Related papers: Hierarchy of Conservation Laws of Diffusion--Conve…
The notions of generating sets of conservation laws of systems of differential equations with respect to symmetry groups and equivalence groups are introduced and applied. This allows us to generalize essentially the procedure of finding…
All possible linearly independent local conservation laws for $n$-dimensional diffusion--convection equations $u_t=(A(u))_{ii}+(B^i(u))_i$ were constructed using the direct method and the composite variational principle. Application of the…
We study local conservation laws of variable coefficient diffusion-convection equations of the form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$. The main tool of our investigation is the notion of equivalence of conservation laws with respect to…
We investigate conservation laws of diffusion-convection equations to construct first-order potential systems corresponding to these equations. We do two iterations of the construction procedure, looking, in the second step, for the…
We show that the so-called hidden potential symmetries considered in a recent paper [Gandarias M., Physica A, 2008, V.387, 2234-2242] are ordinary potential symmetries that can be obtained using the method introduced by Bluman and…
We study conservation laws and potential symmetries of (systems of) differential equations applying equivalence relations generated by point transformations between the equations. A Fokker-Planck equation and the Burgers equation are…
We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible…
A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion-reaction processes in $n\geq 1$ dimensions, where the diffusivity depends on the gradient of the flow. For this class, all…
This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for…
A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with…
The concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on subdiffusion and diffusion-wave equations with the…
A new method for the Lie group classification of differential equations is proposed. It is based of the determination of all possible cases of linear dependence of certain indeterminate appearing in the determining equations of symmetries…
In this paper we consider generalization of procedure of construction of potential systems for systems of partial differential equations with multidimensional spaces of conservation laws. More precisely, for construction of potential…
We determine, by hierarchy, dependencies between higher order linear symmetries which occur when generating them using recursion operators. Thus, we deduce a formula which gives the number of independent generalized symmetries (basis) of…
Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power onlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrodinger…
Conservation laws are formulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not…
The reaction diffusion equation arises in physical situations in problems from population growth, genetics and physical sciences. We consider the generalised Fisher equation in cylindrical coordinates from Lie theory stand point. An…
We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators…
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a…
All low-order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time-dependent. Such equations arise in numerous physical applications and have attracted…