density estimates for a fluid jet model in the heisenberg group abstract in the heisenberg group setting we consider the fluid jet functional h n w 2 q w 1 d and we prove some density estimates for local minima in particular for large r the measure of the minimal jet in balls of radius r centered at the jet is of the order of r q 1 while the measure of the exterior medium is of the order of r q where q 2 n 1 is the homogeneous dimension also we obtain that h n u q in a suitable weak sense at the free boundary u 1

formative processes with applications to the decision problem in set theory i powerset and singleton operators this paper introduces formative processes composed by transitive partitions given a family f of sets a formative process ending in the venn partition e of f is shown to exist sufficient criteria are also singled out for a transitive partition to model via a function from set variables to unions of sets in the partition all set literals modeled by e on the basis of such criteria a procedure is designed that mimics a given formative process by another where sets have finite rank bounded by c e with c a specific computable function as a by product one of the core results on decidability in computable set theory is rediscovered namely the one that regards the satisfiability of unquantified set theoretic formulae involving boolean operators the singleton former and the powerset operator the method described which is able to exhibit a set solution when the answer is affirmative can be extended to solve the satisfiability problem for broader fragments of set theory

an approximate solution based on jacobi polynomials for time fractional convectiondiffusion equation operational matrix of shifted jacobi polynomials is considered solution of time fractional order convectiondiffusion problem is numerically estimated main problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded a system of nonlinear algebraic equations is achieved by approximating the known and unknown functions with the help of shifted jacobi functions problem can be used extensively in science and engineering as in oil reservoir simulations in this article we present a numerical method to numerically solve a time fractional convectiondiffusion equation our method is based on the operational matrices of shifted jacobi polynomials at first problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded then we approximate the known and unknown functions with the help of shifted jacobi functions a system of nonlinear algebraic equations is obtained finally the unknown coefficients are determined by mathematicatm we implemented the proposed method for several examples that they indicate the high accuracy method it should be noted that this method is generalizable to some appropriate problems

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

on stable solutions for boundary reactions a de giorgi type result in dimension 4 1 we prove that every bounded stable solution of begin aligned delta 1 2 u f u 0 qquad text in mathbb r 3 end aligned is a 1d profile i e u x phi e cdot x for some e in mathbb s 2 where phi mathbb r rightarrow mathbb r is a nondecreasing bounded stable solution in dimension one equivalently stable critical points of boundary reaction problems in mathbb r d 1 mathbb r d 1 cap x d 1 ge 0 of the form begin aligned int x d 1 ge 0 frac 1 2 nabla u 2 dx dx d 1 int x d 1 0 f u dx end aligned are 1d when d 3 these equations have been studied since the 1940 s in crystal dislocations also as it happens for the allen cahn equation the associated energies enjoy a gamma convergence result to the perimeter functional in particular when f u u 3 u or equivalently when f u frac 1 4 1 u 2 2 our result implies the analogue of the de giorgi conjecture for the half laplacian in dimension 4 namely that monotone solutions are 1d note that our result is a pde version of the fact that stable embedded minimal surfaces in mathbb r 3 are planes it is interesting to observe that the corresponding statement about stable solutions to the allen cahn equation namely when the half laplacian is replaced by the classical laplacian is still unknown for d 3

multivariable cochain operations and little cubes this paper has two goals the first see theorem 2 15 and remark 2 16 a is to construct a small em chain operad s which acts naturally on the normalized cochains s x of a topological space x this is of interest in view of a theorem of mandell 15 page 44 which states that if 9 is any e chain operad over f the algebraic closure of the field with p elements which acts naturally on s x 0 f then the homotopy category of connected p complete nilpotent spaces of finite type

fractional calculus a different approach to the analysis of viscoelastically damped structures fractional calculus is used to construct stress strain relationships for viscoelastic materials these relationships are used in the finite element analysis of viscoelastically damped structures and closed form solutions to the equations of motion are found the attractive feature of this approach is that very few empirical parameters are required to model the viscoelastic material and calculate the response of the structure for general loading conditions

a wavelet operational method for solving fractional partial differential equations numerically fractional calculus is an extension of derivatives and integrals to non integer orders and a partial differential equation involving the fractional calculus operators is called the fractional pde they have many applications in science and engineering however not only the analytical solution existed for a limited number of cases but also the numerical methods are very complicated and difficult in this paper we newly establish the simulation method based on the operational matrices of the orthogonal functions we formulate the operational matrix of integration in a unified framework by using the operational matrix of integration we propose a new numerical method for linear fractional partial differential equation solving in the method we 1 use the haar wavelet 2 establish a lyapunov type matrix equation and 3 obtain the algebraic equations suitable for computer programming two examples are given to demonstrate the simplicity clarity and powerfulness of the new method

