Exact Solutions of Three Nonclassical Equations, and Their Construction with Maple System

Resumo

Since the second half of the twentieth century, wide studies of Sobolev-type equations are undertaken. These equations contain items that are derivatives with respect to time of the second order derivatives of the unknown function with respect to space variables. They can describe nonstationary processes in semiconductors, in plasm, phenomenons in hydrodynamics and other ones. It is important to notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existense and uniqueness of solutions, their asymptotics and blow-up are known. But there are few results about exact solutions of Sobolev-type equations. There are books and papers about exact solutions of partial equations, but they are devoted mainly to classical equations, where the first or second order derivative with respect to time or the derivative with respect to time of the first order derivative of the unknown function with respect to the space variable is equal to a stationary expression. Therefore it is interesting to study exact solutions of Sobolev-type equations.

In the present paper, three nonclassical nonlinear partial equations are studied. Some results about existense, uniqueness and blow-up of their solutions are already known. Here, we construct some classes of their exact solutions with the help of Maple System. We use the method of travelling waves, the method of short-cut decompositions and construction of solutions of some special forms. Also, we discuss the way of realizing of these investigations with the help of Maple system of computer mathematics.