analytical methods for solving the time fractional swift hohenberg s h equation in this paper the most effective methods the homotopy perturbation method hpm and the differential transform method dtm are applied to obtain the approximate solutions of the nonlinear time fractional swift hohenberg s h equation the basic philosophy of these methods does not involve linearization weak nonlinearity assumptions or perturbation theory numerical solutions for various combinations of the parameters a eigenvalue parameter l length and a fractional index are obtained the solutions of the s h equation are useful for studies of shear thinning effects in non newtonian fluid flows at the end the solutions obtained are also presented graphically

numerical method for non linear wave and diffusion equations by the variational iteration method this paper applies he s variational iteration to the wave equations in an infinite one dimensional medium and some non linear diffusion equations a suitable choice of an initial solution can lead to the needed exact solution by a few iterations copyright 2007 john wiley sons ltd

an efficient algorithm for solving integro differential equations system in this paper an application of he s homotopy perturbation hpm method is applied to solve of system of integro differential equations the results reveal that the hpm is very effective and simple

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

note on the homotopy perturbation method for multivariate vector value oscillatory integrals this paper considers a homotopy perturbation method for approximating multivariate vector value highly oscillatory integrals the asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented numerical examples show the efficiency of the approximation method

new iterative method application to partial differential equations abstract in the present article we implement the new iterative method nim proposed by daftardar gejji and jafari v daftardar gejji h jafari an iterative method for solving non linear functional equations j math anal appl 316 2006 753 763 to solve linear nonlinear partial differential equations of integer and fractional order the results obtained are compared with those obtained by other iterative methods such as adomian decomposition homotopy perturbation and variational iteration methods

application of the variational iteration method to the whitham broer kaup equations explicit traveling wave solutions including blow up and periodic solutions of the whitham broer kaup equations are obtained by the variational iteration method moreover the results are compared with those obtained by the adomian decomposition method revealing that the variational iteration method is superior to the adomian decomposition method

a review of the decomposition method in applied mathematics abstract the decomposition method can be an effective procedure for analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions closure approximations perturbation theory or restrictive assumptions on stochasticitiy

optimal mass transportation in the heisenberg group in this paper we consider the problem of optimal transportation of absolutely continuous masses in the heisenberg group hn in the case when the cost function is either the square of the carnot caratheodory distance or the square of the koranyi norm in both cases we show existence and uniqueness of an optimal transport map in the former case the proof requires a delicate analysis of minimizing geodesics of the group and of the differentiability properties of the squared distance function in the latter case the proof requires some fine properties of bv functions in the heisenberg group

numerical method for solving fractional coupled burgers equations in this paper we use the fractional variational iteration method fvim to solve a time and space fractional coupled burgers equations some numerical examples are presented to show the efficiency of considered method a comparison of the proposed method is made with the exact solution adomain decomposition method adm generalized differential transformation method gdtm and homotopy perturbation method hpm

decomposition method for solving fractional riccati differential equations in this article we implement a relatively new analytical technique the adomian decomposition method for solving fractional riccati differential equations the fractional derivatives are described in the caputo sense in this scheme the solution takes the form of a convergent series with easily computable components the diagonal pade approximants are effectively used in the analysis to capture the essential behavior of the solution the corresponding solutions of the integer order equations are found to follow as special cases of those of fractional order equations some numerical examples are presented to illustrate the efficiency and reliability of the method

the variational iteration method an efficient scheme for handling fractional partial differential equations in fluid mechanics variational iteration method has been used to handle linear and nonlinear differential equations the main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently in this work a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics the fractional derivatives are described in the caputo sense numerical illustrations that include the fractional wave equation fractional burgers equation fractional kdv equation fractional klein gordon equation and fractional boussinesq like equation are investigated to show the pertinent features of the technique comparison of the results obtained by the variational iteration method with those obtained by adomian decomposition method reveals that the first method is very effective and convenient the basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus

the stefan problem for the fisher kpp equation abstract we study the fisher kpp equation with a free boundary governed by a one phase stefan condition such a problem arises in the modeling of the propagation of a new or invasive species with the free boundary representing the propagation front in one space dimension this problem was investigated in du and lin 2010 11 and the radially symmetric case in higher space dimensions was studied in du and guo 2011 10 in both cases a spreading vanishing dichotomy was established namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state or fails to establish and dies out in the long run moreover in the case of spreading the asymptotic spreading speed was determined in this paper we consider the non radially symmetric case in such a situation similar to the classical stefan problem smooth solutions need not exist even if the initial data are smooth we thus introduce and study the weak solution for a class of free boundary problems that include the fisher kpp as a special case we establish the existence and uniqueness of the weak solution and through suitable comparison arguments we extend some of the results obtained earlier in du and lin 2010 11 and du and guo 2011 10 to this general case we also show that the classical aronson weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here

the diffusive logistic model with a free boundary in heterogeneous environment abstract in this paper we study the population dynamics of an invasive species in heterogeneous environment which is modeled by a diffusive logistic equation with free boundary condition to understand the effect of the dispersal rate d and the parameter the ratio of the expansion speed of the free boundary and the population gradient at the expanding front on the dynamics of this model we divide the heterogeneous environment into two cases strong heterogeneous environment and weak heterogeneous environment by choosing d and as variable parameters we derive sufficient conditions for species spreading resp vanishing in the strong heterogeneous environment while in the weak heterogeneous environment we obtain sharp criteria for the spreading and vanishing moreover when spreading happens we give an estimate for the asymptotic spreading speed of the free boundary these theoretical results may have important implications for prediction and prevention of biological invasions

a collocation method for fractional diffusion equation in a long time with chebyshev functions abstract in this paper our aim is to find a new numerical method for diffusion equation with fractional derivative on time and space the employed fractional derivative is in the caputo sense also by employing a class of shifted chebyshev polynomials for the space area and a collection of rational chebyshev functions for the time domain and then using collocation method we obtain an algebraic system of equations the convergence estimate of the new scheme have been concluded finally we evaluate results of this method with other numerical methods

approximate analytical solution of the nonlinear fractional kdv burgers equation in this paper explicit and approximate solutions of the nonlinear fractional kdv burgers equation with time space fractional derivatives are presented and discussed the solutions of our equation are calculated in the form of rabidly convergent series with easily computable components the utilized method is a numerical technique based on the generalized taylor series formula which constructs an analytical solution in the form of a convergent series five illustrative applications are given to demonstrate the effectiveness and the leverage of the present method graphical results and series formulas are utilized and discussed quantitatively to illustrate the solution the results reveal that the method is very effective and simple in determination of solution of the fractional kdv burgers equation

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

the minimal habitat size for spreading in a weak competition system with two free boundaries abstract in this paper we focus on the dynamics for a lotka volterra type weak competition system with two free boundaries where free boundaries which may intersect each other as time evolves are used to describe the spreading of two competing species respectively in the weak competition case the dynamics of this model can be classified into four cases which forms a spreading vanishing quartering the notion of the minimal habitat size for spreading is introduced to determine if species can always spread some sufficient conditions for spreading and vanishing are established also when spreading occurs some rough estimates for spreading speed and the long time behavior of solutions are established

the variational iteration method a highly promising method for solving the system of integro differential equations this paper applies he s variational iteration method for solving two systems of volterra integro differential equations the solution process is illustrated and various physically relevant results are obtained comparison of the obtained results with exact solutions shows that the used method is an effective and highly promising method for various classes of both linear and nonlinear integro differential equations

on the solutions of nonlinear fractional klein gordon equation with modified riemann liouville derivative abstract in this paper fractional variational iteration method fvim is applied for finding approximate analytical solutions of nonlinear fractional klein gordon equation time fractional derivative is described in the riemann liouville sense a new application of fractional variational iteration method fvim is extended to derive analytical solutions in the form of a series for this equation the behavior of the solutions and the effects of different values of fractional order are indicated graphically it is shown that the solutions obtained by the fvim are reliable convenient and effective for strongly nonlinear partial equations with modified riemann liouville derivative

