a lower bound on the partition function for a classical charge symmetric system a lower bound is obtained for the grand canonical partition function and hence for the pressure of a charge symmetric system with positive definite interaction for the coulomb interaction the lower bound on the pressure is the debye huckel approximation

comparison results and estimates on the gradient without strict convexity in this paper we establish a comparison result for solutions to the problem mbox minimize int omega f nabla u x dx quad hbox on u u u 0 in w 1 1 0 omega

the convergence of fully discrete galerkin approximations for the benjamin bona mahony bbm equation abstract in the present paper we analyze a second order in time fully discrete finite element method for the bbm equation the discretization in space is based on the standard galerkin method for the time discretization the crank nicolson scheme is used we also prove the convergence of a linearized galerkin modification scheme

dirichlet problem for demi coercive functionals on considere des fonctionnelles integrales du type f u f du ou f r n 0 est une fonction convexe ayant la propriete suivante il existe a 0 b 0 r n tel que f z a z b z r n

lipschitz regularity for minima without strict convexity of the lagrangian abstract we give in a non smooth setting some conditions under which some of the minimizers of f u x d x g x u x d x among the functions in w 1 1 that lie between two lipschitz functions are lipschitz we weaken the usual strict convexity assumption in showing that if just the faces of the epigraph of a convex function f r n r are bounded and the boundary datum u 0 satisfies a generalization of the bounded slope condition introduced by a cellina then the minima of f u x d x on u 0 w 0 1 1 whenever they exist are lipschitz a relaxation result follows

a meshless local collocation method for time fractional diffusion wave equation abstract in this manuscript we present a radial basis function based local collocation method for solving time fractional diffusion wave equation the advantage of the local collocation method is only the discretization points located in each sub domain surrounding the collocation point need to be considered it also avoids the ill conditioning problem arises due to the use of global collocation method the theoretical stability and convergence of the proposed time semi discrete scheme are also discussed numerical experiments for one dimension and two dimension problems are carried out it is shown that the present method is efficient and numerical results have nice agreements with theoretical result

some new solitonary solutions of the modified benjamin bona mahony equation abstract in this paper we use the exp function method to construct some new soliton solutions of the benjamin bona mahony and modified benjamin bona mahony equations these equations have important and fundamental applications in mathematical physics and engineering sciences the exp function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation this method provides the concise and straightforward solution in a very easier way the results obtained in this paper can be viewed as a refinement and improvement of the previously known results

the finite difference streamline diffusion methods for sobolev equations with convection dominated term in this paper we give two finite difference streamline diffusion fdsd schemes for solving linear sobolev equations with convection dominated term stability and optimal error estimates for them are derived in suitable norms

optimal decay rates of solutions for a multidimensional generalized benjamin bona mahony equation abstract in this paper we study the optimal decay rates of solutions for the generalized benjamin bona mahony equation in multi dimensional space n 3 by using fourier transform and the energy method we obtain the l q 2 q convergence rates of the solutions under the condition that the initial data is small the optimal decay rates obtained in this paper are found to be the same as the decay rate for the heat equation

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

statistical field theory for simple fluids the collective variables representation abstract an alternative representation of an exact statistical field theory for simple fluids based on the method of collective variables is presented the results obtained are examined from the point of another version of theory that was developed recently by performing a hubbard stratonovich transformation of the configurational boltzmann factor j m caillol mol phys 101 2003 1617 the analytical expressions for the pressure and the free energy are derived in two loop approximation for both versions of theory and it is shown that they are indeed equivalent the results yield a new type approximation within an untested approximation scheme

a numerical method for two dimensional schrodinger equation using collocation and radial basis functions in this paper we propose a numerical scheme to solve the two dimensional 2d time dependent schrodinger equation using collocation points and approximating the solution using multiquadrics mq and the thin plate splines tps radial basis function rbf the scheme works in a similar fashion as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme

shape derivatives for minima of integral functionals for omega varying among open bounded sets in mathbb r n r n we consider shape functionals j omega j defined as the infimum over a sobolev space of an integral energy of the kind int omega f nabla u g u f u g u under dirichlet or neumann conditions on partial omega under fairly weak assumptions on the integrands f f and g g we prove that when a given domain omega is deformed into a one parameter family of domains omega varepsilon through an initial velocity field v in w 1 infty mathbb r n mathbb r n v w 1 r n r n the corresponding shape derivative of j j at omega in the direction of v v exists under some further regularity assumptions we show that the shape derivative can be represented as a boundary integral depending linearly on the normal component of v v on partial omega our approach to obtain the shape derivative is new and it is based on the joint use of convex analysis and gamma convergence techniques it allows to deduce as a companion result optimality conditions in the form of conservation laws

grand canonical distribution for multicomponent system in the collective variables method the collective variables method with a reference system is developed for the case of the grand canonical ensemble for a multicomponent continuous system the method is used to investigate phase transitions in a binary system for a binary symmetrical system the relations between microscopic parameters determining the alternation of gas liquid and separation phase transitions are found the functional of the grand partition function of the symmetrical mixture is examined in the framework of parameters containing the separation point the system is described with two sets of collective variables k a set connected with the gas liquid critical point andck a set connected with the separation phenomenon the fourfold basic density measure is constructed inck variable phase space which contains the variablec v connected with the order parameter of the system it is shown that the problem can be reduced to the 3d ising model in an external field

the classical problem of the calculus of variations in the autonomous case relaxation and lipschitzianity of solutions we consider the problem of minimizing b a l x t x t dt x a a x b b under the assumption that the lagrangian l is continuous and satisfies a growth assumption that does not imply superlinear growth we provide a result on the relaxation of the functional and show that a solution to the minimum problem is lipschitzian

a linearly implicit conservative difference scheme for the generalized rosenau kawahara rlw equation this paper concerns the numerical study for the generalized rosenau kawahara rlw equation obtained by coupling the generalized rosenau rlw equation and the generalized rosenau kawahara equation we first derive the energy conservation law of the equation and then develop a three level linearly implicit difference scheme for solving the equation we prove that the proposed scheme is energy conserved unconditionally stable and second order accurate both in time and space variables finally numerical experiments are carried out to confirm the energy conservation the convergence rates of the scheme and effectiveness for long time simulation

on the bounded slope condition and the validity of the euler lagrange equation under the bounded slope condition on the boundary values of a minimization problem for a functional of the gradient of u we show that a continuous minimizer w is in fact lipschitzian an application of this result to prove the validity of the euler lagrange equation for w is presented

continuity properties of solutions to some degenerate elliptic equations abstract we consider a nonlinear possibly degenerate elliptic operator l v div a v b x v where the field a and the function b are unnecessarily strictly monotonic and a satisfies a very mild ellipticity assumption for a given boundary datum we prove the existence of the maximum and the minimum of the solutions and formulate a haar rado type result namely a continuity property for these solutions that may follow from the continuity of in the homogeneous case we formulate some generalizations of the bounded slope condition and use them to obtain the lipschitz or local lipschitz regularity of solutions to l u 0 we prove the global holder regularity of the solutions in the case where is lipschitz

the density functional theory of classical fluids revisited we reconsider the density functional theory of nonuniform classical fluids from the point of view of convex analysis from the observation that the logarithm of the grand partition function log xi phi is a convex functional of the external potential phi it is shown that the kohn sham free energy cal a rho is a convex functional of the density rho log xi phi and cal a rho constitute a pair of legendre transforms and each of these functionals can therefore be obtained as the solution of a variational principle the convexity ensures the unicity of the solution in both cases the variational principle which gives log xi phi as the maximum of a functional of rho is precisely that considered in the density functional theory while the dual principle which gives cal a rho as the maximum of a functional of phi seems to be a new result

meshless rbfs method for numerical solutions of two dimensional high order fractional sobolev equations abstract in this paper meshless rbfs method is proposed to solve two dimensional time fractional sobolev equations the proposed method uses rbfs for approximation of spatial operator finite difference formula of o t 2 0 1 is used for time fractional derivative approximation while rule 0 1 as time stepping scheme for the advancement of solution in time validation of the proposed method is made by considering various test examples from literature simulated results are found in very good agreement with available exact solutions a rigorous comparative analysis made with other methods testifies proposed method s superiority in higher dimensions efficiency and accuracy of rbfs method are examined by varying number of nodes n in the domain of influence time step size t as well as l 2 l and l rms error norms linearized stability analysis of the proposed method is thoroughly discussed and verified numerically to support the analysis

