Related papers: On exact and perturbation solutions to nonlinear e…
The divergence of the thermal conductivity in the thermodynamic limit is thoroughly investigated. The divergence law is consistently determined with two different numerical approaches based on equilibrium and non-equilibrium simulations. A…
We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel…
Asymmetric heat transfer systems, often referred to as thermal diodes or thermal rectifiers, have garnered increasing interest due to their wide range of application possibilities. Most of those previous macroscopic thermal diodes either…
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…
We use the perturbation method to approximately solve subdiffusion-reaction equations. Within this method we obtain the solutions of the zeroth and the first order. The comparison our analytical solutions with the numerical results shown…
It is well recognized that new types of exact travelling wave solutions to nonlinear partial differential equations can be obtained by modifications of the methods which are in hand. In this study, we extend the class of auxiliary equations…
In the presented paper known (up to the beginning of 2008) Lie- and non-Lie exact solutions of different $(1+1)$-dimensional diffusion-convection equations of form $f(x)u_t=(g(x)A(u)u_x)_x+h(x)B(u)u_x$ are collected.
We present a unified description of heat flow in two-terminal hybrid quantum systems. Using simple models, we analytically study nonlinear aspects of heat transfer between various reservoirs: metals, solids, and spin baths, mediated by the…
We consider the problem of heat diffusion in branched systems and networks on the basis of a model described in terms of heat equation on metric graphs. Using the explicit analytical solutions of the latter, evolution of the temperature…
Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gr\"obner bases is used both to find the conditions on $f(u)$ under which symmetries other than the…
This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout…
A method to construct the exact solution of the PDE is presents, which combines the two kind methods(the nonlinear transformation and RQ(Reduction the PDE to a Quadrature problem) method).The nonlinear diffusion equation is chosen to…
We discuss a class of hyperbolic reaction-diffusion equations and apply the modified method of simplest equation in order to obtain an exact solution of an equation of this class (namely the equation that contains polynomial nonlinearity of…
A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations $f(x)u_{tt}=(H(u)u_x)_x+K(u)u_x$, is given, by using a compatibility method and additional equivalence transformations. A number of new…
Sampling equation method is presented to look for exact solutions of nonlinear differential equations. Application of this approach to one of the extensive chaos model is considered. Exact solutions of this model in travelling wave are…
An Eulerian-Lagrangian approach to incompressible fluids that is convenient for both analysis and physics is presented. Bounds on burning rates in combustion and heat transfer in convection are discussed, as well as results concerning…
Based the homogeneous balance method, a general method is suggested to obtain several kinds of exact solutions for some kinds of nonlinear equations. The validity and reliability of the method are tested by applying it to the Bousseneq…
Analytic solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile, are presented. The solutions are a generalization of the well known…
Integro-partial differential equations occur in many contexts in mathematical physics. Typical examples include time-dependent diffusion equations containing a parameter (e.g., the temperature) that depends on integrals of the unknown…
A finite element based computational scheme is developed and employed to assess a duality based variational approach to the solution of the linear heat and transport PDE in one space dimension and time, and the nonlinear system of ODEs of…