fractional variational calculus and the transversality conditions this paper presents the euler lagrange equations and the transversality conditions for fractional variational problems the fractional derivatives are defined in the sense of riemann liouville and caputo the connection between the transversality conditions and the natural boundary conditions necessary to solve a fractional differential equation is examined it is demonstrated that fractional boundary conditions may be necessary even when the problem is defined in terms of the caputo derivative furthermore both fractional derivatives the riemann liouville and the caputo arise in the formulations even when the fractional variational problem is defined in terms of one fractional derivative only examples are presented to demonstrate the applications of the formulations

sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities the goal of this paper is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and in self similar variables the optimal convergence rates to barenblatt self similar profiles and their generalizations it relies on the identification of the optimal constants in some related hardy poincare inequalities and concludes a long series of papers devoted to generalized entropies functional inequalities and rates for nonlinear diffusion equations

rates of decay to equilibria for p laplacian type equations abstract the long time asymptotics for p laplacian type equations t p m div m p 2 m in r n is studied for p 1 and m n p 1 n p 1 the non negative solutions of the equations are shown to behave asymptotically as t like barenblatt type solutions and the explicit rates of decay are established for the convergence of the relative energy the convergence with respect to the wasserstein distances and the convergence with respect to the l 1 norm the rates are proved to be optimal for p 2 the method used is based on mass transportation inequalities

fast diffusion to self similarity complete spectrum long time asymptotics and numerology the complete spectrum is determined for the operator open image in new window on the sobolev space w1 2 rn formed by closing the smooth functions of compact support with respect to the norm open image in new window here the barenblatt profile is the stationary attractor of the rescaled diffusion equation open image in new window in the fast supercritical regime mopen image in new window the same diffusion dynamics represent the steepest descent down an entropy e u on probability measures with respect to the wasserstein distance d2 formally the operator h hess e is the hessian of this entropy at its minimum so the spectral gap h 2 n 1 m found below suggests the sharp rate of asymptotic convergence open image in new window from any centered initial data 0 u 0 x l1 rn with second moments this bound improves various results in the literature and suggests the conjecture that the self similar solution u t x r t n x r t is always slowest to converge the higher eigenfunctions which are polynomials with hypergeometric radial parts and the presence of continuous spectrum yield additional insight into the relations between symmetries of rn and the flow thus the rate of convergence can be improved if we are willing to replace the distance to with the distance to its nearest mass preserving dilation or still better affine image the strange numerology of the spectrum is explained in terms of the number of moments of

combination of epsilon and ritz methods with multiscaling basis for solving a class of fractional optimal control problems abstract this paper presents an approximate direct method for solving a class of fractional optimal control problems by combining the epsilon and ritz methods and utilizing multiscaling basis functions by applying the epsilon method we transform the given optimal control problem into a variational problem then we solve the variational problem approximately by the ritz direct method with multiscaling polynomial basis functions the choice of multiscaling basis functions provides a high rate of convergence for the ritz method the convergence of the derived approximate minimal to the exact one is extensively discussed at the end some illustrative test examples are included to demonstrate the efficiency of the technique

theory of extended solutions for fast diffusion equations in optimal classes of data radiation from singularities this paper is devoted to constructing a general theory of nonnegative solutions for the equation

equilibria of homogeneous functionals in the fair competition regime we consider macroscopic descriptions of particles where repulsion is modelled by non linear power law diffusion and attraction by a homogeneous singular non singular kernel leading to variants of the keller segel model of chemotaxis we analyse the regime in which both homogeneities scale the same with respect to dilations that we coin as fair competition in the singular kernel case we show that existence of global equilibria can only happen at a certain critical value and they are characterised as optimisers of a variant of hls inequalities we also study the existence of self similar solutions for the sub critical case or equivalently of optimisers of rescaled free energies these optimisers are shown to be compactly supported radially symmetric and non increasing stationary solutions of the non linear keller segel equation on the other hand we show that no radially symmetric non increasing stationary solutions exist in the non singular kernel case implying that there is no criticality however we show the existence of positive self similar solutions for all values of the parameter under the condition that diffusion is not too fast we finally illustrate some of the open problems in the non singular kernel case by numerical experiments

phase transitions and symmetry breaking in singular diffusion the long time asymptotics are determined for fast nonlinear diffusion by linearizing otto s gradient descent model at the barenblatt profile the spectrum of the entropy is explicitly determined the dynamics are found to undergo a phase transition in which rotational symmetry is broken as the strength of the nonlinearity is varied

