nonlocal symmetries and formulae for generation of solutions for a class of diffusion convection equations abstract new formulae of nonlocal nonlinear superposition and generation of solutions are proposed for nonlinear diffusion convection equations which are linearizable or are invariant with respect to a generalized hodograph transformation or connected by this transformation we study in what particular ways additional lie symmetries of diffusion convection equations induce nonlocal symmetries of equations obtained from the initial ones by nonlocal transformations the formulae derived are used for the construction of exact solutions

group classification of variable coefficient generalized kawahara equations an exhaustive group classification of variable coefficient generalized kawahara equations is carried out as a result we derive new variable coefficient nonlinear models admitting lie symmetry extensions all inequivalent lie reductions of these equations to ordinary differential equations are performed we also present some examples on the construction of exact and numerical solutions

numerical methods for time periodic solutions of nonlinear parabolic boundary value problems this paper is devoted to a numerical analysis of periodic solutions of a finite difference system which is a discrete version of a class of nonlinear reaction diffusion convection equations under nonlinear boundary conditions three monotone iterative schemes for the finite difference system are presented and it is shown by the method of upper and lower solutions that the sequence of iterations from each of these iterative schemes converges monotonically to either a maximal periodic solution or a minimal periodic solution depending on whether the initial iteration is an upper solution or a lower solution a comparison theorem for the various monotone sequences is given it is also shown that the maximal and minimal periodic solutions of the finite difference system converge to the corresponding maximal and minimal periodic solutions of the reaction diffusion convection equation as the mesh size decreases to zero some error estimates between the theoretical and the computed iterations for each of the three iterative schemes are obtained and a discussion on the numerical stability of these schemes is given also given are some numerical results of a logistic reaction diffusion problem

classification of reduction operators and exact solutions of variable coefficient newell whitehead segel equations abstract a class of the newell whitehead segel equations also known as generalized fisher equations and newell whitehead equations is studied with lie and nonclassical symmetry points of view the classifications of lie reduction operators and of regular nonclassical reduction operators are performed the set of admissible transformations the equivalence groupoid of the class is described exhaustively the criterion of reducibility of variable coefficient newell whitehead segel equations to their constant coefficient counterparts is derived wide families of exact solutions for such variable coefficient equations are constructed

waveform relaxation for reaction diffusion equations abstract we report a new waveform relaxation wr algorithm for general semi linear reaction diffusion equations the superlinear rate of convergence of the new wr algorithm is proved and we also show the advantages of the new approach superior to the classical wr algorithms by the estimation on iteration errors the corresponding discrete wr algorithm for reaction diffusion equations is presented and further the parallelism of the discrete wr algorithm is analyzed moreover the new approach is extended to handle the coupled reaction diffusion equations numerical experiments are carried out to verify the effectiveness of the theoretic work

uniform convexity and smoothness and their applications in finsler geometry we generalize the alexandrov toponogov comparison theorems to finsler manifolds under suitable upper lower resp bounds on the flag and tangent curvatures together with the 2 uniform convexity smoothness resp of tangent spaces we show the 2 uniform convexity smoothness resp of finsler manifolds as applications we prove the almost everywhere existence of the second order differentials of semi convex functions and of c concave functions with the quadratic cost function

group analysis of 2 1 and 3 1 dimensional diffusion convection equations abstract we perform the complete lie group classification of a 2 1 and a 3 1 dimensional classes of non linear diffusion convection equations this classification generalizes and completes existing results in the literature the derived lie symmetries are used for construction of similarity reductions and exact solutions of certain equations from both classes

on the equation y2 x x2 p generators are found for the group of rational points on the title curve for all primes p 5 mod 8 less than 1 000 the rank is always i in accordance with conjectures of selmer and mordell some of the generators are rather large

conservation laws and symmetries of radial generalized nonlinear p laplacian evolution equations abstract a class of generalized nonlinear p laplacian evolution equations is studied these equations model radial diffusion reaction processes in n 1 dimensions where the diffusivity depends on the gradient of the flow for this class all local conservation laws of low order and all lie symmetries are derived the physical meaning of the conservation laws is discussed and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases the symmetries are used to derive exact group invariant solutions from solvable three dimensional subgroups of the full symmetry group which yields a direct reduction of the nonlinear equation to a quadrature the physical and analytical properties of these exact solutions are explored some of which describe moving interfaces and green s functions