a two dimensional haar wavelets method for solving systems of pdes in this paper we modify the idea of haar wavelets method to obtain semi analytical solutions for the systems of three dimensional nonlinear partial differential equations theoretical considerations are discussed to demonstrate the efficiency of the method a test problem is presented the approximate solutions of the system of three dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature the numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions

exponential decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations abstract in this paper we investigate the exponential time decay rate of solutions toward traveling waves for the cauchy problem of generalized benjamin bona mahony burgers equations e u t u t x x u x x u x f u x 0 t 0 x r with prescribed initial data i u 0 x u 0 x u as x here 0 r are constants u are two given constants satisfying u u and the nonlinear function f u c 2 r is assumed to be either convex or concave based on the existence of traveling waves the local stability and the algebraic decay rate to traveling waves of solutions to the cauchy problem e and i established in yin et al 2007 13 we show an exponential decay rate of the solutions to the cauchy problem e and i toward the traveling waves mentioned above by employing the space time weighted energy method which was initiated by kawashima and matsumura in 1985 14 and later elaborated by matsumura and nishihara 1994 15 and nishikawa 1998 16

debye huckel theory for charge symmetric coulomb systems it is proven that the pressure density and correlation functions of a classical charge symmetric coulomb system are asymptotic as the plasma parameter e tends to zero to the approximations predicted by the debye huckel theory these approximations consist of the ideal gas term plus a term of one lower order in e the sine gordon transformation and some new correlation inequalities for the associated functional integrals are used

the numerical simulation of the phase field crystal pfc and modified phase field crystal mpfc models via global and local meshless methods abstract in the current work we consider the two dimensional time dependent phase field crystal pfc and the modified phase field crystal models mpfc to obtain their numerical solutions for this purpose we apply two numerical meshless methods based on radial basis functions rbfs and also two meshless methods which are based on moving least squares method mls four techniques developed in this paper are globally radial basis functions grbfs radial basis functions pseudo spectral rbfs ps moving least squares mls and generalized moving least squares gmls approximations two methods based on rbfs are global and the other methods based on moving least squares are local procedures as is well known the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non regular domains with different choices of grids in high dimensions applying the new methods on spatial domain and also using the semi implicit scheme for time variable yield a linear system of algebraic equations to solve this linear system the lu decomposition algorithm and command backslash in matlab software for gmls method are applied the implementation of boundary conditions in the rbfs ps is applied directly because the boundary conditions of the mentioned problems are periodic some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper moreover in the appendix of the paper a matlab code for gmls method is written

a mesoscopic field theory of ionic systems versus a collective variable approach we establish a link between the two functional approaches a mesoscopic field theory developed recently by ciach and stell 2000 j mol liq 87 253 for the study of ionic models and an exact statistical field theory based on the method of collective variables

mean field theory for coulomb systems we study a classical charge symmetric system with an external charge distributionq in three dimensions in the limit that the plasma parameter zero we prove that ifq is scaled appropriately then the correlation functions converge pointwise to those of an ideal gas in the external mean field x where is given by 2z sinh q this is the mean field equation of debye and huckel the proof uses the sine gordon transformation the mayer expansion and a correlation inequality

a generalization of the stillinger lovett sum rules for the two dimensional jellium in the equilibrium statistical mechanics of classical coulomb fluids the long range tail of the coulomb potential gives rise to the stillinger lovett sum rules for the charge correlation functions for the jellium model of mobile particles of charge q immersed in a neutralizing background the stillinger lovett sum rules give the charge and second moment of the screening cloud around a particle of the jellium in this paper we generalize these sum rules to the screening cloud induced around a pointlike guest charge zq immersed in the bulk interior of the 2d jellium with the coupling constant q2 is the inverse temperature in the whole region of the thermodynamic stability of the guest charge amplitude z 2 the derivation is based on a mapping technique of the 2d jellium at the coupling even positive integer onto a discrete 1d anticommuting field theory we assume that the final results remain valid for all real values of corresponding to the fluid regime the generalized sum rules reproduce for arbitrary coupling the standard z 1 and the trivial z 0 results they are also checked in the debye huckel limit 0 and at the free fermion point 2 the generalized second moment sum rule provides some exact information about possible sign oscillations of the induced charge density in space

exact renormalization group a new method for blocking the action we consider the exact renormalization group for a non canonical scalar field theory in which the field is coupled to the external source in a special nonlinear way the wilsonian action and the average effective action are then simply related by a legendre transformation up to a trivial quadratic form an exact mapping between canonical and non canonical theories is obtained as well as the relations between their flows an application to the theory of liquids is sketched

the lie group method based on radial basis functions for solving nonlinear high dimensional generalized benjamin bona mahony burgers equation in arbitrary domains abstract the aim of this paper is to introduce a new numerical method for solving the nonlinear generalized benjamin bona mahony burgers gbbmb equation this method is combination of group preserving scheme gps with radial basis functions rbfs which takes advantage of two powerful methods one as geometric numerical integration method and the other meshless method thus we introduce this method as the lie group method based on radial basis functions lg rbfs in this method we use kansas approach to approximate the spatial derivatives and then we apply gps method to approximate first order time derivative one of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions to demonstrate this point we solve nonlinear gbbmb equation on various geometric domains in one two and three dimension spaces the results of numerical experiments are compared with analytical solutions and the method presented in dehghan et al 2014 to confirm the accuracy and efficiency of the presented method

a new algorithm based on lucas polynomials for approximate solution of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equation in this paper a new method based on hybridization of lucas and fibonacci polynomials is developed for approximate solutions of 1d and 2d nonlinear generalized benjaminbonamahonyburgers equations firstly time discretization is made by using finite difference approaches after that unknown function and its derivatives are expanded to lucas series based on these series expansion differentiation matrices are derived by utilizing fibonacci polynomials by doing so the solution of the mentioned equations is reduced to the solution of an algebraic system of equations by solving this system of equations the lucas series coefficients are obtained then substituting these coefficients into lucas series expansion approximate solutions can be constructed successively the main goal of this paper is to indicate that lucas polynomial based method is appropriate for 1d and 2d nonlinear problems efficiency and performance of the proposed method are judged on six test problems which consists of the 1d and 2d version of mentioned equation by calculating l2 and l error norms feasibility of the method is verified by obtained accurate results

legendre wavelets based numerical algorithm for simulation of multidimensional benjamin bona mahony burgers and sobolev equations abstract in this work a numerical algorithm is developed with the help of legendre wavelets and quasilinearization method for simulation of multidimensional benjamin bona mahony burgers bbmb and sobolev equations initially non linear equation is linearized by quasilinearization method and then legendre wavelets are used for both space and temporal discretization the algorithm is developed without the aid of finite difference method to preserve the accuracy of the legendre wavelets in numerical experiments time dependent 1d 2d and 3d problems are considered to check the accuracy and efficiency of the algorithm a comparison is made with the analytical and numerical solutions available in the literature it is concluded that the present algorithm without finite difference based legendre method gives better results than finite difference based haar wavelet methods for time dependent 2d bbmb and 2d sobolev equations also numerical experiments reveal that the proposed algorithm produces more accurate results even with comparatively coarser grids

an efficient numerical algorithm for multi dimensional time dependent partial differential equations abstract an efficient and robust numerical scheme based on haar wavelets and finite differences is suggested for the solution of two dimensional time dependent linear and nonlinear partial differential equations pdes excellent feature of the scheme is the conversion of linear and non linear pdes to algebraic equations which are comparatively easy to handle convergence of the scheme which guarantees small error norm as the resolution level increases is also an important part of this work different error norms are computed to check efficiency of the technique computations verify accuracy flexibility and low computational cost of the method

on the validity of the inverse conjecture in classical density functional theory it is shown that the basic assumptions of the classical density functional approach are rigorously correct forh stable systems in the grand canonical ensemble moreover it is established that the set of all single particle densities is convex these results are derived by providing necessary and sufficient conditions for the solution of the classical inverse problem for single particle densities analogous results are obtained for the solution of the higher order correlation inverse problem and the ramifications of these results for the validity of two body decomposition of forces are discussed

a variational method for second order shape derivatives we consider shape functionals obtained as minima on sobolev spaces of classical integrals having smooth and convex densities under mixed dirichlet neumann boundary conditions we propose a new approach for the computation of the second order shape derivative of such functionals yielding a general existence and representation theorem in particular we consider the p torsional rigidity functional for p geq 2

a formally exact field theory for classical systems at equilibrium we propose a formally exact statistical field theory for describing classical fluids with ingredients similar to those introduced in quantum field theory we consider the following essential and related problems i how to find the correct field functional hamiltonian which determines the partition function ii how to introduce in a field theory the equivalent of the indiscernibility of particles and iii how to test the validity of this approach we can use a simple hamiltonian in which a local functional transposes in terms of fields the equivalent of the indiscernibility of particles the diagrammatic expansion and the renormalization of this term are presented this corresponds to a non standard problem in feynman expansion and requires a careful investigation then a non local term associated with an interaction pair potential is introduced in the hamiltonian it has been shown that there exists a mapping between this approach and the standard statistical mechanics given in terms of mayer function expansion we show on three properties the chemical potential the so called contact theorem and the interfacial properties that in the field theory the correlations are shifted on non usual quantities some perspectives of the theory also are given

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

the use of compact boundary value method for the solution of two dimensional schrodinger equation in this paper a high order and accurate method is proposed for solving the unsteady two dimensional schrodinger equation we apply a compact finite difference approximation of fourth order for discretizing spatial derivatives and a boundary value method of fourth order for the time integration of the resulting linear system of ordinary differential equations the proposed method has fourth order accuracy in both space and time variables moreover this method is unconditionally stable due to the favorable stability property of boundary value methods the results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature these results show that the combination of a compact finite difference approximation of fourth order and a fourth order boundary value method gives an efficient algorithm for solving the two dimensional schrodinger equation

numerical solutions of two dimensional sobolev and generalized benjamin bona mahony burgers equations via haar wavelets abstract this work addresses numerical method based on haar wavelets and finite differences to solve two dimensional linear nonlinear sobolev and non linear generalized benjamin bona mahony burgers ngbbmb equations the temporal part is discretized using finite differences while spatial part is approximated by two dimensional haar wavelets with this strategy computing solution of two dimensional pdes reduces to computing solution of linear system of algebraic equations collocation approach is then applied to determine the wavelet coefficients from linear system this paper shows that two dimensional haar wavelets are suitable and effective for two dimensional linear and non linear pdes for validation of the proposed scheme different problems have been solved and error norms l l 2 are computed computation verifies that current scheme has good outcome

on the convergence of the modified rosenau and the modified benjamin bona mahony equations abstract we consider the modified rosenau and the modified benjamin bona mahony equations which contain nonlinear dispersive effects we prove that as the diffusion parameter tends to zero the solutions of the dispersive equations converge to entropy solutions of a scalar conservation law the proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the l p setting

error estimate in fractional differential equations using multiquadratic radial basis functions the purpose of this paper is to investigate the collocation method based on multiquadric mq radial basis functions rbfs for fractional differential equations fdes in the process of doing this a scattered data rbf interpolation is considered a posterior error bound for the smooth solutions of fdes in the proposed method is established in the l 2 norm using an alternate characterization of the native space numerical results are presented which confirm the theoretical prediction of the convergence behavior of the proposed method

the lack of strict convexity and the validity of the comparison principle for a simple class of minimizers abstract let w x a x b be an affine function in r n r n l r n r be convex and w be a local minimizer of i v l v x d x in w 1 1 r with w x w x on in the trace sense then w satisfies the comparison principle from above i e w x w x a e on if and only if a l a does not belong to the relative interior of a n dimensional face of the epigraph of l as a consequence if f is the projection of a bounded face of the epigraph of l the local minimizer w x max x x 0 f satisfies the comparison principle from above if and only if dim f n 1 or x 0