solving two point boundary value problems using combined homotopy perturbation method and green s function method in this paper an algorithm is proposed for the solution of second order boundary value problems with two point boundary conditions the green s function method is applied first to transform the ordinary differential equation into an equivalent integral one which has already satisfied the boundary conditions and then the homotopy perturbation method is used to the resulting equation to construct the numerical solution for such problems numerical examples demonstrate the efficiency and reliability of the algorithm developed it is quite accurate and readily implemented for both linear and nonlinear differential equations with homogeneous and nonhomogeneous boundary conditions furthermore the lower order approximation is of higher accuracy for most cases some other extended applications of this algorithm are also exhibited

application of he s homotopy perturbation method to nonlinear integro differential equations abstract in this paper an application of he s homotopy perturbation method is applied to solve nonlinear integro differential equations the results reveal that the he s homotopy perturbation method is very effective and simple

solution of the complex modified korteweg de vries equation by the projected differential transform method in this paper we first transform the complex modified korteweg de vries cmkdv equation into a real valued system of partial differential equations then the projected differential transformation method is used to compute an approximate solution of the cmkdv equation three illustrative examples are given to show the effectiveness of the method and comparing with the differential transformation method and the variational iteration method

on a free boundary problem for a reaction diffusion advection logistic model in heterogeneous environment abstract in this paper we investigate a reaction diffusion advection equation with a free boundary which models the spreading of an invasive species in one dimensional heterogeneous environments we assume that the species has a tendency to move upward along the resource gradient in addition to random dispersal and the spreading mechanism of species is determined by a stefan type condition investigating the sign of the principal eigenvalue of the associated linearized eigenvalue problem under certain conditions we obtain the sharp criteria for spreading and vanishing via system parameters also we establish the long time behavior of the solution and the asymptotic spreading speed finally some biological implications are discussed

variational iteration method new development and applications variational iteration method has been favourably applied to various kinds of nonlinear problems the main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently in this paper recent trends and developments in the use of the method are reviewed major applications to nonlinear wave equation nonlinear fractional differential equations nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed the confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods variational iteration method is uniquely qualified to address this challenge the flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions this paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations a new iteration formulation is suggested to overcome the shortcoming a very useful formulation for determining approximately the period of a nonlinear oscillator is suggested examples are given to illustrate the solution procedure

variational iteration method for solving burger s and coupled burger s equations by means of variational iteration method the solutions of burger s equation and coupled burger s equations are exactly obtained comparison with the adomian decomposition method is made showing that the former is more effective than the later in this paper he s variational iteration method is introduced to overcome the difficulty arising in calculating adomian polynomials

numerical solutions of the integral equations homotopy perturbation method and adomian s decomposition method in this paper a homotopy perturbation method is proposed to solve non singular integral equations comparisons are made between adomian s decomposition method and the proposed method it is shown adomian s decomposition method is a homotopy only finally by using homotopy perturbation method a new iterative scheme like adomian s decomposition method is proposed for solving the non singular integral equations of the first kind the results reveal that the proposed method is very effective and simple

sub riemannian interpolation inequalities we prove that ideal sub riemannian manifolds i e admitting no non trivial abnormal minimizers support interpolation inequalities for optimal transport a key role is played by sub riemannian jacobi fields and distortion coefficients whose properties are remarkably different with respect to the riemannian case as a byproduct we characterize the cut locus as the set of points where the squared sub riemannian distance fails to be semiconvex answering to a question raised by figalli and rifford geom funct anal 20 1 124 159 2010 as an application we deduce sharp and intrinsic borell brascamp lieb and geodesic brunn minkowski inequalities in the aforementioned setting for the case of the heisenberg group we recover in an intrinsic way the results recently obtained by balogh et al calc var part differ equ 57 2 61 2018 and we extend them to the class of generalized h type carnot groups our results do not require the distribution to have constant rank yielding for the particular case of the grushin plane a sharp measure contraction property and a sharp brunn minkowski inequality

the diffusive logistic equation with a free boundary and sign changing coefficient abstract this short paper concerns a diffusive logistic equation with a free boundary and sign changing coefficient which is formulated to study the spread of an invasive species where the free boundary represents the expanding front a spreading vanishing dichotomy is derived namely the species either successfully spreads to the right half space as time t and survives persists in the new environment or it fails to establish itself and will extinct in the long run the sharp criteria for spreading and vanishing are also obtained when spreading happens we estimate the asymptotic spreading speed of the free boundary