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity ii this article continues the analysis of the preceding article showing that the lyapunov functional introduced by newman can be used to prove the stability of the barenblatt pattle solutions of the porous medium equation

assessing a set of additive utility functions for multicriteria decision making the uta method abstract the purpose of the method presented in this paper is to assess additive utility functions which aggregate multiple criteria in a composite criterion using the information given by a subjective ranking on a set of stimuli or actions weak order comparison judgments and the multicriteria evaluations of these actions it is an ordinal regression method using linear programming to estimate the parameters of the utility function stability and sensitivity analysis leads to the assessment of a set of utility functions by means of post optimality analysis techniques in linear programming finally a simple illustrative example is presented and some extensions of the method are proposed

weighted interpolation inequalities a perturbation approach we study optimal functions in a family of caffarelli kohn nirenberg inequalities with a power law weight in a regime for which standard symmetrization techniques fail we establish the existence of optimal functions study their properties and prove that they are radial when the power in the weight is small enough radial symmetry up to translations is true for the limiting case where the weight vanishes a case which corresponds to a well known subfamily of gagliardo nirenberg inequalities our approach is based on a concentration compactness analysis and on a perturbation method which uses a spectral gap inequality as a consequence we prove that optimal functions are explicit and given by barenblatt type profiles in the perturbative regime

application of taylor series in obtaining the orthogonal operational matrix in this research first we explicitly obtain the relation between the coefficients of the taylor series and jacobi polynomial expansions then we present a new method for computing classical operational matrices derivative integral and product for general jacobi orthogonal functions polynomial and rational this method can be used for many classes of orthogonal functions

on the parabolic kernel of the schrodinger operator etude des equations paraboliques du type q x t t u x t 0 sur une variete riemannienne generale introduction estimations de gradients inegalites de harnack majorations et minorations des solutions fondamentales equation de la chaleur et noyau de green operateur de schrodinger

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

radial fast diffusion on the hyperbolic space we consider positive radial solutions to the fast diffusion equation ut u on the hyperbolic space h for n 2 m ms 1 ms n 2 n 2 by radial we mean solutions depending only on the geodesic distance r from a given point o h we investigate their fine asymptotics near the extinction time t in terms of a separable solution of the form v r t 1 t t 1 1 m v r where v is the unique positive energy solution radial w r t o to v c v 1 m for a suitable c 0 a semilinear elliptic problem thoroughly studied in 29 7 we show that u converges to v in relative error in the sense that u t v t 1 0 as t t in particular the solution is bounded above and below near the extinction time t by multiples of 1 t t 1 1 m e n 1 r m solutions are smooth and bounds on derivatives are given as well in particular sharp convergence results as t t are shown for spatial derivatives again in the form of convergence in relative error

information geometry of q gaussian densities and behaviors of solutions to related diffusion equations this paper presents new geometric aspects of the behaviors of solutions to the porous medium equation pme and its associated equation first we discuss thermostatistical structure with information geometry on a manifold of generalized exponential densities a dualistic relation between the two existing formalisms is elucidated next by equipping the manifold of q gaussian densities with such a structure we derive several physically and geometrically interesting properties of the solutions the manifold is proved invariant and attracting for the evolving solutions which play crucial roles in our analysis we demonstrate that the moment conserving projection of a solution coincides with a geodesic curve on the manifold further the evolutional velocities of the second moments and the convergence rate to the manifold are evaluated in terms of the bregman divergence finally we show that the self similar solution is geometrically special in the sense that it is simultaneously geodesic with respect to the mutually dual two affine connections