finite size corrections for coulomb systems in the debye huckel regime it has been argued that for a finite two dimensional classical coulomb system of characteristic size r in its conducting phase as r the total free energy times the inverse temperature admits an expansion of the form where is the euler characteristic of the manifold where the system exists the first two terms represent the bulk free energy and the surface free energy respectively the coefficients a and b are non universal but the coefficient of ln r is universal it does not depend on the detail of the microscopic constitution of the system particle densities temperature etc by doing the usual legendre transform this universal finite size correction is also present in the grand potential the explicit form of the expansion has been checked for some exactly solvable models for a special value of the coulombic coupling in this paper we present a method for obtaining these finite size corrections in the debye huckel regime it is based on the sine gordon field theory to find an expression for the grand canonical partition function in terms of the spectrum of the laplace operator as an example we find explicitly the grand potential expansion for a coulomb system confined in a disc and in an annulus with ideal conductor walls

characteristic finite element methods for nonlinear sobolev equations1this work is supported by the doctoral fundation of the educational committee of china 1 in this paper the time stepping along characteristic finite element methods for the nonlinear sobolev equation is considered optimal convergence rates in h 1 and l 2 are demonstrated for this scheme numerical results published elsewhere have confirmed that larger time steps are appropriate with these scheme and that the approximations exhibit improved qualitative behavior

on the convergence of difference schemes for the benjamin bona mahony bbm equation abstract in this paper we analyze a crank nicolson type finite difference scheme for the bbm equation we prove the second order convergence in discrete h 1 norm of the difference approximate solutions the existence stability and uniqueness are also discussed in detail at last a linearized difference scheme is given and its convergence is also proved

a new class of polynomial functions for approximate solution of generalized benjamin bona mahony burgers gbbmb equations abstract in this paper a new class of polynomial functions equipped with a parameter is used which is applied to approximate the gbbmb in this method the first time discretization of the gbbmb equations is made by using finite difference approaches then at any spatial point a nonlinear equations system is generated using the collocation points and solving a nonlinear system in the least squares method the coefficients of functions are obtained in some examples a linearization is performed first and then the collocation method is applied

a trickiness of the high temperature limit for number density correlation functions in classical coulomb fluids the debye huckel theory describes rigorously the thermal equilibrium of classical coulomb fluids in the high temperature 0 regime denotes the inverse temperature it is generally believed that the debye huckel theory and the systematic high temperature expansion provide an adequate description also in the region of small strictly positive values of 0 this hypothesis is tested in the present paper on a two dimensional coulomb gas of pointlike unit charges interacting via a logarithmic potential which is equivalent to an integrable sine gordon field model in particular we apply a form factor method to obtain the exact asymptotic large distance behavior of particle correlation functions considered in the charge and number density combinations we first determine the general forms of the leading and subleading asymptotic terms at strictly positive 0 and then evaluate their high temperature 0 forms in the case of the charge correlation function the leading asymptotic term at a strictly positive 0 is also the leading one in the high temperature 0 regime on the contrary the 0 behavior of the number density correlation function is accompanied by an interference between the first two asymptotic terms consequently the large distance behavior of this function exhibits a discontinuity when going from strictly positive values of 0 to the debye huckel limit 0 this is the crucial conclusion of the paper the large distance asymptotics and the high temperature limit do not commute for the density correlation function of the two dimensional coulomb gas

sine gordon theory for the equation of state of classical hard core coulomb systems ii high temperature expansion we perform a high temperature expansion of the grand potential of the restrictive primitive model of electrolytes in the frame of the extended sine gordon theory exposed in the companion paper we recover a result already obtained by stell an lebowitz j chem phys 49 3706 1968 by means of diagrammatic expansions

galerkin finite element method for the rosenau rlw equation finite element galerkin solutions for the rosenau rlw rrlw equation are considered the existence and uniqueness of the weak solution are shown and the semidiscrete galerkin scheme is studied using appropriate projections for a second order accuracy in time we propose the galerkin crank nicolson fully discrete method we also prove the convergence of the extrapolated crank nicolson scheme finally some numerical experiments are given to demonstrate the validity and accuracy of our method

the numerical solution of nonlinear high dimensional generalized benjamin bona mahony burgers equation via the meshless method of radial basis functions in this paper a numerical technique is proposed for solving the nonlinear generalized benjamin bona mahony burgers equation firstly we obtain a time discrete scheme by approximating the first order time derivative via forward finite difference formula then we use kansa s approach to approximate the spatial derivatives we prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method also we show that convergence order of the time discrete scheme is o t we solve the two dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular triangular and circular domains and also the three dimensional case is solved on the cubical and spherical domains the aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations also several test problems including the three dimensional case are given numerical examples confirm the efficiency of the proposed scheme