on the solution of nonlinear fractional order differential equation abstract this paper investigates the uniqueness and dependence of the solution of nonlinear differential equation with fractional order by the fixed points theorem the obtained results include naturally those in open literature for some special cases and a numerical approach to the discussed problem is suggested

numerical solution of a biological population model using he s variational iteration method this paper presents numerical solution of a degenerate parabolic equation arising in the spatial diffusion of biological populations the variational iteration method and adomian decomposition method are used for solving this equation and then numerical results are compared with each other showing that the variational iteration method leads to more accurate results furthermore variational iteration method overcomes the difficulty arising in calculating the adomian s polynomials which is an important advantage over the adomian decomposition method

application of the variational iteration method to the regularized long wave equation this paper applies the variational iteration method to the regularized long wave equation in this method a generalized lagrange multiplier is introduced to construct a correction functional for the discussed problem the multiplier can be identified optimally via variational theory three invariants of the motion are evaluated to determine the conservation properties of the system

application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives two nonlinear anelastic models with fractional derivatives describing the properties of a series of materials as polymers and polycrystalline materials are presented in this paper these models are studied analytically using a variational iteration method the paper clarifies the different ways in which the fractional differentiation operator can be defined a volterra series method of model parameters identification from the experimental data is also presented

spreading vanishing dichotomy in a diffusive logistic model with a free boundary ii abstract we study the diffusive logistic equation with a free boundary in higher space dimensions and heterogeneous environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for simplicity we assume that the environment and the solution are radially symmetric in the special case of one space dimension and homogeneous environment this free boundary problem was investigated in du and lin 2010 10 we prove that the spreading vanishing dichotomy established in du and lin 2010 10 still holds in the more general and ecologically realistic setting considered here moreover when spreading occurs we obtain best possible upper and lower bounds for the spreading speed of the expanding front when the environment is asymptotically homogeneous at infinity these two bounds coincide our results indicate that the asymptotic spreading speed determined by this model does not depend on the spatial dimension

homotopy perturbation technique abstract the homotopy perturbation technique does not depend upon a small parameter in the equation by the homotopy technique in topology a homotopy is constructed with an imbedding parameter p 0 1 which is considered as a small parameter some examples are given the approximations obtained by the proposed method are uniformly valid not only for small parameters but also for very large parameters

a new collocation approach for solving systems of high order linear volterra integro differential equations with variable coefficients this paper contributes an efficient numerical approach for solving the systems of high order linear volterra integro differential equations with variable coefficients under the mixed conditions the method we have used consists of reducing the problem to a matrix equation which corresponds to a system of linear algebraic equations the obtained matrix equation is based on the matrix forms of fibonacci polynomials and their derivatives by means of collocations in addition the method is presented with error numerical results with comparisons are given to demonstrate the applicability efficiency and accuracy of the proposed method the results of the examples indicated that the method is simple and effective and could provide an approximate solution with high accuracy or exact solution of the system of high order linear volterra integro differential equations

a diffusive logistic model with a free boundary in time periodic environment abstract we study the diffusive logistic equation with a free boundary in time periodic environment such a model may be used to describe the spreading of a new or invasive species with the free boundary representing the expanding front for time independent environment in the cases of one space dimension and higher space dimensions with radial symmetry this free boundary problem has been studied in du and lin 2010 12 du and guo 2011 9 in both cases a spreading vanishing dichotomy was established and when spreading occurs the asymptotic spreading speed was determined in this paper we show that the spreading vanishing dichotomy is retained in time periodic environment and we also determine the spreading speed the former is achieved by further developing the earlier techniques and the latter is proved by introducing new ideas and methods

application of he s variational iteration method to nonlinear jaulent miodek equations and comparing it with adm instead of finding a small parameter for solving nonlinear problems through perturbation method a new analytical method called he s variational iteration method vim is introduced to be applied to solve nonlinear jaulent miodek coupled kdv and coupled mkdv equations in this article in this method general lagrange multipliers are introduced to construct correction functionals for the problems the multipliers can be identified optimally via the variational theory the results are compared with exact solutions