fractional order legendre operational matrix of fractional integration for solving the riccati equation with fractional order this paper is devoted to both theoretical and numerical study of riccati equation with fractional order a formulation to the fractional order legendre operational matrix of fractional integration is constructed existence and uniqueness results for the considered problem are provided and proved the fractional derivative is described in the caputo sense some numerical examples are discussed to demonstrate the efficiency and the accuracy of the proposed algorithm

link between solitary waves and projective riccati equations many solitary wave solutions of nonlinear partial differential equations can be written as a polynomial in two elementary functions which satisfy a projective hence linearizable riccati system from that property the authors deduce a method for building these solutions by determining only a finite number of coefficients this method is much shorter and obtains more solutions than the one which consists of summing a perturbation series built from exponential solutions of the linearized equation they handle several examples for the henon heiles hamiltonian system they obtain several exact solutions one of them defines a new solitary wave solution for a coupled system of boussinesq and nonlinear schrodinger equations for a third order dispersive equation with two monomial nonlinearities they isolate all cases where the general solution is single valued

the toda system and clustering interfaces in the allen cahn equation we consider the allen cahn equation varepsilon 2 delta u 1 u 2 u 0 in a bounded smooth domain in mathbb r 2 under zero neumann boundary conditions where varepsilon 0 is a small parameter let 0 be a segment contained in connecting orthogonally the boundary under certain nondegeneracy and nonminimality assumptions for 0 satisfied for instance by the short axis in an ellipse we construct for any given n 1 a solution exhibiting n transition layers whose mutual distances are o varepsilon log varepsilon and which collapse onto 0 as varepsilon to 0 asymptotic location of these interfaces is governed by a toda type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary

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

some aspects of fractional diffusion equations of single and distributed order abstract the time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first order time derivative with a fractional derivative of order 0 1 the fundamental solution for the cauchy problem is interpreted as a probability density of a self similar non markovian stochastic process related to a phenomenon of sub diffusion the variance grows in time sub linearly a further generalization is obtained by considering a continuous or discrete distribution of fractional time derivatives of order less than one then the fundamental solution is still a probability density of a non markovian process that however is no longer self similar but exhibits a corresponding distribution of time scales

new quadrature approach based on operational matrix for solving a class of fractional variational problems this manuscript presents a new numerical approach to approximate the solution of a class of fractional variational problems the presented approach is consisting of using the shifted legendre orthonormal polynomials as basis functions of the operational matrix of fractional derivatives described in the caputo sense and that of fractional integrals described in the sense of riemann liouville with the help of the legendre gauss quadrature formula together with the lagrange multipliers method for converting such fractional variational problems into easier problems that consist of solving an algebraic system in the unknown coefficients the convergence of the proposed method is analyzed finally in order to demonstrate the accuracy of the present method some test problems are introduced with their approximate solutions and comparisons with other numerical approaches

error estimate for the numerical solution of fractional reaction subdiffusion process based on a meshless method in this paper a numerical technique based on a meshless method is proposed for solving the time fractional reaction subdiffusion equation firstly we obtain a time discrete scheme based on a finite difference scheme then we use the meshless galerkin method to approximate the spatial derivatives and obtain a full discrete scheme in the proposed scheme some integrals appear over the boundary and the domain of problem which will be approximated using gauss legendre quadrature rule then we prove that the time discrete scheme is unconditionally stable and convergent using the energy method we show convergence order of the time discrete scheme is o the aim of this paper is to obtain an error estimate and to show convergence for the meshless galerkin method based on the radial basis functions numerical examples confirm the efficiency and accuracy of the proposed scheme

a fractional epidemiological model for computer viruses pertaining to a new fractional derivative abstract in the computer security and for any defensive strategy computer viruses are very significant aspect to understand their expansion and extension is very important component in order to deal with this issue we consider a fractional epidemiological model in this article we analyze moderate epidemiological model to describe computer viruses with an arbitrary order derivative having non singular kernel we obtain the solution of the problem by using an iterative method by using the fixed point theorem the existence of the solution is discussed the uniqueness of the solution is verified we perform some numerical simulations and show graphically to observe the impact of the arbitrary order derivative