numerical simulation of kadomtsev petviashvili benjamin bona mahony equations using finite difference method in this paper the finite difference method is employed to solve kadomtsev petviashvili benjamin bona mahony ii kp bbm ii partial differential equations the time and space variable are discretized by the crank nicholson method and the central difference scheme respectively the consistence and stability are also proved some examples are investigated to verify the efficiency of the present method

a conservative linear difference scheme for the 2d regularized long wave equation abstract in the present work a three level in time linear and conservative implicit finite difference scheme for solving the 2d regularized long wave equation is proposed the existence uniqueness and conservations for mass and energy of the numerical solution are proved by the discrete energy method the new scheme is shown to be second order convergent and unconditionally stable numerical examples are provided to show the present scheme to be efficient and reliable

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

general considerations on the finite size corrections for coulomb systems in the debye huckel regime we study the statistical mechanics of classical coulomb systems in a low coupling regime debye huckel regime in a confined geometry with dirichlet boundary conditions for the electric potential we use a method recently developed by the authors which relates the grand partition function of a coulomb system in a confined geometry with a certain regularization of the determinant of the laplacian on that geometry with dirichlet boundary conditions we study several examples of fully confining geometry in two and three dimensions and semi confined geometries where the system is confined only in one or two directions of the space we also generalize the method to study systems confined in arbitrary geometries with smooth boundary we find a relation between the expansion for small argument of the heat kernel of the laplacian and the large size expansion of the grand potential of the coulomb system this allow us to find the finite size expansion of the grand potential of the system in general we recover known results for the bulk grand potential in two and three dimensions and the surface tension for two dimensional systems we find the surface tension for three dimensional systems for two dimensional systems our general calculation of the finite size expansion gives a proof of the existence a universal logarithmic finite size correction predicted some time ago at least in the low coupling regime for three dimensional systems we obtain a prediction for the curvature correction to the grand potential of a confined system

l p error estimates for shifted surface spline interpolation on sobolev space the accuracy of interpolation by a radial basis function is usually very satisfactory provided that the approximant f is reasonably smooth however for functions which have smoothness below a certain order associated with the basis function no approximation power has yet been established hence the purpose of this study is to discuss the lp approximation order 1 p of interpolation to functions in the sobolev space wpk with k max 0 d 2 d p we are particularly interested in using the shifted surface spline which actually includes the cases of the multiquadric and the surface spline moreover we show that the accuracy of the interpolation method can be at least doubled when additional smoothness requirements and boundary conditions are met

optimal constant shape parameter for multiquadric based rbf fd method radial basis functions rbfs have become a popular method for interpolation and solution of partial differential equations pdes many types of rbfs used in these problems contain a shape parameter and there is much experimental evidence showing that accuracy strongly depends on the value of this shape parameter in this paper we focus on pde problems solved with a multiquadric based rbf finite difference rbf fd method we propose an efficient algorithm to compute the optimal value of the shape parameter that minimizes the approximation error the algorithm is based on analytical approximations to the local rbf fd error derived in 1 we show through several examples in 1d and 2d both with structured and unstructured nodes that very accurate solutions compared to finite differences can be achieved using the optimal value of the constant shape parameter

radial basis functions method for solving the fractional diffusion equations abstract fractional order diffusion equations are generalizations of classical diffusion equations treating super diffusive flow processes the paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions rbfs for the numerical solution of a class of initial boundary value fractional diffusion equations with variable coefficients on a finite domain the space fractional derivatives are defined by using riemann liouville fractional derivative we first provide riemann liouville fractional derivatives for the five kinds of rbfs including the powers gaussian multiquadric matern and thin plate splines in one dimension the time dependent fractional diffusion equation is discretized in space with the rbf collocation method and the remaining system of ordinary differential equations odes is advanced in time with an ode method using a method of lines approach some numerical results are given in order to demonstrate the efficiency and accuracy of the method additionally some physical properties of this fractional diffusion system are simulated which further confirm the effectiveness of our method the stability of the linear systems arising from discretizing riemann liouville fractional differential operator with rbfs is also analysed

