Preface |
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vii | |
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1 | (5) |
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Elements of Symmetry Analysis |
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6 | (31) |
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6 | (15) |
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6 | (8) |
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14 | (1) |
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14 | (7) |
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21 | (16) |
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Representation of a Lie Algebra |
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26 | (3) |
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Properties of Lie Algebras |
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29 | (8) |
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37 | (59) |
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Ordinary and Partial Derivatives |
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37 | (8) |
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45 | (5) |
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50 | (2) |
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52 | (2) |
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54 | (5) |
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59 | (15) |
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59 | (4) |
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63 | (2) |
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65 | (1) |
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Algorithm Used in the Calculus of Variations |
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65 | (4) |
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Euler Operator for q Dependent Variables |
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69 | (2) |
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Euler Operator for q + p Dimensions |
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71 | (3) |
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Prolongation of Vector Fields |
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74 | (22) |
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Symmetries of Ordinary Differential Equations |
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96 | (120) |
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96 | (2) |
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Symmetry Transformations of Functions |
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98 | (25) |
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98 | (5) |
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Infinitesimal Transformations |
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103 | (4) |
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107 | (5) |
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112 | (5) |
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Prolongation of Transformations |
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117 | (6) |
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Symmetry Transformations of Differential Equations |
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123 | (25) |
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Definition of a Symmetry Group |
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123 | (1) |
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Main Properties of Symmetry Groups |
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124 | (1) |
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Calculation of the Infinitesimal Symmetries |
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125 | (14) |
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139 | (9) |
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Analysis of Ordinary Differential Equations |
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148 | (68) |
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148 | (26) |
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Second-Order Ordinary Differential Equations |
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174 | (27) |
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Higher-Order Ordinary Differential Equations |
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201 | (15) |
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Point Symmetries of Partial Differential Equations |
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216 | (149) |
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216 | (1) |
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Lie's Theory Used in MathLie |
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217 | (3) |
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Invariance Based on Frechet Derivatives |
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220 | (2) |
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Application of the Theory |
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222 | (35) |
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Calculation of Prolongations |
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223 | (6) |
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Derivation of Determining Equations |
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229 | (6) |
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Interactive Solution of Determining Equations |
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235 | (8) |
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243 | (14) |
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Similarity Reduction of Partial Differential Equations |
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257 | (25) |
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282 | (83) |
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282 | (1) |
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The Earthworm's New Year Problem |
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282 | (7) |
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Single Flux Line in Superconductors |
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289 | (7) |
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The Korteweg-de Vries Equation and its Generalizations |
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296 | (8) |
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Stokes' Solution of the Creeping Flow |
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304 | (7) |
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Two-Dimensional Boundary-Layer Flows: Group Classification |
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311 | (12) |
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323 | (7) |
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330 | (10) |
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340 | (6) |
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346 | (9) |
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The First Atomic Explosion |
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355 | (10) |
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Non-Classical Symmetries of Partial Differential Equations |
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365 | (27) |
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365 | (1) |
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Mathematical Background of the Non-classical Method |
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366 | (4) |
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Applications of the Non-classical Method |
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370 | (22) |
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370 | (7) |
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377 | (6) |
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The Fokker-Planck Equation |
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383 | (9) |
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Potential Symmetries of Partial Differential Equations |
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392 | (12) |
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392 | (1) |
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Basics of Potential Symmetries |
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393 | (1) |
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Calculation of Potential Symmetries |
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394 | (4) |
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Applications of Potential Symmetries |
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398 | (6) |
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A Non-linear Reaction Diffusion Equation |
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398 | (1) |
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Cylindrical Korteweg-de Vries Equation |
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399 | (3) |
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402 | (2) |
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Approximate Symmetries of Partial Differential Equations |
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404 | (20) |
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404 | (1) |
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405 | (1) |
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One-Parameter Approximation Group |
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405 | (2) |
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Approximate Group Generator |
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407 | (1) |
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The Determining Equations and an Algorithm of Calculation |
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408 | (2) |
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410 | (14) |
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410 | (9) |
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Perturbed Korteweg-de Vies Equation |
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419 | (5) |
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424 | (33) |
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424 | (1) |
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Elements of Generalized Symmetries |
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425 | (2) |
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Algorithm for Calculation of Generalized Symmetries |
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427 | (1) |
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428 | (5) |
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428 | (2) |
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Potential Burgers Equation |
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430 | (1) |
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Generalized Korteweg-de Vries Equations |
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431 | (1) |
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Coupled System of Wave Equations |
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432 | (1) |
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Second-Order ODEs and the Euler-Lagrange Equation |
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433 | (4) |
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Generalized Symmetries and Second-Order ODEs |
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434 | (2) |
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436 | (1) |
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Algorithm for Conservation Laws of Second-Order ODEs |
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437 | (1) |
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Examples for Second-Order ODEs |
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438 | (19) |
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438 | (8) |
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Two-Dimensional Quartic Oscillators |
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446 | (6) |
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452 | (5) |
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Solution of Coupled Linear Partial Differential Equations |
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457 | (26) |
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457 | (1) |
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General Canonical Form of PDEs |
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458 | (13) |
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Application of the General Canonical Form Algorithm |
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462 | (9) |
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471 | (4) |
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472 | (1) |
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Integrating ODEs and Pseudo-ODEs |
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473 | (1) |
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473 | (1) |
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474 | (1) |
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Simplification of Equations |
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475 | (4) |
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475 | (1) |
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476 | (1) |
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Reducing the Number of Dependent Variables |
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477 | (2) |
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479 | (4) |
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Liouville-Type Equation of Quantum Gravity Theory |
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480 | (3) |
Appendix |
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483 | (10) |
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Marius Sophus Lie: A Mathematician's Life |
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483 | (4) |
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List of Key Symbols Used in Mathematica |
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487 | (1) |
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488 | (5) |
References |
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493 | (10) |
Index for MathLie and Mathematica Functions |
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503 | (2) |
Subject Index |
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505 | |