Related papers: Equivalence Groups and Differential Invariants for…
A novel symmetry method for finding exact solutions to nonlinear PDEs is illustrated by applying it to a semilinear reaction-diffusion equation in multi-dimensions. The method uses a separation ansatz to solve an equivalent first-order…
Potential equivalence transformations (PETs) are effectively applied to a class of nonlinear diffusion-convection equations. For this class all possible potential symmetries are classified and a theorem on connection of them with point ones…
The exact renormalization group is applied to a nonlinear diffusion equation with a discontinuous diffusion coefficient. The generating functional of the solution for the initial-value problem of nonlinear diffusion equations is first…
This paper systematically explains how to apply the invariant subspace method using variable transformation for finding the exact solutions of the (k+1)-dimensional nonlinear time-fractional PDEs in detail. More precisely, we have shown how…
Group classification of systems of two coupled nonlinear reaction-diffusion equation with a diagonal diffusion matrix is carried out. Symmetries of diffusion systems with singular diffusion matrix and additional first order derivative terms…
We discuss how point transformations can be used for the study of integrability, in particular, for deriving classes of integrable variable-coefficient differential equations. The procedure of finding the equivalence groupoid of a class of…
The theory of group classification of differential equations is analyzed, substantially extended and enhanced based on the new notions of conditional equivalence group and normalized class of differential equations. Effective new techniques…
While the energy-dependent neutron diffusion equation is widely employed in nuclear engineering, its status as an approximation to the transport equation is not yet completely understood, and several different approximations are in use to…
We find the group of equivalence transformations for equations of the form $y''= A(x)y' + F(y),$ where $A$ and $F$ are arbitrary functions. We then give a complete group classification of these families of equations, using a direct method…
Equivalence transformations play one of the important roles in continuum mechanics. These transformations reduce the original equations to simpler forms. One of the classes of nonlocal equivalence transformations is the class of reciprocal…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
Models of diffusive processes that occur on evolving domains are frequently employed to describe biological and physical phenomena, such as diffusion within expanding tissues or substrates. Previous investigations into these models either…
The differential constraints are applied to obtain explicit solutions of nonlinear diffusion equations. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the determining…
The purpose of this paper is to present a class of particular solutions of a C(2,1) conformally invariant nonlinear Klein-Gordon equation by symmetry reduction. Using the subgroups of similitude group reduced ordinary differential equations…
Complete descriptions of the Lie symmetries of a class of nonlinear reaction-diffusion equations with gradient-dependent diffusivity in one and two space dimensions are obtained. A surprisingly rich set of Lie symmetry algebras depending on…
Using techniques of the theory of semigroups of linear operators we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the…
Complete sets of bases of differential invariants, operators of invariant differentiation and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of…
A method is proposed of obtaining (2+1)-dimensional non- linear equations with non-analytic dispersion relations. Bilocal formalism is shown to make it possible to represent these equations in a form close to that for their counterparts in…
A mathematical model for the discrete nonlinear fragmentation (collision-induced breakage) equation with diffusion is studied. The existence of global weak solutions is established in arbitrary spatial dimensions without assuming a strictly…
A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence…