd Nichtlineare Mechanik (DE-588)4042095-4 gnd
topic_facet	Mécanique non linéaire - Mathématiques Équations différentielles non linéaires - Solutions numériques Mathematik Differential equations, Nonlinear Numerical solutions Nonlinear mechanics Mathematics Numerische Mathematik Nichtlineare partielle Differentialgleichung Nichtlineare Mechanik
url	http://www.e-streams.com/es0710/es0710*9.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014734968&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT poljaninandrejd handbookofnonlinearpartialdifferentialequations AT zajcevvalentinf handbookofnonlinearpartialdifferentialequations AT poljaninandrejd nonlinearpartialdifferentialequations AT zajcevvalentinf nonlinearpartialdifferentialequations

Holdings

There is no print copy available.

Interlibrary loan Place Request Caution: Not in THWS collection! Indexes

MARC

Record in the Search Index

There is no print copy available.

Similar Items