Nonlocal Symmetries and Generation of Solutions for Partial Differential Equations

The paper describes a number of simple but quite effective methods for constructing exact solutions of PDEs, that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for construc...

In this work, we construct the general solution to the Heat Equation (HE) and to many tensor structures associated to the Heat Equation, such as Symmetries, Lagrangians, Poisson Brackets (PB) and Lagrange Brackets, using newly devised techniques that may be applied to any linear equation (e.g., Schroedinger Equation in field theory, or the small-oscillations problem in mechanics). In particular, we improve a time-independent PB found recently which defines a Hamiltonian Struc...

The method of equivariant moving frames on multi-space is used to construct symmetry preserving finite difference schemes of partial differential equations invariant under finite-dimensional symmetry groups. Invariant numerical schemes for a heat equation with a logarithmic source and the spherical Burgers equation are obtained. Numerical tests show how invariant schemes can be more accurate than standard discretizations on uniform rectangular meshes.

N-dimensional B\"acklund transformation (BT), Cole-Hopf transformation and Auto-B\"acklund transformation (Auto-BT) of n-dimensional Burgers system are derived by using simplified homogeneous balance (SHB). By the Auto-BT, another solution of the n-dimensional Burgers system can be obtained provided that a particular solution of the Burgers system is given. Since the particular solution of n-dimensional Burgers system can be given easily by the Cole-Hopf transformation, then ...

A detailed analysis of the invariant point transformations for the first four partial differential equations which belong to the Complex Burgers` Hierarchy is performed. Moreover, a detailed application of the reduction process through the Lie point symmetries is presented while we construct similarity solutions. We conclude that the differential equations of our consideration are reduced to first-order equations such as the Abel, Riccati and to a linearisable second-order di...

We present new periodic, kink-like and soliton-like travelling wave solutions to the hyperbolic generalization of Burgers equation. To obtain them, we employ the classical and generalized symmetry methods and the ansatz-based approach

We derive the solution representation for a large class of nonlocal boundary value problems for linear evolution PDEs with constant coefficients in one space variable. The prototypical such PDE is the heat equation, for which problems of this form model physical phenomena in chemistry and for which we formulate and prove a full result. We also consider the third order case, which is much less studied and has been shown by the authors to have very different structural properti...

We develop a method for generating solutions to large classes of evolutionary partial differential systems with nonlocal nonlinearities. For arbitrary initial data, the solutions are generated from the corresponding linearized equations. The key is a Fredholm integral equation relating the linearized flow to an auxiliary linear flow. It is analogous to the Marchenko integral equation in integrable systems. We show explicitly how this can be achieved through several examples i...

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of evolution equations having non-trivial Lie symmetry. Next, we modify the above method to generate more general nonlocal symmetries, which yields a purely algebraic approach to classifying nonlocal symmetries of evolution type systems. Seve...

In the present paper the classical point symmetry analysis is extended from partial differential to functional differential equations with functional derivatives. In order to perform the group analysis and deal with the functional derivatives we extend the quantities such as infinitesimal transformations, prolongations and invariant solutions. For the sake of example the procedure is applied to the continuum limit of the heat equation. The method can further lead to significa...