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 n
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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. In this chapter, we will show a very helpful role of computer algebra systems in the implementation of a whole solution process. In particular, following the analytical-numerical approach, we will construct analytical formulas of numerical methods (method of lines, spectral collocation method), obtain numerical and graphical solutions of nonlinear problems (the initial boundary value problem for the Burgers equation and the nonlinear first-order system, the initial boundary value problem describing nonlinear parametrically excited standing waves in a vertically oscillating rectangular container in Eulerian coordinates), and compare the numerical solutions with other types of solutions obtained in the book (e.g., numerical and asymptotic solutions obtained in Eulerian and Lagrangian coordinates).
7.1
Method of Lines
The method of lines, proposed by E. N. Sarmin and L. A. Chudov [131] in 1963, allows us to solve initial boundary value problems for linear and nonlinear PDEs. The first step in the solution process consists in the construction and analysis of a numerical method for the given PDEs by applying finite differences in all but one dimension and leaving the other variable continuous. Then we can solve the resulting system of ODEs (by applying the classical numerical methods), visualize the solutions obtained, analyze them, and compare with other types of solutions or experimental data. There exist the semi-analytical method of lines ([149],
I. Shingareva and C. Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, DOI 10.1007/978-3-7091-0517-7_7, © Springer-Verlag/Wien 2011
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Analytical-Numerical Approach
[164]) for finding semi-analytical solutions of linear evolution equations (where the discretization process results in a system of ODEs, which is solved by applying the matrix exponential method) and the numerical method of lines for finding numerical solutions of linear and nonlinear evolution equations. In scientific literature, there are many papers discussing various aspects of the method of lines for various types of PDEs, e.g., see important works in this field written by W. E. Schiesser ([134], [135], [137], [138]). In this section, following the analytical-numerical approach, we describe a whole solution process based on the numerical method of lines for evolution equations (in particular, the Burgers equation and the nonlinear first-order system). It should be noted that in some cases the method of lines can be applied to elliptic equations (e.g., see [136], Problem 6.15) by adding a time derivative to an elliptic PDE, applying finit
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