the variational iteration method for cauchy problems in this paper the variational iteration method is used to solve the cauchy problem some examples are given to elucidate the solution procedure and reliability of the obtained results the variational iteration algorithm leads to exact solutions in the present study

the spreading front of invasive species in favorable habitat or unfavorable habitat abstract spatial heterogeneity and habitat characteristic are shown to determine the asymptotic profile of the solution to a reaction diffusion model with free boundary which describes the moving front of the invasive species a threshold value r 0 fr d t is introduced to determine the spreading and vanishing of the invasive species we prove that if r 0 fr d t 0 1 for some t 0 0 the spreading must happen while if r 0 fr d 0 1 the spreading is also possible our results show that the species in the favorable habitat can establish itself if the diffusion is slow or the occupying habitat is large in an unfavorable habitat the species dies out if the initial value of the species is small however big initial number of the species is benefit for the species to survive when the species spreads in the whole habitat the asymptotic spreading speed is given some implications of these theoretical results are also discussed

variational iteration method for solving the epidemic model and the prey and predator problem in this article two problems are considered the first one is the problem of the spread of a non fatal disease in a population which is assumed to have constant size over the period of the epidemic and the other one is the problem of prey and predator the variational iteration method is employed to compute an approximation to the solutions of the systems of nonlinear ordinary differential equations governing on these problems the results are compared with the results obtained by adomian decomposition and power series methods some plots are presented to show the reliability and simplicity of the method

variational iteration method for solving nonlinear boundary value problems abstract in this paper he s variational iteration method is implemented to give approximate and analytical solutions for a class of boundary value problems the variational iteration method which produces the solutions in terms of convergent series requiring no linearization or small perturbation numerical examples are given and comparisons are made with the adomian decomposition method the fact that this method solves nonlinear equations without using adomian polynomials can be considered as an advantage of this method over adomian decomposition method

variational approach to solitary wave solution of the generalized zakharov equation he s semi inverse method is applied to search for the solitary wave solution of the generalized zakharov equation gze the solution process reveals that the method is easy and straightforward

multiple spreading phenomena for a free boundary problem of a reaction diffusion equation with a certain class of bistable nonlinearity abstract this paper deals with a free boundary problem for diffusion equation with a certain class of bistable nonlinearity which allows two positive stable equilibrium states as an ode model this problem models the invasion of a biological species and the free boundary represents the spreading front of its habitat our main interest is to study large time behaviors of solutions for the free boundary problem we will completely classify asymptotic behaviors of solutions and in particular observe two different types of spreading phenomena corresponding to two positive stable equilibrium states moreover it will be proved that if the free boundary expands to infinity an asymptotic speed of the moving free boundary for large time can be uniquely determined from the related semi wave problem

an approximate method for solving a class of weakly singular volterra integro differential equations abstract in this paper we present a new approach to resolve linear and nonlinear weakly singular volterra integro differential equations of first or second order by first removing the singularity using taylor s approximation and then transforming the given first or second order integro differential equations into an ordinary differential equation such as the well known legendre degenerate hypergeometric euler or abel equations in such a manner that adomian s asymptotic decomposition method can be applied which permits convenient resolution of these equations some examples with closed form solutions are studied in detail to further illustrate the proposed technique and the results obtained demonstrate this approach is indeed practical and efficient

the homotopy perturbation method for nonlinear oscillators with discontinuities in the paper the homotopy perturbation method is applied to the nonlinear oscillators with discontinuities only one iteration leads to high accuracy of the solutions

application of he s methods to nonlinear chemistry problems in this paper he s variational iteration method vim and homotopy perturbation method hpm are implemented for solving analytically two systems of ordinary differential equations that often appear in chemical applications in both methods the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions the solution procedure is of utter simplicity and the obtained solution is of high accuracy

generalized continuous time random walks master equations and fractional fokker planck equations continuous time random walks which generalize random walks by adding a stochastic time between jumps provide a useful description of stochastic transport at mesoscopic scales the continuous time random walk model can accommodate certain features such as trapping which are not manifest in the standard macroscopic diffusion equation the trapping is incorporated through a waiting time density and a fractional diffusion equation results from a power law waiting time a generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force here we have derived the master equations for continuous time random walks with space and time dependent forcing for two cases when the force is evaluated at the start of the waiting time and at the end of the waiting time the differences persist in low order spatial continuum approximations however the two processes are shown to be governed by the same fokker planck equations in the diffusi

