Algebraic Approach
Algebraic approach, based on transformation methods, is the most powerful analytic tool for studying nonlinear partial differential equations. Although the first exact solutions of PDEs have been determined in the 18th century (works by Cauchy, Euler, Ham
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Algebraic approach, based on transformation methods, is the most powerful analytic tool for studying nonlinear partial differential equations. Although the first exact solutions of PDEs have been determined in the 18th century (works by Cauchy, Euler, Hamilton, Jacobi, Lagrange, Monge), the most important results were obtained by S. Lie at the end of 19th century (see [91]). Nowadays, the continuous transformation groups (or symmetries, or Lie groups), proposed by Lie, and other transformations can be computed with the help of computer algebra systems, Maple and Mathematica. In general, transformations (with respect to a given nonlinear PDE) can be divided into two parts: transformations of the independent variables, dependent variables; transformations of the independent variables, dependent variables, and their derivatives. In this chapter, first we will consider various types of transformations, e.g., point and contact transformations, transformations relating differential equations, linearizing and bilinearizing transformations. Transformation methods allow us to find transformations (under which a nonlinear PDE is invariant) and new variables (independent, dependent), with respect to which the differential equations become more simpler, e.g., linear. Transformations can convert a solution of a nonlinear PDE to the same or another solution of this equation, the invariant solutions can be found by symmetry reductions, rewriting the equation in new variables. Nonlinear PDEs can be written in Cauchy– Kovalevskaya form, or normal form, or canonical form, or linear form after a point or contact transformation. Then we will consider the two wide classes of methods for finding exact solutions to nonlinear PDEs and nonlinear systems. The first class of methods, called reductions, consists in finding the reductions to a differential equation in a lesser number of independent variables. Among them, we will discuss traveling wave reductions, ansatz methods, self-
I. Shingareva and C. Lizárraga-Celaya, Solving Nonlinear Partial Differential Equations with Maple and Mathematica, DOI 10.1007/978-3-7091-0517-7_2, © Springer-Verlag/Wien 2011
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Algebraic Approach
similar reductions. The second class of methods, called separation of variables, allow us to construct an exact solution to a given nonlinear PDE as a combination of yet undetermined functions of fewer variables, and in the solution process to determine these simpler functions. We will consider selective problems in which it is possible to perform ordinary separation of variables, partial separation of variables, generalized separation of variables, and functional separation of variables. Then, we will consider another class of methods, called the classical method of finding symmetries of nonlinear PDEs that allow us to obtain transformation groups, under which the PDEs are invariant, and new variables (independent and dependent), with respect to which differential equations become more simpler. For a class of the nonlinear second-order PDEs of general form in two inde
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