on the lavrentiev phenomenon for multiple integral scalar variational problems abstract we prove the non occurrence of lavrentiev gaps between lipschitz and sobolev functions for functionals of the form i u f u u u when r n r is lipschitz and belongs to a wide class of open bounded sets in r n containing lipschitz domains the lagrangian f is assumed to be either convex in both variables or a sum of functions f s a s g b s with g convex and s a s g 0 b s satisfying a non oscillatory condition at infinity we thus derive the non occurrence of the lavrentiev phenomenon for unnecessarily convex functionals of the gradient no growth conditions are assumed

new periodic and soliton solutions for the generalized bbm and burgers bbm equations abstract in this paper we consider the benjamin bona mahony equation bbm and we obtain new exact solutions for it by using a generalization of the well known tanh coth method new periodic and soliton solutions for the generalized bbm and burgers bbm equations are formally derived

nonlinear stability of generalized benjamin bona mahony burgers shock profiles in several dimensions abstract this paper is concerned with nonlinear stability of planar viscous shock profiles of the generalized benjamin bona mahony burgers equations in two dimensions our analysis is motivated by j goodman s work 1989 4 on the nonlinear stability of scalar viscous shock profiles in two dimensions and is based on some new decay estimates on the planar viscous shock profiles of the generalized benjamin bona mahony burgers equations

the functional of the grand partition function for the investigation of the liquid gas critical point an approach for the description of a spatially homogeneous system of interacting particles in the vicinity of the critical point is proposed an explicit expression for the functional integral form of the partition function in the grand canonical ensemble is obtained the functional integral is defined on a set of the collective variables cv k the variable k corresponds to the density vibration mode with the wave vector k it is shown that in the vicinity of the critical point for all k larger than some b the cv are distributed with the gaussian measure density in the long wave region k b the surfaces of the cumulants consisting of the correlation functions of some reference system the reference system is a model system of particles with repulsive pair interactions possess wide plateaus in the vicinity of the point k 0 it is shown that due to these two facts the homogeneous system can be put in correspondance with a certain lattice system with spacing c b the problem of the calculation of the partition function can be reduced to the calculation of the functional for a three dimensional ising model with an external field

evaluation of mixed crank nicolson scheme and tau method for the solution of klein gordon equation abstract numerical method based on the crank nicolson scheme and the tau method is proposed for solving nonlinear klein gordon equation nonlinear klein gordon equation is reduced by crank nicolson scheme to the system of ordinary differential equations then tau method is used to solve this system by using interpolating scaling functions and operational matrix of derivative the order of convergence is proposed and some numerical examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

sine gordon theory for the equation of state of classical hard core coulomb systems iii loopwise expansion we present an exact field theoretical representation of an ionic solution made of charged hard spheres the action of the field theory is obtained by performing a hubbard stratonovich transform of the configurational boltzmann factor it is shown that the stillinger lovett sum rules are satisfied if and only if all the field correlation functions are short range functions the mean field gaussian and two loops approximations of the theory are derived and discussed the mean field approximation for the free energy constitutes an exact lower bound for the exact free energy while the mean field pressure is an exact upper bound the one loop order approximation is shown to be identical with the random phase approximation of the theory of liquids finally at the two loop order and in the pecular case of the restricted primitive model one recovers results obtained in the framework of the mode expansion theory

solving pdes with the aid of two dimensional haar wavelets two dimensional haar wavelets are applied for solution of the partial differential equations pdes the proposed method is mathematically simple and fast to demonstrate the efficiency of the method two test problems solution of the diffusion and poisson equations are discussed computer simulation showed that the method guarantees the necessary exactness already for a small number of grid points

numerical solution of the nonlinear klein gordon equation using radial basis functions the nonlinear klein gordon equation is used to model many nonlinear phenomena in this paper we propose a numerical scheme to solve the one dimensional nonlinear klein gordon equation with quadratic and cubic nonlinearity our scheme uses the collocation points and approximates the solution using thin plate splines tps radial basis functions rbf the implementation of the method is simple as finite difference methods the results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme