دانلود کتاب Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

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Contents......Page 6
Preface......Page 10
0.1. Brief Introduction to Lie Group Analysis of Differential Equations......Page 18
0.2. Preliminaries: Heuristic Approach in Examples......Page 73
1 Finite Differences and Transformation Groups in Space of Discrete Variables......Page 82
1.1. The Taylor Group and Introduction of Finite-Difference Derivatives......Page 83
1.2. Difference Analog of the Leibniz Rule and Discrete Differentiation Formulas......Page 90
1.3. Invariant Difference Meshes......Page 92
1.4. Transformations Preserving the Geometric Meaning of Finite-Difference Derivatives; Prolongation Formulas......Page 108
1.5. Newton’s Group and Lagrange’s Formula......Page 115
1.6. Commutation Properties and Factorization of Group Operators on Uniform Difference Meshes......Page 120
1.7. Finite-Difference Integration and Prolongation of the Mesh Space to Nonlocal Variables......Page 127
1.8. Change of Variables in the Mesh Space......Page 131
2.1. An Invariance Criterion for Finite-Difference Equations on the Difference Mesh......Page 136
2.2. Symmetry Preservation in Difference Modeling: Method of Finite-Difference Invariants......Page 146
2.3. Examples of Construction of Difference Models Preserving the Symmetry of the Original Continuous Models......Page 150
3.1. First-Order Invariant Difference Equations and Lattices......Page 160
3.2. Invariant Second-Order Difference Equations and Lattices......Page 163
4.1. Symmetry Preserving Difference Schemes for the Nonlinear Heat Equation with a Source......Page 182
4.2. Symmetry Preserving Difference Schemes for the Linear Heat Equation......Page 201
4.3. Invariant Difference Models for the Burgers Equation......Page 214
4.4. Invariant Difference Model of the Heat Equation with Heat Flux Relaxation......Page 222
4.5. Invariant Difference Model of the Korteweg–de Vries Equation......Page 224
4.6. Invariant Difference Model of the Nonlinear Schroding¨ er Equation......Page 230
5.1. Second-Order Ordinary Delay Differential Equations......Page 232
5.2. Partial Delay Differential Equations......Page 235
5.3. Symmetry of Differential-Difference Equations......Page 237
6.1. Discrete Representation of Euler’s Operator......Page 242
6.2. Criterion for the Invariance of Difference Functionals......Page 246
6.3. Invariance of Difference Euler Equations......Page 250
6.4. Variation of Difference Functional and Quasi-Extremal Equations......Page 252
6.5. Invariance of Global Extremal Equations and Properties of Quasi-Extremal Equations......Page 259
6.6. Conservation Laws for Difference Equations......Page 262
6.7. Noether-Type Identities and Difference Analog of Noether’s Theorem......Page 265
6.8. Necessary and Suf cient Conditions for Global Extremal Equations to Be Invariant......Page 271
6.9. Applications of Lagrangian Formalism to Second-Order Difference Equations......Page 273
6.10. Moving Mesh Schemes for the Nonlinear Schroding¨ er Equation......Page 287
7.1. Discrete Legendre Transform......Page 302
7.2. Variational Statement of the Difference Hamiltonian Equations......Page 304
7.3. Symplecticity of Difference Hamiltonian Equations......Page 306
7.4. Invariance of the Hamiltonian Action......Page 308
7.5. Difference Hamiltonian Identity and Noether-Type Theorem for Difference Hamiltonian Equations......Page 309
7.6. Invariance of Difference Hamiltonian Equations......Page 310
7.7. Examples......Page 312
8 Discrete Representation of Ordinary Differential Equations with Symmetries......Page 318
8.1. The Discrete Representation of ODE as a Series......Page 319
8.2. Three-Point Exact Schemes for Nonlinear ODE......Page 324
Bibliography......Page 332
Index......Page 344