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

the use of fake algebraic riccati equations for co channel demodulation this paper describes a method for nonlinear filtering based on an adaptive observer which guarantees the local stability of the linearized error system a fake algebraic riccati equation is employed in the calculation of the filter gain the design procedure attempts to produce a stable filter at the expense of optimality this contrasts with the extended kalman filter ekf which attempts to preserve optimality via its linearization procedure at the expense of stability a passivity approach is applied to deduce stability conditions for the filter error system the performance is compared with an ekf for a co channel frequency demodulation application

a new method for determining the solution of riccati differential equations abstract this paper presents a new and efficient approach for determining the solution of riccati differential equation the riccati equation is first converted to a second order linear ordinary differential equation and then to a volterra integral equation by solving the resulting volterra equation by means of taylor s expansion the approximate solution of riccati differential equation is obtained which can be achieved by symbolic computation the accuracy of approximate solution can be further improved with the increase of the order of approximations an error analysis is given test examples demonstrate the effectiveness of the method a comparison between the present results with previous results is made inferring that the suggested method is not only enough accurate but also quite stable

pitfalls in fast numerical solvers for fractional differential equations we consider the problem of implementing fast algorithms for the numerical solution of initial value problems of the form x t f t x t x 0 x0 where x is the derivative of x of order in the sense of caputo and 0 1 we review some of the existing methods and explain their respective strengths and weaknesses we identify and discuss potential problems in the development of generally applicable schemes

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

fractional spectral and pseudo spectral methods in unbounded domains theory and applications abstract this paper is intended to provide exponentially accurate galerkin petrov galerkin and pseudo spectral methods for fractional differential equations on a semi infinite interval we start our discussion by introducing two new non classical lagrange basis functions nlbfs 1 and nlbfs 2 which are based on the two new families of the associated laguerre polynomials galfs 1 and galfs 2 obtained recently by the authors in 28 with respect to the nlbfs 1 and nlbfs 2 two new non classical interpolants based on the associated laguerre gauss and laguerre gauss radau points are introduced and then fractional pseudo spectral differentiation and integration matrices are derived convergence and stability of the new interpolants are proved in detail several numerical examples are considered to demonstrate the validity and applicability of the basis functions to approximate fractional derivatives and integrals of some functions moreover the pseudo spectral galerkin and petrov galerkin methods are successfully applied to solve some physical ordinary differential equations of either fractional orders or integer ones some useful comments from the numerical point of view on galerkin and petrov galerkin methods are listed at the end

local minimisers and singular perturbations we construct local minimisers to certain variational problems the method is quite general and relies on the theory of convergence the approach is demonstrated through the model problem it is shown that in certain nonconvex domains n and for e small there exist nonconstant local minimisers u e satisfying u e 1 except in a thin transition layer the location of the layer is determined through the requirement that in the limit u e u 0 the hypersurface separating the states u 0 1 and u 0 1 locally minimises surface area generalisations are discussed with for example vector valued u and anisotropic perturbations replacing u 2

stable transition layers in a semilinear diffusion equation with spatial inhomogeneities in n dimensional domains we prove existence and establish the asymptotic behavior as e 0 of stable stationary solutions to the equation ut e d x u 1 u2 u a x for t x r where rn n 2 with neumann boundary condition the function a x c0 satisfies 1 a x 1 and vanishes on some hypersurfaces the results generalize to n dimensional domains and to variable diffusivity earlier paper by angenent et al j differential equations 67 1987 212

muntz legendre spectral collocation method for solving delay fractional optimal control problems abstract in this paper a numerical method is applied for solving delay fractional optimal control problems dfocps the fractional derivative is described in the caputo sense since the fractional derivative of muntz polynomials can be expressed in terms of the same polynomials those polynomials can accurately represent properties of fractional calculus in some situations such as in the frequency response based analysis of control systems containing a time delay it is necessary to substitute exponential function with an approximation in the form of a rational function the most common approximation is the pade approximation at the first step using pade approximation the delay problem is transformed to a non delay problem next using the operational matrix of the fractional derivative of muntz polynomials and pseudospectral method fractional optimal control problem focp is reduced to a nonlinear programming problem some numerical examples are given to illustrate the effectiveness of the proposed scheme

iterative reproducing kernel hilbert spaces method for riccati differential equations this paper presents iterative reproducing kernel hilbert spaces method irkhsm to obtain the numerical solutions for riccati differential equations with constant and variable coefficients representation of the exact solution is given in the w 2 2 0 x reproducing kernel space numerical solution of riccati differential equations is acquired by interrupting the n term of the exact solution also the error of the numerical solution is monotone decreasing in terms of the norm of w 2 2 0 x the outcomes from numerical examples show that the present iterative algorithm is very effective and convenient

a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation abstract an efficient numerical technique is proposed to solve one and two dimensional space fractional tempered fractional diffusion wave equations the space fractional is based on the riemann liouville fractional derivative at first the temporal direction is discretized using a second order accurate difference scheme then a classic galerkin finite element is employed to obtain a full discrete scheme furthermore for the time discrete and the full discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method finally two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique

stable solutions of the allen cahn equation in dimension 8 and minimal cones abstract for all n 1 we are interested in bounded solutions of the allen cahn equation u u u 3 0 which are defined in all r n 1 and whose zero set is asymptotic to a given minimal cone in particular in dimension n 1 8 we prove the existence of stable solutions of the allen cahn equation whose zero sets are not hyperplanes

analysis of regularized long wave equation associated with a new fractional operator with mittag leffler type kernel abstract in this work we aim to present a new fractional extension of regularized long wave equation the regularized long wave equation is a very important mathematical model in physical sciences which unfolds the nature of shallow water waves and ion acoustic plasma waves the existence and uniqueness of the solution of the regularized long wave equation associated with atangana baleanu fractional derivative having mittag leffler type kernel is verified by implementing the fixed point theorem the numerical results are derived with the help of an iterative algorithm in order to show the effects of various parameters and variables on the displacement the numerical results are presented in graphical and tabular form

an efficient numerical algorithm for the fractional drinfeld sokolov wilson equation abstract the fundamental purpose of the present paper is to apply an effective numerical algorithm based on the mixture of homotopy analysis technique sumudu transform approach and homotopy polynomials to obtain the approximate solution of a nonlinear fractional drinfeld sokolov wilson equation the nonlinear drinfeld sokolov wilson equation naturally occurs in dispersive water waves the uniqueness and convergence analysis are shown for the suggested technique the convergence of the solution is fixed and managed by auxiliary parameter the numerical results are shown graphically results obtained by the application of the technique disclose that the suggested scheme is very accurate flexible effective and simple to use

jacobi collocation method for the approximate solution of some fractional order riccati differential equations with variable coefficients abstract this paper presents a computational method for the approximate solution of arbitrary order non linear fractional riccati differential equations with variable coefficients proposed computational method is a combination of the operational matrix of integration method and the collocation method associated with the jacobi polynomials convergence analysis of the proposed method is provided numerical results for different fractional orders of the riccati differential equations are discussed figures and tables are used to show the numerical results derived from the proposed computational method for particular cases of jacobi polynomials such as the legendre polynomials the chebyshev polynomials of the second kind the chebyshev polynomials of the third kind the chebyshev polynomial of the fourth kind and the gegenbauer or ultraspherical polynomials numerical results from the proposed methods are compared from those derived by using the existing analytical and numerical methods it is observed that the results from the proposed method are more accurate maximum absolute error and the root mean square error tables are given for all five kinds of polynomials for comparison purposes

optimal profiles in a phase transition model with a saturating flux abstract it is well known that for the allen cahn equation the minimizing transition in an infinite cylinder r is one dimensional and unique up to a translation in the first variable we analyze in this paper the existence and symmetry of optimal profiles for transitions in a similar phase separation model with a saturating flux this amounts to consider transitions in the space of bv functions as we consider the area integral instead of the dirichlet energy to penalize the creation of wild interfaces

an efficient approach for solving the riccati equation with fractional orders the present study introduces a novel and simple analytical method for the solution of fractional order riccati differential equation in this approach the solution considered as a taylor series expansion converges rapidly to the nonlinear problem new homotopy perturbation method nhpm depends only on two components of the homotopy series the method is illustrated by applications and the results obtained are compared with those of the exact solution moreover comparing the methodology with some known techniques shows that the present approach is relatively easy and efficient

a new method based on legendre polynomials for solutions of the fractional two dimensional heat conduction equation abstract in this paper we develop a new scheme for numerical solutions of the fractional two dimensional heat conduction equation on a rectangular plane our main aim is to generalize the legendre operational matrices of derivatives and integrals to the three dimensional case by the use of these operational matrices we reduce the corresponding fractional order partial differential equations to a system of easily solvable algebraic equations the method is applied to solve several problems the results we obtain are compared with the exact solutions and we find that the error is negligible

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9