Approach your problems from the right end and begin with the answers. Then one day, perhaps you will find the final answer. "The Hermit Clad In Crane Feathers" In R. van Gullk's The Chinese Haze Hurders. It Isn't that they can't see the solution. It IS that they can't see the problem. G. K. Chesterton. The Scandal of Father Brown. "The POint of a Pin." Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of k now ledge of m athemat i cs and re I ated fie I ds does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, COding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And In addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely Integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the eXisting classificatIOn schemes.
I: A Summary of the Decomposition Method.- 1: The Decomposition Method.- 1.1 Introduction.- 1.2 Summary of the Decomposition Method.- 1.3 Generation of the An Polynomials.- 1.4 The An for Differential Nonlinear Operators.- 1.5 Convenient Computational Forms for the An Polynomials.- 1.6 Calculation of the An Polynomials for Composite Nonlinearities.- 1.7 New Generating Schemes - the Accelerated Polynomials.- 1.8 Convergence of the An Polynomials.- 1.9 Euler's Transformation.- 1.9.1 Solution of a Differential Equation by Decomposition.- 1.9.2 Application of Euler Transform to Decomposition Solution.- 1.9.3 Numerical Comparison.- 1.9.4 Solution of Linearized Equation.- 1.10 On the Validity of the Decomposition Solution.- 2: Effects of Nonlinearity and Linearization.- 2.1 Introduction.- 2.2 Effects on Simple Systems.- 2.3 Effects on SOlution for the General Case.- 3: Research on Initial and Boundary Conditions for Differential and Partial Differential Equations.- II: Applications to the Equations of Physics.- 4: The Burger's Equation.- 5: Heat Flow and Diffusion.- 5.1 One-Dimensional Case.- 5.2 Two-Dimensional Case.- 5.3 Three-Dimensional Case.- 5.4 Some Examples.- 5.5 Heat Conduction in an Inhomogeneous Rod.- 5.6 Nonlinear Heat Conduction.- 5.7 Heat Conduction Equation with Discontinuous Coefficients.- 5.8 Nonlinear Boundary Conditions.- 5.9 Comparisons.- 5.10 Uncoupled Equations with Coupled Conditions.- 6: Nonlinear Oscillations in Physical Systems.- 6.1 Oscillatory Motion.- 6.2 Pendulum Problem.- 6.3 The Duffing and Van der Pol Oscillators.- 7: The KdV Equation.- 8: The Benjamin-Ono Equation.- 9: The Sine-Gordon Equation.- 10: The Nonlinear Schrodinger Equation and the Generalized Schrodinger Equation.- 10.1 Nonlinear Schrodinger Equation.- 10.2 Generalized Schrodinger Equation.- 10.3 Schrodinger's Equation with a Quartic Potential.- 11: Nonlinear Plasmas.- 12: The Tricomi Problem.- 13: The Initial-Value Problem for the Wave Equation.- ChaDter 14: Nonlinear Dispersive or Dissipative Waves.- 14.1 Wave Propagation in Nonlinear Media.- 14.2 Dissipative Wave Equations.- 15: The Nonlinear Klein-Gordon Equation.- 16: Analysis of Model Equations of Gas Dynamics.- 17: A New Approach to the Efinger Model for a Nonlinear Quantum Theory for Gravitating Particles.- 18: The Navier-Stokes Equations.- Epilogue.