New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System

The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well.

Generalized Camassa–Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively.

On the Derivation of Nonclassical Symmetries of the Black–Scholes Equation via an Equivalence Transformation

The nonclassical symmetries method is a powerful extension of the classical symmetries method for finding exact solutions of differential equations. Through this method, one is able to arrive at new exact solutions of a given differential equation, i.e., solutions that are not obtainable directly as invariant solutions from classical symmetries of the equation. The challenge with the nonclassical symmetries method, however, is that governing equations for the admitted nonclassical symmetries are typically coupled and nonlinear and therefore difficult to solve. In instances where a given equation is related to a simpler one via an equivalent transformation, we propose that nonclassical symmetries of the given equation may be obtained by transforming nonclassical symmetries of the simpler equation using the equivalence transformation. This is what we illustrate in this paper. We construct four nontrivial nonclassical symmetries of the Black–Scholes equation by transforming nonclassical symmetries of the heat equation. For completeness, we also construct invariant solutions of the Black–Scholes equation associated with the determined nonclassical symmetries.

Nonclassical symmetries and analytic solutions to Kawahara equation

The focus of this paper is on Kawahara equation (KE) which models diverse variety of waves enforced in many branches of applied sciences. Symmetries to the considered equation are successfully obtained by practicing nonclassical method. According to nonclassical symmetries, KE has been reduced to numerous ODEs. Further, these ODEs are treated such that new analytic solutions are determined by applying power series, ansätz and [Formula: see text]-expansion method.

Nonclassical Symmetry of Differential Equations

In this paper, we discuss the difference between classical and nonclassical symmetries. In addition, we found the non-classical symmetry of the Benjamin Bona Mahony Equation (BBM). Finally, we found a new exact solution to a Benjamin Bona Mahony Equation (BBM) using nonclassical symmetry.

Some Connections between Classical and Nonclassical Symmetries of a Partial Differential Equation and Their Applications

Essential connections between the classical symmetry and nonclassical symmetry of a partial differential equations (PDEs) are established. Through these connections, the sufficient conditions for the nonclassical symmetry of PDEs can be derived directly from the inconsistent conditions of the system determining equations of the classical symmetry of the PDE. Based on the connections, a new algorithm for determining the nonclassical symmetry of a PDEs is proposed. The algorithm make the determination of the nonclassical symmetry easier by adding compatibility extra equations obtained from system of determining equations of the classical symmetry to the system of determining equations of the nonclassical symmetry of the PDE. The findings of this study not only give an alternative method to determine the nonclassical symmetry of a PDE, but also can help for better understanding of the essential connections between classical and nonclassical symmetries of a PDE. Concurrently, the results obtained here enhance the efficiency of the existing algorithms for determining the nonclassical symmetry of a PDE. As applications of the given algorithm, a nonclassical symmetry classification of a class of generalized Burgers equations and the nonclassical symmetries of a KdV-type equations are given within a relatively easier way and some new nonclassical symmetries have been found for the Burgers equations.

A Potential Constraints Method of Finding Nonclassical Symmetry of PDEs Based on Wu’s Method

Solving nonclassical symmetry of partial differential equations (PDEs) is a challenging problem in applications of symmetry method. In this paper, an alternative method is proposed for computing the nonclassical symmetry of PDEs. The method consists of the following three steps: firstly, a relationship between the classical and nonclassical symmetries of PDEs is established; then based on the link, we give three principles to obtain additional equations (constraints) to extend the system of the determining equations of the nonclassical symmetry. The extended system is more easily solved than the original one; thirdly, we use Wu’s method to solve the extended system. Consequently, the nonclassical symmetries are determined. Due to the fact that some constraints may produce trivial results, we name the candidate constraints as “potential” ones. The method gives a new way to determine a nonclassical symmetry. Several illustrative examples are given to show the efficiency of the presented method.

Symmetry Reduction and Analytical Solutions of Potential Kdv Equation

In this paper, classical and nonclassical symmetries of the potential KdV equation are considered. A catalogue of symmetry reductions for potential KdV equation is obtained using the classical Lie method and nonclassical method due to Bluman and Cole.[1] The Lie algebra consists of five finite parameter Lie group transformations; two being the scaling symmetry and the others being translations. By using the nonclassical method, four finite parameter group transformations are obtained. Two different types of solutions, approximate and exact solutions, are found by using the symmetries. Using the classical symmetries, only approximate series solutions are constructed, but using the nonclassical symmetries, both the exact solution and the approximate solution were found. The advantage of nonclassical symmetries is that the exact solution of the KdV equation can be found with it. Perturbation method is also used for approximate solutions.