a new operational matrix for solving fractional order differential equations abstract fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations for that reason we need a reliable and efficient technique for the solution of fractional differential equations this paper deals with the numerical solution of a class of fractional differential equations the fractional derivatives are described in the caputo sense our main aim is to generalize the legendre operational matrix to the fractional calculus in this approach a truncated legendre series together with the legendre operational matrix of fractional derivatives are used for numerical integration of fractional differential equations the main characteristic behind the approach using this technique is that it reduces such problems to those of solving a system of algebraic equations thus greatly simplifying the problem the method is applied to solve two types of fractional differential equations linear and nonlinear illustrative examples are included to demonstrate the validity and applicability of the presented technique

a space time fractional optimal control problem analysis and discretization we study a linear quadratic optimal control problem involving a parabolic equation with fractional diffusion and caputo fractional time derivative of orders s in 0 1 and gamma in 0 1 respectively the spatial fractional diffusion is realized as the dirichlet to neumann map for a nonuniformly elliptic operator thus we consider an equivalent formulation with a quasi stationary elliptic problem with a dynamic boundary condition as state equation the rapid decay of the solution to this problem suggests a truncation that is suitable for numerical approximation we consider a fully discrete scheme piecewise constant functions for the control and for the state first degree tensor product finite elements in space and a finite difference discretization in time we show convergence of this scheme and under additional data regularity derive a priori error estimates for the case s in 0 1 and gamma 1

numerical solution of a class of fractional optimal control problems via the legendre orthonormal basis combined with the operational matrix and the gauss quadrature rule abstract a numerical direct method for solving a general class of fractional optimal control problems focps is presented in the discussed focp the fractional derivative in the dynamical system is considered in the caputo sense to solve the problem first the focp is transformed into an equivalent variational problem then using the legendre orthonormal basis the problem is reduced to the problem of solving a system of algebraic equations with the aid of an operational matrix of riemann liouville fractional integration gauss quadrature formula and newton s iterative method for solving a system of algebraic equations the problem is solved approximately approximations achieved by this method satisfy all the initial conditions of the problem which is an important property the convergence of the method is extensively discussed and finally some illustrative examples are included to demonstrate the applicability of the new technique

sharp short and long time l bounds for solutions to porous media equations with homogeneous neumann boundary conditions abstract we study a class of nonlinear diffusion equations whose model is the classical porous media equation on domains r n n 3 with homogeneous neumann boundary conditions we improve the known results in such model case proving sharp l q 0 l regularizing properties of the evolution for short time and sharp long time l bounds for convergence of solutions to their mean value the generality of the discussion allows to consider almost at the same time weighted versions of the above equation provided an appropriate weighted sobolev inequality holds in fact we show that the validity of such weighted sobolev inequality is equivalent to the validity of a suitable l q 0 l bound for solutions to the associated weighted porous media equation the long time asymptotic analysis relies on the assumed weighted sobolev inequality only and allows to prove uniform convergence to the mean value with the rate predicted by linearization in such generality

transcendental bernstein series for solving nonlinear variable order fractional optimal control problems abstract this paper deals with finding an approximate solution of variable order fractional optimal control problems v focps based on lagrange multiplier optimization technique in combination with the new operational matrix of variable order vo fractional derivatives to carry out the proposed approach we firstly develop the well known bernstein polynomials to the new series of functions namely transcendental bernstein series tbs then we implement these basis functions to approximate the solutions of v focps in fact the series expansion in tbs with unknown free coefficients and control parameters is the novel idea for solving the fractional systems with less terms of approximation the convergence analysis of our proposed method will be guaranteed by proving a new theorem for the tbs to investigate the practical computational efficiency and accuracy of the presented method some numerical examples are provided the experimental results confirm the applicability of the method and a good agreement between the approximate and exact solutions

a moving mesh finite difference scheme that preserves scaling symmetry for a class of nonlinear diffusion problems abstract a moving mesh finite difference scheme based on local conservation is presented for a class of scale invariant second order nonlinear diffusion problems with moving boundaries that a preserves the scaling properties and b is exact at the nodes for initial conditions sampled from similarity solutions details are presented for one dimensional problems the extension to multidimensions is described and the exactness property is confirmed for two radially symmetric moving boundary problems the porous medium equation and a simplistic glacier equation in addition the accuracy of the scheme is also tested for non self similar initial conditions by computing relative errors in the approximate solution in the l norm and the approximate boundary position indicating superlinear convergence

wavelets method for solving fractional optimal control problems in this paper an efficient and accurate computational method based on the legendre wavelets lws is proposed for solving a class of fractional optimal control problems focps in the proposed method the focp under consideration is reduced to a system of nonlinear algebraic equations which can be simply solved to this end the fractional derivative of the state variable and the control variable are expanded by the lws with unknown coefficients then the operational matrix of the riemann liouville fractional integration with some properties of the lws are employed to achieve a nonlinear algebraic equation in place of the performance index and a linear system of algebraic equations in place of the dynamical system in terms of the unknown coefficients finally the method of constrained extrema which consists of adjoining the constraint equations derived from the given dynamical system to the performance index by a set of undetermined lagrange multipliers is applied as a result the necessary conditions of optimality are derived as a system of algebraic equations in the unknown coefficients of the state variable control variable and lagrange multipliers furthermore the efficiency of the proposed method is shown for some concrete examples the results reveal that the proposed method is very accurate and efficient

best matching barenblatt profiles are delayed the growth of the second moments of the solutions of fast diffusion equations is asymptotically governed by the behavior of self similar solutions however at next order there is a correction term which amounts to a delay depending on the nonlinearity and on a distance of the initial data to the set of self similar barenblatt solutions this distance can be measured in terms of a relative entropy to the best matching barenblatt profile this best matching barenblatt function determines a scale in new variables based on this scale which are given by a self similar change of variables if and only if the initial datum is one of the barenblatt profiles the typical scale is monotone and has a limit coming back to original variables the best matching barenblatt profile is delayed compared to the self similar solution with same initial second moment as the initial datum such a delay is a new phenomenon which has to be taken into account for instance when fitting experimental data

a comparative study of interactive tradeoff cutting plane methods for momp abstract in this paper four interactive cutting plane methods developed for multiple objective mathematical programming problems are compared to one another using randomly selected problems tradeoff cutting plane methods converge to the best compromise solution by iteratively reducing the objective space by cutting planes the cutting planes are constructed using partial preference information obtained from the decision maker the performance of the four methods is evaluated and compared to one another by solving the test problems under i four classes of preference functions in addition to the original preference functions to simulate the nonexisting decision maker ii different precision levels in the decision maker s comparative accuracy iii decision space expanded or contracted and iv inconsistent responses by the decision maker based on the comparative study some guidelines for performing a comparative study of multicriteria interactive methods are presented

a lyapunov functional for the evolution of solutions to the porous medium equation to self similarity i we employ the transformation properties and the integral invariants of the porous medium equation in m dimensions to construct a lyapunov functional which characterizes the evolution of the profile to its self similar solution

global positivity estimates and harnack inequalities for the fast diffusion equation abstract we investigate local and global properties of positive solutions to the fast diffusion equation u t u m in the range d 2 d m 1 corresponding to general nonnegative initial data for the cauchy problem posed in the whole euclidean space r d we prove sharp local positivity estimates weak harnack inequalities and elliptic harnack inequalities we use them to derive sharp global positivity estimates and a global harnack principle for the mixed initial and boundary value problem posed in a bounded domain of r d with homogeneous dirichlet condition we prove weak and elliptic harnack inequalities our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow

a numerical technique for solving fractional optimal control problems this paper presents a numerical method for solving a class of fractional optimal control problems focps the fractional derivative in these problems is in the caputo sense the method is based upon the legendre orthonormal polynomial basis the operational matrices of fractional riemann liouville integration and multiplication along with the lagrange multiplier method for the constrained extremum are considered by this method the given optimization problem reduces to the problem of solving a system of algebraic equations by solving this system we achieve the solution of the focp illustrative examples are included to demonstrate the validity and applicability of the new technique

a numerical technique for solving a class of fractional variational problems this paper presents a numerical method for solving a class of fractional variational problems fvps with multiple dependent variables multi order fractional derivatives and a group of boundary conditions the fractional derivative in the problem is in the caputo sense in the presented method the given optimization problem reduces to a system of algebraic equations using polynomial basis functions an approximate solution for the fvp is achieved by solving the system the choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed we extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique

special fast diffusion with slow asymptotics entropy method and flow on a riemannian manifold we consider the asymptotic behaviour of positive solutions u t x of the fast diffusion equation u t delta u m m rm div u m 1 nabla u posed for x in mathbb r d t 0 with a precise value for the exponent m d 4 d 2 the space dimension is d 3 so that m 1 and even m 1 for d 3 this case had been left open in the general study blanchet et al in arch rat mech anal 191 347 385 2009 since it requires quite different functional analytic methods due in particular to the absence of a spectral gap for the operator generating the linearized evolution the linearization of this flow is interpreted here as the heat flow of the laplace beltrami operator of a suitable riemannian manifold mathbb r d bf g with a metric g which is conformal to the standard mathbb r d metric studying the pointwise heat kernel behaviour allows to prove suitable gagliardo nirenberg inequalities associated with the generator such inequalities in turn allow one to study the nonlinear evolution as well and to determine its asymptotics which is identical to the one satisfied by the linearization in terms of the rescaled representation which is a nonlinear fokker planck equation the convergence rate turns out to be polynomial in time this result is in contrast with the known exponential decay of such representation for all other values of m

on general multiple linear quadratic control problems general multiple linear quadratic control problems are studied where the overall objective is a nonlinear function of multiple quadratic performance indices a multilevel solution procedure is proposed to obtain an analytical linear control law for this nonseparable control problem the auxiliary lagrangian problem formulated at the lower level is of a parametric linear quadratic structure in which the weighting vector is adjusted by the upper level until a derived optimal condition is met

an approximate method for numerically solving fractional order optimal control problems of general form in this article we discuss fractional order optimal control problems focps and their solutions by means of rational approximation the methodology developed here allows us to solve a very large class of focps linear nonlinear time invariant time variant siso mimo state input constrained free terminal conditions etc by converting them into a general rational form of optimal control problem ocp the fractional differentiation operator used in the focp is approximated using oustaloup s approximation into a state space realization form the original problem is then reformulated to fit the definition used in general purpose optimal control problem ocp solvers such as riots 95 a solver created as a matlab toolbox illustrative examples from the literature are reproduced to demonstrate the effectiveness of the proposed methodology and a free final time ocp is also solved

the cauchy problem for ut δu m when 0 m 1 this paper deals with the cauchy problem for the nonlinear diffusion equation u t u u m 1 0 on 0 x rn u 0 u0 when 0 m 1 fast diffusion case we prove that there exists a global time solution for any locally integrable function u0 hence no growth condition at infinity for u0 is required moreover the solution is shown to be unique in that class behavior at infinity of the solution and l loc regularizing effects are also examined when m max n 2 n 0 1

quantitative a priori estimates for fast diffusion equations with caffarelli kohn nirenberg weights harnack inequalities and holder continuity abstract we study a priori estimates for a class of non negative local weak solution to the weighted fast diffusion equation u t x x u m with 0 m 1 posed on cylinders of 0 t r n the weights x and x with n and 2 n 2 n can be both degenerate and singular and need not belong to the class a 2 a typical assumption for this kind of problems this range of parameters is optimal for the validity of a class of caffarelli kohn nirenberg inequalities which play the role of the standard sobolev inequalities in this more complicated weighted setting the weights that we consider are not translation invariant and this causes a number of extra difficulties and a variety of scenarios for instance the scaling properties of the equation change when considering the problem around the origin or far from it we therefore prove quantitative with computable constants upper and lower estimates for local weak solutions focussing our attention where a change of geometry appears such estimates fairly combine into forms of harnack inequalities of forward backward and elliptic type as a consequence we obtain holder continuity of the solutions with a quantitative even if non optimal exponent our results apply to a quite large variety of solutions and problems the proof of the positivity estimates requires a new method and represents the main technical novelty of this paper our techniques are flexible and can be adapted to more general settings for instance to a wider class of weights or to similar problems posed on riemannian manifolds possibly with unbounded curvature in the linear case m 1 we also prove quantitative estimates recovering known results in some cases and extending such results to a wider class of weights