numerical treatment for solving the perturbed fractional pdes using hybrid techniques in this article an accurate numerical approach is introduced in this approach we mixed between the fractional finite difference method and the restrictive taylor approximation rta the proposed method is implemented to solve numerically the perturbed fractional partial differential equations fpdes special attention is given to study the stability and consistency of the method by means of gerschgorin theorem and using the stability matrix analysis two numerical examples are given and compared with the exact solution

a riemannian interpolation inequality a la borell brascamp and lieb a concavity estimate is derived for interpolations between l 1 m mass densities on a riemannian manifold the inequality sheds new light on the theorems of prekopa leindler borell brascamp and lieb that it generalizes from euclidean space due to the curvature of the manifold the new riemannian versions of these theorems incorporate a volume distortion factor which can however be controlled via lower bounds on ricci curvature the method uses optimal mappings from mass transportation theory along the way several new properties are established for optimal mass transport and interpolating maps on a riemannian manifold

on some free boundary problems of the prey predator model abstract in this paper we investigate some free boundary problems for the lotka volterra type prey predator model in one space dimension the main objective is to understand the asymptotic behavior of the two species prey and predator spreading via a free boundary we prove a spreading vanishing dichotomy namely the two species either successfully spread to the entire space as time t goes to infinity and survive in the new environment or they fail to establish and die out in the long run the long time behavior of solution and criteria for spreading and vanishing are also obtained finally when spreading successfully we provide an estimate to show that the spreading speed if exists cannot be faster than the minimal speed of traveling wavefront solutions for the prey predator model on the whole real line without a free boundary

a fractional variational iteration method for solving fractional nonlinear differential equations recently fractional differential equations have been investigated by employing the famous variational iteration method however all the previous works avoid the fractional order term and only handle it as a restricted variation a fractional variational iteration method was first proposed in g c wu e w m lee fractional variational iteration method and its application phys lett a 374 2010 2506 2509 and gave a generalized lagrange multiplier in this paper two fractional differential equations are approximately solved with the fractional variational iteration method

jacobian determinant inequality on corank 1 carnot groups with applications abstract we establish a weighted pointwise jacobian determinant inequality on corank 1 carnot groups related to optimal mass transportation akin to the work of cordero erausquin mccann and schmuckenschlager in this setting the presence of abnormal geodesics does not allow the application of the general sub riemannian optimal mass transportation theory developed by figalli and rifford and we need to work with a weaker notion of jacobian determinant nevertheless our result achieves a transition between euclidean and sub riemannian structures corresponding to the mass transportation along abnormal and strictly normal geodesics respectively the weights appearing in our expression are distortion coefficients that reflect the delicate sub riemannian structure of our space as applications entropy brunn minkowski and borell brascamp lieb inequalities are established on carnot groups

on the numerical solution of linear and nonlinear volterra integral and integro differential equations abstract sinc bases are developed to approximate the solutions of linear and nonlinear volterra integral and integro differential equations properties of these sinc bases and some operational matrices are first presented these properties are then used to reduce the integral and integro differential equations to systems of linear or nonlinear algebraic equations numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems the examples include convolution type singular as well as singularly perturbed problems

bakry emery curvature and model spaces in sub riemannian geometry we prove comparison theorems for the sub riemannian distortion coefficients appearing in interpolation inequalities these results which are equivalent to a sub laplacian comparison theorem for the sub riemannian distance are obtained by introducing a suitable notion of sub riemannian bakry emery curvature the model spaces for comparison are variational problems coming from optimal control theory as an application we establish the sharp measure contraction property for 3 sasakian manifolds satisfying a suitable curvature bound

