Carlos Lizárraga-Celaya

Analytical-Numerical Approach

One of the main points (related to computer algebra systems) is based on the implementation of a whole solution process, 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 analyzing and comparing the numerical solution obtained with other types of solutions.

Approximate Analytical Approach

In this chapter, we will follow the approximate analytical approach for solving nonlinear PDEs. We consider the most important recently developed methods and traditional methods to find approximate analytical solutions of nonlinear PDEs and nonlinear systems. We will apply the Adomian decomposition method (ADM) and perturbation methods to solve nonlinear PDEs (e.g., the Burgers equation, the Klein–Gordon equation, the Fisher equation, etc.) and nonlinear systems.

General Analytical Approach. Integrability

In this chapter, following the general analytical approach, we consider the basic concepts, ideas, and the most important methods for solving analytically nonlinear partial differential equations with the aid of Maple and Mathematica. In particular, we will consider the concepts of integrability, the Painleve integrability, complete integrability for evolution equations, the Lax pairs, the variational principle.

Geometric-Qualitative Approach

In this chapter, following a geometric-qualitative approach to partial differential equations, we will consider important methods and concepts concerning quasilinear and nonlinear PDEs (in two independent variables) and solutions of classical and generalized Cauchy problems (with continuous and discontinuous initial data), namely, the Lagrange method of characteristics and its generalizations, the concepts of solution surfaces (or integral surfaces), general solutions, discontinuous or weak solutions, solution profiles at infinity, complete integrals, the Monge cone, characteristic directions.

Special Functions and Orthogonal Polynomials

14.50–15.15 Michael Schlosser Macdonald Polynomials in the Light of Basic Hypergeometric Series 15.15–15.40 Heung Yeung Lam Sixteen Eisenstein Series 16.00–16.25 Nicholas Witte Semi-classical Orthogonal Polynomials and the Painlevé-Garnier Systems 16.25–16.50 Howard S. Cohl Fourier Expansions of the Fundamental Solution for Powers of the Laplacian in R 16.50–17.15 Richard Askey The First Addition Formula and Some of What Came Later