By Inna Shingareva, Carlos Lizárraga-Celaya (auth.)
The emphasis of this paintings is on developing kinds of suggestions (exact, approximate analytical, numerical, graphical) of diverse nonlinear PDEs effectively, simply, and fast. The reader can research a large choice of options and resolve a number of nonlinear PDEs incorporated and lots of different differential equations, simplifying and reworking the equations and options, arbitrary capabilities and parameters, offered within the book). various comparisons and relationships among numerous different types of ideas, various equipment and ways are supplied, the consequences received in Maple and Mathematica, enables a deeper figuring out of the subject
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Additional resources for Solving Nonlinear Partial Differential Equations with Maple and Mathematica
Example text
18 Eikonal equation. Boundary value problem. Exact solution. Let us consider the boundary value problem for the nonlinear eikonal equation (ux )2 +(uy )2 =1, u(x0 , y0 )=0, where {x ∈ R, y ∈ R}. Applying Maple predefined functions, solve the boundary value problem for the eikonal equation and verify that the solution obtained, u(x, y)=(x0 −x) 1−c22 +c2 (y−y0 ) (Sol1), is exact solution of the boundary value problem. Prove√that the complete integral of this equation takes the form u(x, y)=ay− 1−a2 x+b (Sol2).
These functions allow us to solve nonlinear systems and obtain solutions automatically and develop new methods and procedures for constructing new solutions. As before, we will consider the most important functions for finding analytical solutions of a given nonlinear system. 2 Embedded Analytical Methods 33 Note. ICs are initial conditions; DepVars is a set or list of dependent variables (indicating the solving ordering that is useful for nonlinear PDE systems, where slight changes in the solving ordering can lead to different solutions or make an unsolvable (by pdsolve) a solvable equation).
3 A contact transformation is equivalent to a point transformation that acts on the space of the independent variables, the dependent variable, and its first derivatives, and can be extended to point transformations that act on the space of the independent variables, the dependent variable, and its derivatives to any finite order. Sophus Lie considered contact transformations and contact symmetries of PDEs [91]. , it is a transformation that maps any solution of the PDE into another solution of the same PDE.