non uniform dispersal of logistic population models with free boundaries in a spatially heterogeneous environment abstract in many cases the movement of species within a region depends on the availability of food and other resources necessary for their survival starvation driven diffusion sdd is a non uniform dispersal strategy that increases the motility of biological organisms in unfavorable environments i e a species moves more frequently in search of food if resources are insufficient 4 in this study the proposed model represents the dispersion of an invasive species undergoing sdd where the free boundary represents the expanding front we observed that the spreading vanishing dichotomy which holds in the linear dispersal model also holds in the model undergoing sdd in the case that spreading of species occurs it is shown that the asymptotic spreading speed of the moving front is uniquely determined in relation to the semi wave speed moreover our results are compared with the results of the linear dispersal model to investigate the advantages of this strategic dispersal with respect to survival in new environments i e the conditions that allowed the species undergoing sdd to spread and the random diffusers to vanish are monitored

ricci curvature for metric measure spaces via optimal transport we dene a notion of a measured length space x having nonnegative n ricci curvature for n 2 1 1 or having1 ricci curvature bounded below byk fork2 r the denitions are in terms of the displacement convexity of certain functions on the associated wasserstein metric space p2 x of probability measures we show that these properties are preserved under measured gromov hausdor limits we give geometric and analytic consequences this paper has dual goals one goal is to extend results about optimal transport from the setting of smooth riemannian manifolds to the setting of length spaces a second goal is to use optimal transport to give a notion for a measured length space to have ricci curvature bounded below we refer to 11 and 44 for background material on length spaces and optimal transport respectively further bibliographic notes on optimal transport are in appendix f in the present introduction we motivate the questions that we address and we state the main results to start on the geometric side there are various reasons to try to extend notions of curvature from smooth riemannian manifolds to more general spaces a fairly general setting is that of length spaces meaning metric spaces x d in which the distance between two points equals the inmum of the lengths of curves joining the points in the rest of this introduction we assume that x is a compact length space alexandrov gave a good notion of a length space having curvature bounded below by k with k a real number in terms of the geodesic triangles in x in the case of a riemannian manifold m with the induced length structure one recovers the riemannian notion of having sectional curvature bounded below by k length spaces with alexandrov curvature bounded below by k behave nicely with respect to the gromovhausdor topology on compact metric spaces modulo isometries they form

the brunn minkowski inequality in 1978 osserman 124 wrote an extensive survey on the isoperimetric inequality the brunn minkowski inequality can be proved in a page yet quickly yields the classical isoperimetric inequality for important classes of subsets of rn and deserves to be better known this guide explains the relationship between the brunn minkowski inequality and other inequalities in geometry and analysis and some applications

relative controllability of fractional dynamical systems with distributed delays in control this paper is concerned with the global relative controllability of fractional dynamical systems with multiple delays in control for finite dimensional spaces sufficient conditions for controllability results are obtained using schauder s fixed point theorem and the controllability grammian matrix which is defined by the mittag leffler matrix function an example is provided to illustrate the theory

analytical solution of a fractional diffusion equation by variational iteration method in the present paper the analytical approximate solution of a fractional diffusion equation is deduced with the help of powerful variational iteration method by using an initial value the explicit solutions of the equation for different cases have been derived which accelerate the rapid convergence of the series solution the present method performs extremely well in terms of efficiency and simplicity numerical results for different particular cases of the problem are presented graphically

semilinear fractional integro differential equations with compact semigroup abstract in this paper we study the local and global existence of mild solutions to a class of fractional integro differential equations in an arbitrary banach space associated with operators generating compact semigroup on the banach space

spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection diffusion equations abstract in this study we consider free boundary problems for nonlinear advection diffusion equations of the form u t u x x u x f u for t 0 g t x h t where x g t and x h t are free boundaries this problem may be used to describe the spreading of a biological or chemical species where the free boundaries represent the expanding fronts when f is a logistic nonlinearity it has been shown that the asymptotic spreading speeds of the two fronts h t and g t are different due to the advection term in this study for monostable bistable and combustion types nonlinearities we give much sharper estimates of the different spreading speeds of the fronts and we also prove that the solution converges to a semi wave in c 2 norm as t when spreading occurs we develop new approaches and extend a previous result to the problem with the advection term

