Exact Solutions Of Partial Equations Research Articles

Exact Solutions of a Nonclassical Nonlinear Equation of the Fourth Order

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, phenomena in hydrodinamics and other ones. Notice that wide studies of qualitative properties of solutions of Sobolev-type equations exist. Namely, results about existence 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, a fourth order nonlinear partial equation is studied. Three classes of its exact solutions are built. They are expressed in terms of special functions (solutions of some ordinary differential equations). For two of these classes subsets that can be expressed in elementary functions are built, for the third one subsets that can be described in elementary functions and an implicit function (without a quadrature) are built.

Elzaki Substitution Method for Solving Nonlinear Partial Differential Equations with Mixed Partial Derivatives Using Adomian Polynomial

In this paper we apply a new method, named Elzaki Substitution Method to solve nonlinear homogeneous and nonhomogeneous partial differential equations with mixed partial derivatives, which is based on Elzaki Transform. The proposed method introduces also Adomian polynomials and the nonlinear terms can be handled by the use of this polynomials. The proposed method worked perfectly to find the exact solutions of partial equations with mixed partial derivatives without any need of linearization or discretization in comparison with other methods such as Method of Separation of Variables (MSV) and Variation Iteration Method (VIM). Some illustrative examples are given to demonstrate the applicability and efficiency of proposed method.

New Optical Solutions of the Fractional Gerdjikov-Ivanov Equation With Conformable Derivative

Finding the exact solution of partial equations is one of the most challenging problems in mathematics. In most cases, there is no exact solution to this set of equations. In these cases, the use of numerical and approximate methods is inevitable. Nevertheless, the exact PDE solver methods are always preferred because they present the solution directly without any restrictions to use. This article aims to examine the perturbed Gerdjikov-Ivanov equation in an exact approach point of view. This equation plays a significant role, and has many important applications in nonlinear fiber optics and photonic crystal fibers. To this end, firstly, we obtain some novel optical solutions of the equation via a newly proposed analytical method called generalized exponential rational function method. In order to understand the dynamic behavior of these solutions, several graphs are plotted for some specific choices for the parameters. To the best of our knowledge, these two techniques have never been tested for the equation in the literature. The findings of this article may have a high significance application while handling the other nonlinear PDEs.

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

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.