Related papers: Enhanced group classification of nonlinear diffusi…
The group classification of variable coefficient quasilinear reaction-diffusion equations $u_t=u_{xx}+h(x)B(u)$ is carried out exhaustively. This became possible due to usage of a conditional equivalence group found in the course of the…
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…
We investigate non-equilibrium critical phenomena using a nonperturbative renormalization group method. Reaction-diffusion processes are described by a scale dependent effective action which evolution is governed by very generic flow…
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…
We study the class of 3-dimensional nonlinear 2-hessian equations mentioned in the text. We perform preliminary group classification on 2-hessian equation. In fact, we find additional equivalence transformation on the space (x,y,z,u,f),…
Group classification of the generalized complex Ginzburg-Landau equations is presented. An approach to group classification of systems of reaction-diffusion equations with general diffusion matrix is developed.
Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid…
The Gradient Scheme framework provides a unified analysis setting for many different families of numerical methods for diffusion equations. We show in this paper that the Gradient Scheme framework can be adapted to elasticity equations, and…
The gradient discretisation method (GDM) is a generic framework for designing and analysing numerical schemes for diffusion models. In this paper, we study the GDM for the porous medium equation, including fast diffusion and slow diffusion…
We propose a new method for constructing exact solutions to nonlinear delay reaction--diffusion equations of the form $$ u_t=ku_{xx}+F(u,w), $$ where $u=u(x,t)$, $w=u(x,t-\tau)$, and $\tau$ is the delay time. The method is based on…
Group classification of a class of nonlinear fin equations is carried out exhaustively. Additional equivalence transformations and conditional equivalence groups are also found. They allow to simplify results of classification and further…
Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric…
In this paper we present an efficient numerical approach based on the Renormalization Group method for the computation of self-similar dynamics. The latter arise, for instance, as the long-time asymptotic behavior of solutions to nonlinear…
We carry out the complete group classification of the class of (1+1)-dimensional linear Schr\"odinger equations with complex-valued potentials. After introducing the notion of uniformly semi-normalized classes of differential equations, we…
In this paper, we develop a modified nonlinear dynamic diffusion (DD) finite element method for convection-diffusion-reaction equations. This method is free of stabilization parameters and is capable of precluding spurious oscillations. We…
The tempered fractional diffusion equation could be recognized as the generalization of the classic fractional diffusion equation that the truncation effects are included in the bounded domains. This paper focuses on designing the high…
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…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies…
By using the Malliavin calculus and solving a control problem, Bismut type derivative formulae are established for a class of degenerate diffusion semigroups with non-linear drifts. As applications, explicit gradient estimates and Harnack…