Global superconvergence of finite element methods for parabolic inverse problems

In this article we transform a large class of parabolic inverse problems into a nonclassical parabolic equation whose coefficients consist of trace type functionals of the solution and its derivatives subject to some initial and boundary conditions. For this nonclassical problem, we study finite element methods and present an immediate analysis for global superconvergence for these problems, on basis of which we obtain a posteriori error estimators. This research was supported in part by the Shahid Beheshti University, the National Basic Research Program of China (2007CB814906), the National Natural Science Foundation of China (10471103 and 10771158), Social Science Foundation of the Ministry of Education of China (Numerical methods for convertible bonds, 06JA630047), Tianjin Natural Science Foundation (07JCYBJC14300).     

Composite spectral method for solution of the diffusion equation with specification of energy

Many physical subjects are modeled by nonclassical parabolic boundary value problems with nonlocal boundary conditions replacing the classic boundary conditions. In this article, we introduce a new numerical method for solving the one‐dimensional parabolic equation with nonlocal boundary conditions. The approximate proposed method is based upon the composite spectral functions. The properties of composite spectral functions consisting of terms of orthogonal functions are presented and are utilized to reduce the problem to some algebraic equations. The method is easy to implement and yields very accurate result. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Numerical procedures for the determination of the leading coefficient a (x)

The aim of this paper is to study the parabolic inverse problem of determination of the leading coefficient in the heat equation with an extra condition at the terminal. After introducing a new variable, we reformulate the problem as a nonclassical parabolic equation along with the initial and boundary conditions. The finite difference methods, backward Euler and Crank–Nicolson schemes are studied. The results of a numerical example are presented, which demonstrate the efficiency and rapid convergence of the methods. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical approximation to the solution of an inverse heat conduction problem

The aim of this article is to study the parabolic inverse problem of determination of the leading coefficient in the heat equation with an extra condition at the terminal. After introducing a new variable, we reformulate the problem as a nonclassical parabolic equation along with the initial and boundary conditions. The uniqueness and continuous dependence of the solution upon the data are demonstrated, and then finite difference methods, backward Euler and Crank–Nicolson schemes are studied. The results of some numerical examples are presented to demonstrate the efficiency and the rapid convergence of the methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

An iterative procedure for determining an unknown spacewise-dependent coefficient in a parabolic equation

An inverse problem for the determination of an unknown spacewise-dependent coefficient in a parabolic equation is considered. The problem is reformulated as a nonclassical parabolic equation along with the initial and boundary conditions. The iterative fixed point projection method is applied to solve the reformulated problem. The comparison analysis of proposed method with a least square method and some numerical examples are presented.     

Limiting behavior of a global attractor for lattice nonclassical parabolic equations

We prove the upper semicontinuity of the global attractor corresponding to a class of lattice nonclassical parabolic equations.     

On a fully coupled nonlinear parabolic problem modelling miscible compressible displacement in porous media

We study a model describing a compressible and miscible displacement in a porous medium. It consists of a coupled system of nonlinear parabolic partial differential equations. Using nonclassical estimates and renormalization tools, we prove the existence of relevant weak solutions for the problem. This is the first existence result obtained for a transport model containing both the coupling due to the compressibility assumption and the coupling due to the concentration dependent viscosity.     

On a singular two dimensional nonlinear evolution equation with nonlocal conditions

This paper deals with a nonclassical initial boundary value problem for a two dimensional parabolic equation with Bessel operator. We prove the existence and uniqueness of the weak solution of the given nonlinear problem. We start by solving the associated linear problem. After writing this latter in its operator form, we establish an a priori bound from which we deduce the uniqueness of the strong solution. For the solvability of the associated linear problem, we prove that the range of the operator generated by the considered problem is dense. On the basis of the obtained results of the linear problem, we apply an iterative process to establish the existence and uniqueness of the nonlinear problem.     