optimal transportation under nonholonomic constraints we study monge s optimal transportation problem where the cost is given by an optimal control cost we prove the existence and uniqueness of an optimal map under certain regularity conditions on the lagrangian absolute continuity of the measures with respect to lebesgue and most importantly the absence of sharp abnormal minimizers in particular this result is applicable in the case of subriemannian manifolds with a 2 generating distribution and cost given by d 2 where d is the subriemannian distance also we discuss some properties of the optimal plan when abnormal minimizers are present finally we consider some examples of displacement interpolation in the case of the grushin plane

on the geometry of metric measure spaces ii we introduce a curvature dimension condition cd k n for metric measure spaces it is more restrictive than the curvature bound underline text curv left m text d m right k introduced in sturm k t 2006 on the geometry of metric measure spaces i acta math 196 65 131 which is recovered as the borderline case cd k the additional real parameter n plays the role of a generalized upper bound for the dimension for riemannian manifolds cd k n is equivalent to text ric m left xi xi right k left xi right 2 and dim m n the curvature dimension condition cd k n is stable under convergence for any triple of real numbers k n l the family of normalized metric measure spaces m d m with cd k n and diameter l is compact condition cd k n implies sharp version of the brunn minkowski inequality of the bishop gromov volume comparison theorem and of the bonnet myers theorem moreover it implies the doubling property and local scale invariant poincare inequalities on balls in particular it allows to construct canonical dirichlet forms with gaussian upper and lower bounds for the corresponding heat kernels

approximate analytical solution for seepage flow with fractional derivatives in porous media abstract in this paper a new and more exact model for seepage flow in porous media with fractional derivatives has been proposed which has modified the well known darcy law and overcome the continuity assumption of seepage flow a new kind of analytical method of nonlinear problem called the variational iteration method is described and used to give approximate solutions of the problem the results show that the proposed iteration method requiring no linearization or small perturbation is very effective and convenient

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

convergence and error estimation of homotopy perturbation method for fredholm and volterra integral equations abstract in this paper an application of the homotopy perturbation method for solving fredholm and volterra integral equations of the second kind is presented discussed method consists in constructing the functional series sum of which determines the function giving the solution of considered problem conditions under which the constructed series is convergent are formulated and proved in the paper in general the composed series is rather fast convergent thanks to which calculation of few first terms ensures a very good approximation of the sought solution estimation of errors of approximate solution obtained by taking the partial sum of the series is also elaborated in the paper

exact solutions for nonlinear integral equations by a modified homotopy perturbation method in this paper we apply a new modified homotopy perturbation method to find exact solutions for nonlinear integral equations a reliable modification of the homotopy perturbation method is proposed and the modified method is employed to solve the nonlinear integral equations the results are compared with those obtained by the original homotopy perturbation method some examples are given to illustrate the ability and reliability of the modified method the results reveal that the modified method is very simple and effective

fourth order integro differential equations using variational iteration method in this paper the variational iteration method proposed by ji huan he is applied to solve both linear and nonlinear boundary value problems for fourth order integro differential equations the numerical results obtained with minimum amount of computation are compared with the exact solutions to show the efficiency of the method the results show that the variational iteration method is of high accuracy more convenient and efficient for solving integro differential equations

solving fredholm integro differential equations using reproducing kernel hilbert space method in this study the numerical solution of fredholm integro differential equation is discussed in a reproducing kernel hilbert space a reproducing kernel hilbert space is constructed in which the initial condition of the problem is satisfied the exact solution ux is represented in the form of series in the space w 2 2 a b in the mean time the n term approximate solution u n x is obtained and is proved to converge to the exact solution u x furthermore we present an iterative method for obtaining the solution in the space w 2 2 a b some examples are displayed to demonstrate the validity and applicability of the proposed method the numerical result indicates that the proposed method is straightforward to implement efficient and accurate for solving linear and nonlinear fredholm integro differential equations

note on a two species competition diffusion model with two free boundaries abstract in guo and wu 2015 and wu 2015 the authors studied a two species competition diffusion model with two free boundaries these two free boundaries describing the spreading fronts of two competing species respectively may intersect each other as time evolves the existence uniqueness and long time behavior of global solution have been established in this note we discuss the conditions for spreading and vanishing and more accurate limits of u v as t when spreading occurs some new results and simpler proofs will be provided