potential symmetries of nonlinear diffusion convection equations in this paper potential symmetries are sought for the nonlinear diffusion convection equations the functional forms of f u and k u that admit such symmetries are completely classified

on linearization of hyperbolic equations using differential invariants abstract in this paper we consider the general class of hyperbolic equations u x t f x t u u x u t we use equivalence transformations to derive differential invariants for this class and for two subclasses then we employ these invariants to construct equations that can be linearized via local mappings further applications of the differential invariants are given

waveform relaxation methods for functional differential systems of neutral type abstract we investigate continuous time and discretized waveform relaxation iterations for functional differential systems of neutral type it is demonstrated that continuous time iterations converge linearly for neutral equations and superlinearly when the right hand side is independent of the history of the derivative of the solution the error bounds for discretized iterations are also obtained and some implementation aspects are discussed numerical results are presented which indicate a potential speedup of this technique as compared with the classical approach based on discrete variable methods

lie symmetries and invariants for a 2d nonlinear heat equation abstract the paper generalizes previous results on the 2d ricci flow equation in a conformal gauge it investigates a general form of the 2d nonlinear heat equation and it points out all possible cases where lie type symmetries associated invariant quantities and similarity solutions can appear the connection with the already known results for the 1d problem is obtained

spectra and pseudospectra of waveform relaxation operators we describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators since the spectrum of the finite interval waveform relaxation operator is only a singleton the spectrum of the infinite interval operator is typically used to describe the behavior of waveform relaxation algorithms here we show that the pseudospectrum is a more useful tool for analyzing convergence of waveform iterations on finite intervals and prove that the pseudospectrum of the infinite interval operator is in fact the limit of the pseudospectra of the finite interval operators

some non self similar solutions to a nonlinear diffusion equation the author generates exact solutions to the nonlinear diffusion equation ut del u 1 2 del u which are not similarity solutions some applications and generalizations are indicated

group classification of 1 1 dimensional schrodinger equations with potentials and power nonlinearities we perform the complete group classification in the class of nonlinear schrodinger equations of the form i t xx v t x 0 where v is an arbitrary complex valued potential depending on t and x is a real nonzero constant we construct all the possible inequivalent potentials for which these equations have nontrivial lie symmetries using a combination of algebraic and compatibility methods the proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics

waveform relaxation methods for periodic differential functional systems the convergence of waveform relaxation methods applied to a periodic differential functional system is studied the continuous dependence is also discussed

waveform relaxation for functional differential equations the convergence of waveform relaxation techniques for solving functional differential equations is studied new error estimates are derived that hold under linear and nonlinear conditions for the right hand side of the equation sharp error bounds are obtained under generalized time dependent lipschitz conditions the convergence of the waveform method and the quality of the a priori error bounds are illustrated by means of extensive numerical data obtained by applying the method of lines to three partial functional differential equations

convergence of waveform relaxation methods for differential algebraic systems this paper gives sufficient conditions for existence and uniqueness of solutions and for the convergence of picard iterations and more general waveform relaxation methods for differential algebraic systems of neutral type the results are obtained by the contraction mapping principle on banach spaces with weighted norms and by the use of the perron frobenius theory of nonnegative and nonreducible matrices it is demonstrated that waveform relaxation methods are convergent faster than the classical picard iterations

differential invariants of a class of lagrangian systems with two degrees of freedom abstract we consider systems of euler lagrange equations with two degrees of freedom and with lagrangian being quadratic in velocities for this class of equations the generic case of the equivalence problem is solved with respect to point transformations using lie s infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems we describe certain types of lagrangian systems in terms of their invariants the results are illustrated by several examples

enhanced group analysis and conservation laws of variable coefficient reaction diffusion equations with power nonlinearities a class of variable coefficient 1 1 dimensional nonlinear reaction diffusion equations of the general form f x ut g x unux x h x um is investigated different kinds of equivalence groups are constructed including ones with transformations which are nonlocal with respect to arbitrary elements for the class under consideration the complete group classification is performed with respect to convenient equivalence groups generalized extended and conditional ones and with respect to the set of all local transformations usage of different equivalences and coefficient gauges plays the major role for simple and clear formulation of the final results the corresponding set of admissible transformations is described exhaustively then using the most direct method we classify local conservation laws some exact solutions are constructed by the classical lie method

