This volume shows how to construct solutions of numerous nonlinear PDEs quickly and easily. The text discusses comparisons and relationships between the varying solutions, methods and approaches.
The emphasis of the book is given to how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. The reader can learn to understand and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, lend a deeper understanding of the subject.
Among a big number of CAS, we choose the two systems, Maple and Mathematica, that are used worldwide by students, research mathematicians, scientists, and engineers. As in the our previous books, we propose the idea to use in parallel both systems, Maple and Mathematica, since in many research problems frequently it is required to compare independent results obtained by using different computer algebra systems, Maple and/or Mathematica, at all stages of the solution process. One of the main points (related to CAS) is based on the implementation of a whole solution method (e. g. starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and comparing the numerical solution obtained with other types of solutions considered in the book, e. g. with asymptotic solution).
Inhaltsverzeichnis
1 Introduction 1. 1 Basic Concepts 2 Algebraic Approach 2. 1 Point Transformations 2. 2 Contact Transformations 2. 3 Transformations Relating Differential Equations 2. 4 Linearizing and Bilinearizing Transformations 2. 5 Reductions of Nonlinear PDEs 2. 6 Separation of Variables 2. 7 Transformation Groups 2. 8 Nonlinear Systems 3 Geometric-Qualitative Approach 3. 1 Method of Characteristics 3. 2 Generalized Method of Characteristics 3. 3 Qualitative Analysis 4 General Analytical Approach. Integrability 4. 1 Painlevé Test and Integrability 4. 2 Complete Integrability. Evolution Equations 4. 3 Nonlinear Systems. Integrability Conditions 5 Approximate Analytical Approach 5. 1 Adomian Decomposition Method 5. 2 Asymptotic Expansions. Perturbation Methods 6 Numerical Approach 6. 1 Embedded Numerical Methods 6. 2 Finite DifferenceMethods 7 Analytical-Numerical Approach 7. 1 Method of Lines 7. 2 Spectral Collocation Method; A Brief Description of Maple A. 1 Introduction A. 2 Basic Concepts A. 3 Maple Language B Brief Description of Mathematica B. 1 Introduction B. 2 Basic Concepts B. 3 Mathematica Language; References, Index