Solution of a semilinear parabolic equation with an unknown control function using the decomposition procedure of Adomian

The investigation of nonclassical parabolic initial‐boundary value problems, which involve an integral over the spatial domain of a function of the desired solution, is of considerable concern. In this article a parabolic partial differential equation subject to energy overspecification is studied. This problem is appeared in modeling of many physical phenomena. The Adomian decomposition method, which is an efficient method for solving various class of problems, is employed for solving this model. This method provides an analytical solution in terms of an infinite convergent power series. Some examples are reported to support the simplicity of the decomposition procedure of Adomian. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Nonclassical symmetries of a class of Burgers' systems

The nonclassical symmetries of a class of Burgers' systems are considered. This study was initialized by Cherniha and Serov with a restriction on the form of the nonclassical symmetry operator. In this paper we remove this restriction and solve the determining equations to show that (1) a new form of a Burgers' system exists that admits a nonclassical symmetry and (2) a Burgers' system exists that is linearizable.     

EXACT SOLUTIONS TO NONLINEAR WAVE EQUATION

In this paper,we use an invariant set to construct exact solutions to a nonlinear wave equation with a variable wave speed. Moreover,we obtain conditions under which the equation admits a nonclassical symmetry. Several different nonclassical symmetries for equations with different diffusion terms are presented.     

Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions

In this work, the method of radial basis functions is used for finding the solution of an inverse problem with source control parameter. Because a much wider range of physical phenomena are modelled by nonclassical parabolic initial-boundary value problems, theoretical behavior and numerical approximation of these problems have been active areas of research. The radial basis functions (RBF) method is an efficient mesh free technique for the numerical solution of partial differential equations. The main advantage of numerical methods which use radial basis functions over traditional techniques is the meshless property of these methods. In a meshless method, a set of scattered nodes are used instead of meshing the domain of the problem. The results of numerical experiments are presented and some comparisons are made with several well-known finite difference schemes.     

Nonclassical symmetry of nonlinear diffusion equation and factorization method

We present the nonclassical symmetry of a nonlinear diffusion equation whose a nonlinear term is an arbitrary function. Generally, there is no guarantee that we can always determine the nonclassical symmetries admitted by the given equation because of the nonlinearity included in the determining equations. Accordingly, constructing invariant solutions is also generally difficult. In this paper, we apply the factorization method to nonclassical symmetry analysis for the nonlinear diffusion equation. Applying this method simplifies the determining equations and leads to their invariant solution automatically.     

Symmetry reductions for a dissipation-modified KdV equation

In this paper we give a group classification for a dissipation-modified Korteweg-de Vries equation by means of the Lie method of the infinitesimals. We prove that, by using the nonclassical method, we get several new solutions which are unobtainable by Lie classical symmetries. We obtain nonclassical symmetries that reduce the dissipation-modified Korteweg-de Vries equation to ordinary equations with the Painlevé property. These solutions have not been derived elsewhere by the singular manifold method.     

Solutions through nonclassical potential symmetries for a generalized inhomogeneous nonlinear diffusion equation

In this paper, we consider a class of generalized diffusion equations which are of great interest in mathematical physics. For some of these equations that model fast diffusion, nonclassical and nonclassical potential symmetries are derived. These symmetries allow us to increase the number of solutions. These solutions are unobtainable neither from classical nor from classical potential symmetries. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonclassical symmetry reductions for an inhomogeneous nonlinear diffusion equation

In this paper we consider a class of generalised diffusion equations which are of great interest in mathematical physics. For some of these equations model, fast diffusion nonclassical symmetries are derived. We find the connection between classes of nonclassical symmetries of the equation and of an associated system. These symmetries allow us to increase the number of solutions. Some of these solutions are unobtainable by classical symmetries and exhibit an interesting behaviour.     

Applying a new algorithm to derive nonclassical symmetries

In this paper we extend the procedure described for Bîlă and Niesen in [Bîlă N, Niesen J. On a new procedure for finding nonclassical symmetries. J Symbol Comp 2004;38:1523–33], to obtain the determining equations of the nonclassical symmetries associated with a partial differential equation system, to a different case. We offer some examples of how our method works. By using this procedure we obtain a new nonclassical symmetry for the 2 + 1-dimensional shallow water wave equation.     

Nonclassical symmetries as special solutions of heir-equations

In Phys. D 78 (1994) 124, we have found that iterations of the nonclassical symmetries method give rise to new nonlinear equations, which inherit the Lie point symmetry algebra of the given equation. In the present paper, we show that special solutions of the right-order heir-equation correspond to classical and nonclassical symmetries of the original equations. An infinite number of nonlinear equations which possess nonclassical symmetries are derived.