on the convergence of waveform relaxation methods for differential functional systems of equations in this paper the convergence of a waveform relaxation method applied to an initial value problem for the volterra functional differential system is discussed it is shown that the method is convergent under the assumption that the splitting function satisfies only the one side lipschitz condition with respect to some arguments and the lipschitz condition with respect to the others the conditions given in the paper also guarantee the existence and uniqueness of the solution to the initial problem discussed in the paper the convergence of the perturbed continuous time waveform relaxation method is also discussed

second order differential invariants of a family of diffusion equations an equivalence transformation algebra for a class of nonlinear diffusion equations is found after having obtained the second order differential invariants with respect to we get some results which allow us to linearize a subclass of the equations considered

convergence of the parallel chaotic waveform relaxation method for stiff systems in this paper we establish several algorithms for parallel chaotic waveform relaxation methods for solving linear ordinary differential systems based on some given models under some different assumptions on the coefficient matrix a and its multisplittings we obtain corresponding sufficient conditions of convergence of the algorithms also a discussion on convergence speed comparison of synchronous and asynchronous algorithms is given

the hardy morrey hardy john nirenberg inequalities involving distance to the boundary abstract we strengthen the classical inequality of c b morrey concerning the optimal holder continuity of functions in w 1 p when p n by replacing the l p modulus of the gradient with the sharp hardy difference involving distance to the boundary when p n we do the same strengthening in the integral form of a well known inequality due to f john and l nirenberg

lie symmetry analysis and exact solutions of the quasigeostrophic two layer problem the quasigeostrophic two layer model is of superior interest in dynamic meteorology since it is one of the easiest ways to study baroclinic processes in geophysical fluid dynamics the complete set of point symmetries of the two layer equations is determined an optimal set of one and two dimensional inequivalent subalgebras of the maximal lie invariance algebra is constructed on the basis of these subalgebras we exhaustively carry out group invariant reduction and compute various classes of exact solutions wherever possible reference to the physical meaning of the exact solutions is given in particular the well known baroclinic rossby wave solutions in the two layer model are rediscovered

sobolev and isoperimetric inequalities with monomial weights abstract we consider the monomial weight x 1 a 1 x n a n in r n where a i 0 is a real number for each i 1 n and establish sobolev isoperimetric morrey and trudinger inequalities involving this weight they are the analogue of the classical ones with the lebesgue measure dx replaced by x 1 a 1 x n a n d x and they contain the best or critical exponent which depends on a 1 a n more importantly for the sobolev and isoperimetric inequalities we obtain the best constant and extremal functions when a i are nonnegative integers these inequalities are exactly the classical ones in the euclidean space r d with no weight when written for axially symmetric functions and domains in r d r a 1 1 r a n 1

periodic waveform relaxation solutions of nonlinear dynamic equations we present a simple theorem to safeguard the convergence of waveform relaxation wr solutions of a dynamic system described by nonlinear ordinary differential equations odes with a periodic constraint namely if a basic expression of certain constants issued from the system is less than one the proposed wr algorithm is convergent to the exact solution it is the first time that wr is used to treat periodic solutions of nonlinear dynamic systems a numerical example is provided to confirm the theoretic work of the paper

singular invariant equation for the 1 1 fokker planck equation semi invariants for the linear parabolic equations with two independent variables time variable t and space variable x and one dependent variable u are derived under the transformation of the independent variables by using the infinitesimal method we also obtain the joint invariant equation for the above mentioned equation under equivalence transformation in fact we prove a necessary and sufficient condition for a 1 1 parabolic equation to be reducible via a local equivalence transformation to the one dimensional classical heat equation this result provides practical criteria for reduction finally examples of 1 1 fokker planck equations from applications are given to verify the results obtained

group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations a complete group classification of a class of variable coefficient 1 1 dimensional telegraph equations f x utt h u ux x k u ux is given by using a compatibility method and additional equivalence transformations a number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained furthermore the possible additional equivalence transformations between equations from the class under consideration are investigated exact solutions of special forms of these equations are also constructed via classical lie method and generalized conditional transformations local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations

lie symmetry solutions for anomalous diffusion an anomalous diffusion equation is solved by the method of lie symmetries the nonlinear diffusion equation includes fractal dimensions and power law dependence on the radial variable and the diffusion function the lie symmetries appropriate boundary and initial conditions and a first integral replace an ansatz that was previously used to find a class of analytic solutions the mean squared displacement of the diffusion varies in time t by an exponent that is easily found by the lie symmetries and boundary conditions

finsler hardy kato s inequality abstract we prove an improved version of the trace hardy inequality so called kato s inequality on the half space in finsler context the resulting inequality extends the former one obtained by 3 in euclidean context also we discuss the validity of the same type of inequalities on open cones

a weighted anisotropic sobolev type inequality and its applications to hardy inequalities in this paper we focus our attention on an embedding result for a weighted sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function f starting from this type of inequalities we prove some refined hardy type inequalities

differential constraints and exact solutions of nonlinear diffusion equations 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 equations used in the search for classical lie symmetries

lie group classification of the nonlinear generalized kompaneets equations with two functional parameters we study the nonlinear generalized kompaneets equations gkes with two functional parameters by using the lie ovsiannikov algorithm we carried out the group classification of the equations it is shown that the kernel algebra of the full groups of the gkes is the one dimensional lie algebra we obtained nine non equivalent up to transformations from the equivalence group nonlinear gkes that allow wider invariance algebras than the kernel one we found a number of exact solutions of those gkes which have the maximal symmetry properties

the distance function from the boundary in a minkowski space let the space be endowed with a minkowski structure that is is the gauge function of a compact convex set having the origin as an interior point and with boundary of class and let be the asymmetric distance associated to given an open domain of class let be the minkowski distance of a point from the boundary of we prove that a suitable extension of to which plays the role of a signed minkowski distance to is of class in a tubular neighborhood of and that is of class outside the cut locus of that is the closure of the set of points of nondifferentiability of in in addition we prove that the cut locus of has lebesgue measure zero and that can be decomposed up to this set of vanishing measure into geodesics starting from and going into along the normal direction with respect to the minkowski distance we compute explicitly the jacobian determinant of the change of variables that associates to every point outside the cut locus the pair where denotes the unique projection of on and we apply these techniques to the analysis of pdes of monge kantorovich type arising from problems in optimal transportation theory and shape optimization

on sor waveform relaxation methods waveform relaxation is a numerical method for solving large scale systems of ordinary differential equations on parallel computers it differs from standard iterative methods in that it computes the solution on many time levels or along a continuous time interval simultaneously this paper deals with the acceleration of the standard waveform relaxation method by successive overrelaxation sor techniques in particular different sor acceleration schemes based on multiplication with a scalar parameter or convolution with a time dependent function are described and theoretically analyzed the theory is applied to a one dimensional and two dimensional model problem and checked against results obtained by numerical experiments

sharp uniform convexity and smoothness inequalities for trace norms we prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the schatten trace ideals c p which are the analogs of the lebesgue spaces l p in non commutative integration the inequalities are all precise analogs of results which had been known in l p but were only known in c p for special values of p in the course of our treatment of uniform convexity and smoothness inequalities for c p we obtain new and simple proofs of the known inequalities for l p

analysis of two parareal algorithms for time periodic problems the parareal algorithm which permits us to solve evolution problems in a time parallel fashion has created a lot of attention over the past decade the algorithm has its roots in the multiple shooting method for boundary value problems which in the parareal algorithm is applied to initial value problems with a particular coarse approximation of the jacobian matrix it is therefore of interest to formulate parareal type algorithms for time periodic problems which also couple the end of the time interval with the beginning and to analyze their performance in this context we present and analyze two parareal algorithms for time periodic problems one with a periodic coarse problem and one with a nonperiodic coarse problem an interesting advantage of the algorithm with the nonperiodic coarse problem is that no time periodic problems need to be solved during the iteration since on the time subdomains the problems are not time periodic either we prove for both linear and nonlinear problems convergence

complete group classification of a class of nonlinear wave equations preliminary group classification became a prominent tool in the symmetry analysis of differential equations due to the paper by ibragimov torrisi and valenti j math phys 32 2988 2995 1991 10 1063 1 529042 in this paper the partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one dimensional lie symmetry extensions related to a fixed finite dimensional subalgebra of the infinite dimensional equivalence algebra of the class under consideration in the present paper we implement the complete group classification of the same class up to both usual and general point equivalence using the algebraic method of group classification this includes the complete preliminary group classification of the class and finding those lie symmetry extensions which are not associated with subalgebras of the equivalence algebra the complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite dimensi