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

a general finite element formulation for fractional variational problems abstract this paper presents a general finite element formulation for a class of fractional variational problems fvps the fractional derivative is defined in the riemann liouville sense for fvps the euler lagrange and the transversality conditions are developed in the fractional finite element formulation ffef presented here the domain of the equations is divided into several elements and the functional is approximated in terms of nodal variables minimization of this functional leads to a set of algebraic equations which are solved using a numerical scheme three examples are considered to show the performance of the algorithm results show that as the number of discretization is increased the numerical solutions approach the analytical solutions and as the order of the derivative approaches an integer value the solution for the integer order system is recovered for unspecified boundary conditions the numerical solutions satisfy the transversality conditions this indicates that for the class of problems considered the numerical solutions can be obtained directly from the functional and there is no need to solve the fractional euler lagrange equations thus the formulation extends the traditional finite element approach to fvps

haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations haar wavelet operational matrix has been widely applied in system analysis system identification optimal control and numerical solution of integral and differential equations in the present paper we derive the haar wavelet operational matrix of the fractional order integration and use it to solve the fractional order differential equations including the bagley torvik ricatti and composite fractional oscillation equations the results obtained are in good agreement with the existing ones in open literatures and it is shown that the technique introduced here is robust and easy to apply

an interactive weighted tchebycheff procedure for multiple objective programming the procedure samples the efficient set by computing the nondominated criterion vector that is closest to an ideal criterion vector according to a randomly weighted tchebycheff metric using filtering techniques maximally dispersed representatives of smaller and smaller subsets of the set of nondominated criterion vectors are presented at each iteration the procedure has the advantage that it can converge to non extreme final solutions especially suitable for multiple objective linear programming the procedure is also applicable to integer and nonlinear multiple objective programs

convex sobolev inequalities derived from entropy dissipation we study families of convex sobolev inequalities which arise as entropy dissipation relations for certain linear fokker planck equations extending the ideas recently developed by the first two authors a refinement of the bakry emery method is established which allows us to prove non trivial inequalities even in situations where the classical bakry emery criterion fails the main application of our theory concerns the linearized fast diffusion equation in dimensions d 1 which admits a poincare but no logarithmic sobolev inequality we calculate bounds on the constants in the interpolating convex sobolev inequalities and prove that these bounds are sharp on a specified range in dimension d 1 our estimates improve the corresponding results that can be obtained by the measure theoretic techniques of barthe and roberto as a by product we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by denzler and mccann in further applications of our method we prove convex sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy and the lasota model for blood cell production

iterative parametric minimax method for a class of composite optimization problems this paper considers a class of composite optimization problems that are often difficult to solve directly due to large dimension nonlinearity nonseparability and or nonconvexity of the problem an iterative parametric minimax method is proposed in which the original optimization problem is embedded into a weighted minimax formulation the resulting auxiliary parametric optimization problems at the lower level often have simple structures that are readily tackled by efficient solution strategies such as the decomposition scheme in dynamic programming and in the primal dual method the analytical expression of the partial derivatives of systems performance indices with respect to the weighting vector in the parametric minimax formulation is derived the gradient method can be thus adopted at the upper level to adjust the value of the weighting vector the solution of the weighted minimax formulation converges to the optimal solution of the original problem in a multilevel iteration process an application of the proposed iterative parametric minimax method is demonstrated in constrained reliability optimization problems

gradient projection and local region search for multiobjective optimisation this paper presents a new method for multiobjective optimisation based on gradient projection and local region search the gradient projection is conducted through the identification of normal vectors of an efficient frontier the projection of the gradient of a nonlinear utility function onto the tangent plane of the efficient frontier at a given efficient solution leads to the definition of a feasible local region in a neighbourhood of the solution within this local region a better efficient solution may be sought to implement such a gradient based local region search scheme a new auxiliary problem is developed if the utility function is given explicitly this search scheme results in an iterative optimisation algorithm capable of general nonseparable multiobjective optimisation otherwise an interactive decision making algorithm is developed where the decision maker dm is expected to provide local preference information in order to determine trade off directions and step sizes optimality conditions for the algorithms are established and the convergence of the algorithms is proven a multiobjective linear programming molp problem is taken for example to demonstrate this method both graphically and analytically a nonlinear multiobjective water quality management problem is finally examined to show the potential application of the method to real world decision problems

