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).
1 General Analytical Approach 1.1 Classification of PDEs 1.2 Analytical Solutions 1.3 Transformations of Nonlinear PDEs 1.4 Traveling-Wave Solutions 1.5 Self-Similar Solutions 1.6 Separable Solutions 1.7 Differential constraints 1.8 Applications of the Painlevé Test 2 Geometric Approach 2.1 Method of Characteristics 2.2 Generalized Method of Characteristics 2.3 Method of Characteristics and Conservation Laws 2.4 Direction Fields 2.5 Integral Surfaces 2.6 Poincaré Sections 3 Algebraic Approach 3.1 One-Parameter Group of Transformations 3.2 Group Analysis for Nonlinear PDEs 3.3 Invariant Solutions 4 Transform Approach 4.1 Inverse Scattering Transform Approach 5 Systems of nonlinear PDEs. Analytical approach 5.1 Overdetermined systems 5.2 Pfaffian equations and overdetermined systems 5.3 Transformations of nonlinear systems 5.4 Traveling wave solutions 5.5 Generalized separation of variables 6 Approximate Analytical Approach 6.1 Asymptotic traveling wave solution 6.2 Asymptotic solution of Cauchy problem 6.3 Asymptotic solution of systems of nonlinear PDEs 7 Qualitative Approach 7.1 Traveling wave solutions 7.2 Asymptotic solutions 8 Numerical Approach 8.1 Numerical and Graphical Solutions 8.2 Finite Difference Methods 9 Analytical-Numerical Approach 9.1 Spectral Collocation Methods 9.2 Finite Element Methods 9.3 Boundary Element Methods Supplement A. Foundations of Maple Basic Concepts Maple Language Supplement B. Foundations of Mathematica Basic Concepts Mathematica Language References General Index Maple Index Mathematica Index
Carlos Lizárraga-Celaya(switch to Inna K. Shingareva)