Enhanced group analysis and conservation laws of variable coefficient reaction–diffusion equations with power nonlinearities

Abstract

A class of variable coefficient (1 + 1)-dimensional nonlinear reaction-diffusion equations of the general form f (x)u t = (g(x)u n u x ) x + h(x)u m is investigated. Different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements. For the class under consideration the complete group classification is performed with respect to convenient equivalence groups (generalized extended and conditional ones) and with respect to the set of all local transformations. Usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results. The corresponding set of admissible transformations is described exhaustively. Then, using the most direct method, we classify local conservation laws. Some exact solutions are constructed by the classical Lie method.

Lie and Conditional Symmetries of a Class of Nonlinear (1 + 2)-Dimensional Boundary Value Problems

“…Now, we apply Definition 2 to BVPs (18)- (20) in order to obtain correctly-specified constraints when this problem is conditionally invariant under Operator (21). Obviously, the first item is fulfilled by the correct choice of the operator.…”

“…Because the BVP in question involves the condition at infinity (20), we also need to examine Items (d)-(f). Let us consider the following change of variables (substitution of (17) does not work in the case of zero Neumann conditions):…”

Laplace type invariants for variable coefficient mKdV equations

“…Although, the direct method involves considerable computational difficulties, it has the benefit of finding the most general equivalence group and also unfolds all form-preserving [12] (also known as admissible [13]) transformations admitted by this class of equations. For recent applications of the direct method one can refer, for example, to references [36][37][38][39]. More detailed description and examples of both methods can be found in [40].…”

Conservation laws and associated Lie point symmetries admitted by the transient heat conduction problem for heat transfer in straight fins

“…The conservation laws (51) and (52) were derived in [28]. If consideration is performed over real field, then (51) and (52) are conservation laws for Eq.…”