invariants of two and three dimensional hyperbolic equations abstract we consider linear hyperbolic equations of the form u t t i 1 n u x i x i i 1 n x i x 1 x n t u x i t x 1 x n t u t u x 1 x n t u we derive equivalence transformations which are used to obtain differential invariants for the cases n 2 and n 3 motivated by these results we present the general results for the n dimensional case it appears at least for n 2 that this class of hyperbolic equations admits differential invariants of order one but not of order two we employ the derived invariants to construct interesting mappings between equivalent equations

computing periodic solutions of linear differential algebraic equations by waveform relaxation we propose an algorithm which is based on the waveform relaxation wr approach to compute the periodic solutions of a linear system described by differential algebraic equations for this kind of two point boundary problems we derive an analytic expression of the spectral set for the periodic wr operator we show that the periodic wr algorithm is convergent if the supremum value of the spectral radii for a series of matrices derived from the system is less than 1 numerical examples where discrete waveforms are computed with a backward difference formula further illustrate the correctness of the theoretical work in this paper

space time domain decomposition for parabolic problems we analyze a space time domain decomposition iteration for a model advection diffusion equation in one and two dimensions the discretization of this iteration is the block red black variant of the waveform relaxation method and our analysis provides new convergence results for this scheme the asymptotic convergence rate is super linear and it is governed by the diffusion of the error across the overlap between subdomains hence it depends on both the size of this overlap and the diffusion coefficient in the equation however it is independent of the number of subdomains provided the size of the overlap remains fixed the convergence rate for the heat equation in a large time window is initially linear and it deteriorates as the number of subdomains increases the duration of the transient linear regime is proportional to the length of the time window for advection dominated problems the convergence rate is initially linear and it improves as the the ratio of advection to diffusion increases moreover it is independent of the size of the time window and of the number of subdomains numerical calculations illustrate our analysis

new results on group classification of nonlinear diffusion convection equations using a new method and additional conditional and partial equivalence transformations we performed group classification in a class of variable coefficient 1 1 dimensional nonlinear diffusion convection equations of the general form f x ut d u ux x k u ux we obtain new interesting cases of such equations with the density f localized in space which have non trivial invariance algebra exact solutions of these equations are constructed we also consider the problem of investigation of the possible local transformations for an arbitrary pair of equations from the class under consideration i e of describing all the possible partial equivalence transformations in this class

convergence analysis of waveform relaxation methods for neutral differential functional systems in this paper the problems of convergence and superlinear convergence of continuous time waveform relaxation method applied to volterra type systems of neutral functional differential equations are discussed under a lipschitz condition with time and delay dependent right hand side imposed on the so called splitting function more suitable conditions about convergence and superlinear convergence of continuous time wr method are obtained we also investigate the initial interval acceleration strategy for the practical implementation of the continuous time waveform relaxation method i e discrete time waveform relaxation method it is shown by numerical results that this strategy is efficacious and has the essential acceleration effect for the whole computation process

multigrid waveform relaxation on spatial finite element meshes the continuous time case the waveform relaxation method and its multigrid acceleration are studied as solution procedures for the system of ordinary differential equations obtained by finite element discretisation of a linear parabolic initial boundary value problem the convergence properties of the continuous time algorithm are theoretically investigated on finite length and infinite length time intervals in addition quantitative convergence estimates and numerical results are presented for one dimensional and two dimensional model problems

timestep acceleration of waveform relaxation dynamic iteration methods for treating certain classes of linear systems of differential equations are considered it is shown that the discretized picard lindelof waveform relaxation iteration can be accelerated by solving the defect equations with a larger timestep or by using a recursive procedure based on a succession of increasing timesteps a discussion of convergence is presented including analysis of a discrete smoothing property maintained by symmetric multistep methods applied to linear wave equations numerical experiments indicate that the method can speed convergence

reductions and symmetries for a generalized fisher equation with a diffusion term dependent on density and space abstract in this work a generalized fisher equation with a space density diffusion term is analyzed by applying the theory of symmetry reductions for partial differential equations the study of this equation is relevant in terms of its applicability in cell dynamics and tumor invasion therefore classical lie symmetries admitted by the equation are determined in addition by using the multipliers method we derive some nontrivial conservation laws for this equation finally we obtain a direct reduction of order of the ordinary differential equations associated and a particular solution

classification of potential symmetries of generalised inhomogeneous nonlinear diffusion equations we consider the class of generalised nonlinear diffusion equations f x ut g x unux x which are of considerable interest in mathematical physics we classify the nonlocal symmetries which are known as potential symmetries for these equations it turns out that potential symmetries exist only if the parameter n takes the values 2 or 23 also certain relations must be satisfied by the functions f x and g x for the cases where we obtain infinite parameter potential symmetries linearising mappings are constructed furthermore we employ the potential symmetries to derive similarity solutions

on point transformations of evolution equations two evolution equations are considered of the form ut hi x t u ux uxx i 1 2 which are related by the general point transformation x p x t u t q x t u u r x t u it is shown that the time transformation must necessarily be of the form t q t and that if hi are polynomials in the spatial derivatives of u then x p x t

group classification and symmetry reduction of variable coefficient nonlinear diffusion convection equation based on classical lie group method the group for a general class of variable coefficient nonlinear diffusion convection equation in 1 1 dimensions is obtained new symmetries are found seven models have been studied

a unified approach to improved l p hardy inequalities with best constants we present a unified approach to improved l p hardy inequalities in r n we consider hardy potentials that involve either the distance from a point or the distance from the boundary or even the intermediate case where the distance is taken from a surface of codimension 1 k n in our main result we add to the right hand side of the classical hardy inequality a weighted l p norm with optimal weight and best constant we also prove nonhomogeneous improved hardy inequalities where the right hand side involves weighted l q norms q p

a geometric characterization of a sharp hardy inequality abstract in this paper we prove that the distance function of an open connected set in r n 1 with a c 2 boundary is superharmonic in the distribution sense if and only if the boundary is weakly mean convex we then prove that hardy inequalities with a sharp constant hold on weakly mean convex c 2 domains moreover we show that the weakly mean convexity condition cannot be weakened we also prove various improved hardy inequalities on mean convex domains along the line of brezis and marcus 1997 7

differential invariants for systems of linear hyperbolic equations in this paper we consider a general class of systems of two linear hyperbolic equations motivated by the existence of the laplace invariants for the single linear hyperbolic equation we adopt the problem of finding differential invariants for the system we derive the equivalence group of transformations for this class of systems the infinitesimal method which makes use of the equivalence group is employed for determining the desired differential invariants we show that there exist four differential invariants and five semi invariants of first order applications of systems that can be transformed by local mappings to simple forms are provided

group classification of heat conductivity equations with a nonlinear source we suggest a systematic procedure for classifying partial differential equations pdes invariant with respect to low dimensional lie algebras this procedure is a proper synthesis of the infinitesimal lie method the technique of equivalence transformations and the theory of classification of abstract low dimensional lie algebras as an application we consider the problem of classifying heat conductivity equations in one variable with nonlinear convection and source terms we have derived a complete classification of nonlinear equations of this type admitting nontrivial symmetry it is shown that there are 3 7 28 and 12 inequivalent classes of pdes of the type considered that are invariant under the one two three and four dimensional lie algebras correspondingly furthermore we prove that any pde belonging to the class under study and admitting the symmetry group of a dimension higher than four is locally equivalent to a linear equation this classification is compared with existing group classifications of nonlinear heat conductivity equations and one of the conclusions is that all of them can be obtained within the framework of our approach furthermore a number of new invariant equations are constructed which have rich symmetry properties and therefore may be used for mathematical modelling of say nonlinear heat transfer processes

an adaptive wavelet method for nonlinear differential algebraic equations abstract a multi resolution collocation method with specially designed spline wavelet is presented to numerically solve a system of nonlinear differential algebraic equations of index one in circuit simulation as the wavelet collocation method is applied to such system we will have to solve large nonlinear algebraic equations for the algebraic equations we give a modified newton type iterative method which bases on two wavelet adaptive techniques we prove the convergence the stability and the complexity on the adaptive method reported here the new approach could effectively reduce computational costs and storage requirements to large extent numerical experiments are given to further illustrate the method s principle

monotone waveform relaxation for systems of nonlinear differential algebraic equations we present a monotone waveform relaxation algorithm which produces very tight upper and lower bounds of the transient response of a class of systems described by nonlinear differential algebraic equations daes that satisfy certain lipschitz conditions the choice of initial iteration is critical and we give two methods of finding it we show that the class of systems in which monotone convergence of waveform relaxation is possible is actually larger than previously reported numerical experiments are given to confirm the monotonicity of convergence of the algorithm

on form preserving point transformations of partial differential equations new identities are presented relating arbitrary order partial derivatives of and for the general point transformation these identities are used to study the nature of those point transformations which preserve the general form of a wide class of 1 1 partial differential equations these results facilitate the search for point symmetries both discrete and continuous lie and assist the search for point transformations which reduce equations to canonical but similar form a simple test for the existence of hodograph type transformations between equations of similar form is given

nonnegative solutions of fractional functional differential equations in this paper we discuss the existence and monotone iterative method of nonnegative solutions for fractional functional differential equations the main conclusion is that the nonnegative solutions can be derived from the monotone iterative method which starts off with a nonnegative upper solution or the zero function under different conditions our approach of obtaining nonnegative solutions is feasible for computational purposes

invariants of a family of third order ordinary differential equations we consider a family of third order ordinary differential equations odes with the cubic nonlinearity in second order derivative which is closed with respect to contact transformations using lie s infinitesimal method we construct the basis of differential invariants for this family of equations and the operators of invariant differentiation invariants provide a simple way of finding the equations which may be equivalent to a given nonlinear ode and the transformation connecting two equivalent equations some examples are given to illustrate the results obtained in the paper

on dynamic iteration methods for solving time periodic differential equations dynamic iteration or more popularly waveform relaxation has received a great deal of attention as an efficient parallel method for solving large systems of ordinary differential equations of initial value type by using a multigrid acceleration the method has been applied successfully to parabolic initial boundary value problems in this paper the applicability of the method is further extended to incorporate the class of time periodic ordinary and partial differential equations the authors discuss the convergence characteristics of both the continuous time and the discrete time time periodic iteration the efficacy of the new procedure is illustrated by an example

a robust finite element solver for a multiharmonic parabolic optimal control problem this paper presents the analysis of a distributed parabolic optimal control problem in a multiharmonic setting in particular the desired state is assumed to be multiharmonic after eliminating the control from the optimality system we arrive at the reduced optimality system for the state and the co state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation due to the linearity the state and the co state are multiharmonic as well we discretize the fourier coefficients by the finite element method this leads to a large system of algebraic equations which fortunately decouples into smaller systems each of them defining the cosine and sine fourier coefficients for the state and co state with respect to a single frequency for these smaller systems we construct preconditioners resulting in a fast converging minimal residual solver with a parameter independent convergence rate all these systems can be solved totally in parallel

waveform relaxation and comparison methods for multipoint boundary value problems waveform relaxation and comparison methods are used to obtain sufficient conditions for the existence and uniqueness of solutions for differential functional equations with r point boundary conditions it is shown that the waveform relaxation method converges to this solution some examples are added to verify corresponding results

submodels of model of nonlinear diffusion in the inhomogeneous medium involving absorption we study the five parameter model describing the process of nonlinear diffusion in an inhomogeneous medium in the presence of absorption for which the differential equation of the model admits a continuous lie group of transformations acting on the set of its solutions we found six submodels of the original model of nonlinear diffusion with different symmetry properties of these six submodels the five submodels with transient absorption for which the absorption coefficient depends on time according to a power law represent the greatest interest with a mathematical point of view and with the point of view of physical applications for each of these nonlinear submodels we obtained formulas for producing new solutions that contain arbitrary constants and we found all invariant submodels all essentially distinct invariant solutions describing these invariant submodels are found in an explicit form or are reduced to finding the solution of nonlinear integral equations the presence of the arbitrary constants in the integral equations that determine these solutions provide new opportunities for analytical and numerical study of boundary value problems for the received submodels and thus for the original model of nonlinear diffusion for the received invariant submodels we studied diffusion processes for which at the initial moment of the time at a fixed point is specified as a concentration and its gradient or as a concentration and its velocity solving of boundary value problems describing these processes is reduced to the solving of nonlinear integral equations we established the existence and uniqueness of solutions of these boundary value problems under some additional conditions the obtained results can be used to study the diffusion of substances diffusion of conduction electrons and other particles diffusion of physical fields and propagation of heat in inhomogeneous medium and also to study a turbulence leith model differential approximation to wave turbulence