asymptotic behaviour of solutions of the fast diffusion equation near its extinction time abstract let n 3 0 m n 2 n 1 0 m 1 n 2 n m and 2 1 1 m for any 0 we will prove the existence and uniqueness for 1 n 2 n m of radially symmetric singular solution g c r n 0 of the elliptic equation v m v x v 0 v 0 in r n 0 satisfying lim x 0 x g x 1 1 m when is sufficiently large we prove the higher order asymptotic behaviour of radially symmetric solutions of the above elliptic equation as x we also obtain an inversion formula for the radially symmetric solution of the above equation as a consequence we will prove the extinction behaviour of the solution u of the fast diffusion equation u t u m in r n 0 t near the extinction time t 0

improved poincare inequalities although the hardy inequality corresponding to one quadratic singularity with optimal constant does not admit any extremal function it is well known that such a potential can be improved in the sense that a positive term can be added to the quadratic singularity without violating the inequality and even a whole asymptotic expansion can be build with optimal constants for each term this phenomenon has not been much studied for other inequalities our purpose is to prove that it also holds for the gaussian poincare inequality the method is based on a recursion formula which allows to identify the optimal constants in the asymptotic expansion order by order we also apply the same strategy to a family of hardy poincare inequalities which interpolate between hardy and gaussian poincare inequalities

on singular diffusion equations with applications to self organized criticality we consider solutions of the singular diffusion equation t um 1ux x m 0 associated with the flux boundary condition limx um 1ux x 0 the evolutions defined by this problem depend on both m and we prove existence and stability of traveling wave solutions parameterized by each traveling wave is stable in its appropriate evolution these traveling waves are in l1 for 1 m 0 but have non integrable tails for m 1 we also show that these traveling waves are the same as those for the scalar conservation law ut f u x uxx for a particular nonlinear convection term f u f u m 1993 john wiley sons inc

asymptotics of the fast diffusion equation via entropy estimates we consider non negative solutions of the fast diffusion equation ut um with m 0 1 in the euclidean space mathbb r d d 3 and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t for m mc d 2 d or as t approaches the extinction time when m mc for a class of initial data we prove that the solution converges with a polynomial rate to a self similar solution for t large enough if m mc or close enough to the extinction time if m mc such results are new in the range m mc where previous approaches fail in the range mc m 1 we improve on known results

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

error bounds for gaussian quadrature and weighted l 1 polynomial approximation error bounds for gaussian quadrature are given in terms of the number of quadrature points and smoothness properties of the function whose integral is being approximated an intermediate step invol

multiple criteria decision making multiattribute utility theory the next ten years management science and decision science have grown exponentially since midcentury two closely related fields central to this growth are multiple criteria decision making mcdm and multiattribute utility theory maut this paper comments on the history of mcdm and maut and discusses topics we believe are important in their continued development and usefulness to management science over the next decade our aim is to identify exciting directions and promising areas for future research

nonlinear diffusions extremal properties of barenblatt profiles best matching and delays abstract in this paper we consider functionals based on moments and nonlinear entropies which have a linear growth in time in case of source type solutions to the fast diffusion or porous medium equations that are also known as barenblatt solutions as functions of time these functionals have convexity properties for generic solutions so that their asymptotic slopes are extremal for barenblatt profiles the method relies on scaling properties of the evolution equations and provides a simple and direct proof of sharp gagliardo nirenberg sobolev inequalities in scale invariant form the method also gives refined estimates of the growth of the second moment and as a consequence establishes the monotonicity of the delay corresponding to the best matching barenblatt solution compared to the barenblatt solution with same initial second moment here the notion of best matching is defined in terms of a relative entropy

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

a new algorithm for the solution of nonlinear two dimensional volterra integro differential equations of high order abstract in this manuscript a new numerical approach is developed for approximating the solution of the one dimensional and two dimensional volterra integro differential equations 2d vides of high order new operational matrices of integration based on shifted jacobi polynomials are presented and used in combination with the collocation method to convert the 2d vide into a system of algebraic equations in addition the convergence analysis of the suggested approach is investigated the efficiency of the algorithm is demonstrated by means of several examples and the results are compared with those given using other numerical schemes

preference modelling by estimating local utility functions for multiobjective optimization abstract this paper is intended to design goal programming models for capturing the decision maker s dm s preference information and for supporting the search for the best compromise solutions in multiobjective optimization at first a linear goal programming model is built to estimate piecewise linear local utility functions based on pairwise comparisons of efficient solutions as well as objectives the interactive step trade off method istm is employed to generate a typical subset of efficient solutions of a multiobjective problem another general goal programming model is then constructed to embed the estimated utility functions in the original multiobjective problem for utility optimization using ordinary nonlinear programming algorithms this technique consisting of the istm method and the newly investigated search process facilitates the identification and elimination of possible inconsistent information which may exist in the dm s preferences it also provides various ways to carry out post optimality analysis to test the robustness of the obtained best solutions a modified nonlinear multiobjective management problem is taken as example to demonstrate the technique

the cauchy problem for ut δum when 0 m 1 on considere le probleme de cauchy pour l equation de diffusion non lineaire u t u u m 1 0 sur 0 r n u 0 u 0 quand 0 m 1 on montre qu il existe une solution temporelle globale pour une fonction localement integrable u 0 on montre l unicite dans cette classe

hypocoercivity for linear kinetic equations conserving mass we develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law local mass conservation is assumed at the level of the collision kernel while transport involves a confining potential so that the solution relaxes towards a unique equilibrium state our goal is to evaluate in an appropriately weighted l 2 norm the exponential rate of convergence to the equilibrium the method covers various models ranging from diffusive kinetic equations like vlasov fokker planck equations to scattering models like the linear boltzmann equation or models with time relaxation collision kernels corresponding to polytropic gibbs equilibria including the case of the linear boltzmann model in this last case and in the case of vlasov fokker planck equations any linear or superlinear growth of the potential is allowed

long time asymptotics for the porous medium equation the spectrum of the linearized operator we compute the complete spectrum of the displacement hessian operator which is obtained from the confined porous medium equation by linearization around its stationary attractor the barenblatt profile on a formal level the operator is conjugate to the hessian of the entropy via similarity transformation we show that the displacement hessian can be understood as a self adjoint operator and find that its spectrum is purely discrete the knowledge of the complete spectrum and the explicit information about the corresponding eigenfunctions give new insights on the convergence and higher order asymptotics of solutions to the porous medium equation towards its attractor more precisely the inspection of the eigenfunctions allows to identify symmetries in rn with flows whose rates of convergence are faster than the uniform translation governed bound the present work complements the analogous study of denzler mccann for the fast diffusion equation

integrating dea oriented performance assessment and target setting using interactive molp methods data envelopment analysis dea and multiple objective linear programming molp are tools that can be used in management control and planning whilst these two types of model are similar in structure dea is directed to assessing past performances as part of management control function and molp to planning future performance targets this paper is devoted to investigating equivalence models and interactive tradeoff analysis procedures in molp such that dea oriented performance assessment and target setting can be integrated in a way that the decision makers preferences can be taken into account in an interactive fashion three equivalence models are investigated between the output oriented dual dea model and the minimax reference point formulations namely the super ideal point model the ideal point model and the shortest distance model these models can be used to support efficiency analysis in the same way as the conventional dea model does and also support tradeoff analysis for setting target values by individuals or groups a case study is conducted to illustrate how dea oriented efficiency analysis can be conducted using the molp methods and how such performance assessment can be integrated into an interactive procedure for setting realistic target values

very slow stabilization for a nonlinear fokker planck equation abstract we study the asymptotic behaviour near extinction of positive solutions of the fast diffusion equation with critical and subcritical exponents by a suitable rescaling the equation is transformed to a nonlinear fokker planck equation we show that the rate of convergence to regular and singular steady states of the transformed equation can be arbitrarily slow for suitable initial data our results reveal the difference in the slow rates occurring in the critical and the subcritical case and they yield a new class of extinction rates for the fast diffusion equation

rate of convergence to barenblatt profiles for the fast diffusion equation we study the asymptotic behaviour of positive solutions of the cauchy problem for the fast diffusion equation as t approaches the extinction time we find a continuum of rates of convergence to a self similar profile these rates depend explicitly on the spatial decay rates of the initial data