global stability of endemic equilibrium of an epidemic model with birth and death on complex networks abstract we study global stability of endemic equilibrium of an epidemic model with birth and death on complex networks under some conditions the local asymptotic stability of the endemic equilibrium was established by zhang and jin 2011 for correlated networks and the global asymptotic stability was obtained by chen and sun 2014 for uncorrelated networks in this work we remove those conditions and prove by constructing a lyapunov function that the endemic equilibrium is globally asymptotically stable numerical simulations are also presented to illustrate the feasibility of the result

discontinuous spectral element methods for time and space fractional advection equations we develop spectral element methods for a time and space fractional advection equation of the form prescript 0 mathcal d tau t u x t theta prescript 0 mathcal d nu x u x t f x t of order tau in 0 1 nu in 0 1 subject to dirichlet initial boundary conditions we present two spectrally accurate and efficient methods for global discretization of both temporal and spatial terms instead of employing traditional low order time integration methods to this end we first develop a petrov galerkin in time and discontinuous galerkin in space pg dg method where we carry out the time integration using a single time domain spectral method sm and we perform the spatial discretization using the discontinuous spectral hp element method dsem this scheme also leads to a more efficient time integration when the time derivative is integer order i e tau 1 we develop the sm dsem scheme based on a new spectral theory for fractional sturm liouville problems

equilibrium problem for elastic body with delaminated t shape inclusion abstract we analyze an equilibrium problem for 2d elastic body with a t shape thin inclusion in presence of damage a part of the inclusion is elastic and the other part is a rigid one a delamination of the inclusion from the elastic body is assumed thus forming a crack between the elastic body and the inclusion nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces the damage is characterized by a positive parameter the paper provides an asymptotic analysis of the solutions as the damage parameter tends to infinity and to zero a passage to infinity of a rigidity parameter of the elastic part of the inclusion is also analyzed junction conditions are determined at the connection point between the elastic and rigid parts of the inclusion an existence theorem is proved for an inverse problem of finding displacement fields and the damage and rigidity parameters provided that a displacement of the tip point of the inclusion is known

analytical solution for the time fractional telegraph equation by the method of separating variables in this paper a method of separating variables is effectively implemented for solving a time fractional telegraph equation tfte we discuss and derive the analytical solution of the tfte with three kinds of nonhomogeneous boundary conditions namely dirichlet neumann and robin boundary conditions

a novel numerical method for the time variable fractional order mobile immobile advection dispersion model evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations recently researchers have found that many physical processes exhibit fractional order behavior that varies with time or space the continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable in this paper we consider the mobile immobile advection dispersion model with the coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint a novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated as for the convergence of the numerical method we only consider a special case i e the time fractional derivative is independent of the time variable t the case where the time fractional derivative depends on both the time variable t and the space variable x will be considered in a future work finally numerical examples are provided to show that the implicit difference approximation is computationally efficient

the analysis of an epidemic model on networks abstract the paper consider an epidemic model with birth and death on networks we derive the epidemic threshold r0 dependent on birth rate b death rate d natural death and from the infectious disease and natural death and cure rate and the stability of the equilibriums the disease free equilibrium and endemic equilibrium are analysed finally the effects of various immunization schemes are studied and compared we show that both targeted and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness for active immunization the threshold is easier to apply practically to illustrate our theoretical analysis some numerical simulations are also included

epidemic thresholds of the susceptible infected susceptible model on networks a comparison of numerical and theoretical results recent work has shown that different theoretical approaches to the dynamics of the susceptible infectedsusceptible sis model for epidemics lead to qualitatively different estimates for the position of the epidemic threshold in networks here we present large scale numerical simulations of the sis dynamics on various types of networks allowing the precise determination of the effective threshold for systems of finite size n we compare quantitatively the numerical thresholds with theoretical predictions of the heterogeneous mean field theory and of the quenched mean field theory we show that the latter is in general more accurate scaling with n with the correct exponent but often failing to capture the correct prefactor

global stability of virus spreading in complex heterogeneous networks various networks are possessed of an obvious heterogeneity in the connectivity properties and it is of practical significance to study epidemic spreading in networks of this kind pastor satorras and vespignani established the dynamical mean field reaction rate equations for the spreading of infections in complex heterogeneous networks based on the well known sis model and figured out an epidemic threshold lambda c such that if lambda effective spreading rate is above lambda c the infection spreads and becomes endemic the significance of this result is far reaching however the authors have not found a strict mathematical proof of their conclusion in the literature in this paper we approach this problem by proving that if lambda is above lambda c the infection spreads and approaches the unique positive stationary point of the reaction rate equations as long as there exist infected nodes in the network initially i e the virus infection process is globally stable

epidemic dynamics in finite size scale free networks many real networks present a bounded scale free behavior with a connectivity cutoff due to physical constraints or a finite network size we study epidemic dynamics in bounded scale free networks with soft and hard connectivity cutoffs the finite size effects introduced by the cutoff induce an epidemic threshold that approaches zero at increasing sizes the induced epidemic threshold is very small even at a relatively small cutoff showing that the neglection of connectivity fluctuations in bounded scale free networks leads to a strong overestimation of the epidemic threshold we provide the expression for the infection prevalence and discuss its finite size corrections the present paper shows that the highly heterogeneous nature of scale free networks does not allow the use of homogeneous approximations even for systems of a relatively small number of nodes

numerical analysis of a two parameter fractional telegraph equation abstract in this paper we consider the two parameter fractional telegraph equation of the form c d t 0 1 u t x c d x 0 1 u t x c d t 0 u t x u t x 0 here c d t 0 c d t 0 1 c d x 0 1 are operators of the caputo type fractional derivative where 0 1 and 0 1 a numerical method is introduced to solve this fractional telegraph equation and stability conditions for the numerical method are obtained numerical examples are given in the final section of the paper

a shape topological control problem for nonlinear crack defect interaction the antiplane variational model we consider the shape topological control of a singularly perturbed variational inequality the geometry dependent state problem that we address in this paper concerns a heterogeneous medium with a micro object defect and a macro object crack modeled in two dimensions the corresponding nonlinear optimization problem subject to inequality constraints at the crack is considered within a general variational framework for the reason of asymptotic analysis singular perturbation theory is applied resulting in the topological sensitivity of an objective function representing the release rate of the strain energy in the vicinity of the nonlinear crack the antiplane strain energy release rate is expressed by means of the mode iii stress intensity factor that is examined with respect to small defects such as microcracks holes and inclusions of varying stiffness the result of shape topological control is useful either for arrests or rise of crack growth

second order approximations for variable order fractional derivatives fractional calculus allows variable order of fractional operators which can be exploited in diverse physical and biological applications where rates of change of the quantity of interest may depend on space and or time in this paper we derive two second order approximation formulas for the variable order fractional time derivatives involved in anomalous diffusion and wave propagation we then present numerical tests that verify the theoretical estimates of convergence rate and also simulations of anomalous sub diffusion and super diffusion that demonstrate new localized diffusion rates that depend on the curvature of the variable order function finally we perform simulations of wave propagation in a truncated domain to demonstrate how erroneous wave reflections at the boundaries can be eliminated by super diffusion and also simulations of the burgers equation that serve as a testbed for studying the loss and recovery of monotonicity using again variable rate diffusion as a function of space and or time taken together our results demonstrate that variable order fractional derivatives can be used to model the physics of anomalous transport with spatiotemporal variability but also as new effective numerical tools that can deal with the long standing issues of outflow boundary conditions and monotonicity of integer order pdes

two high order numerical algorithms for solving the multi term time fractional diffusion wave equations in this paper we apply a high order difference scheme and galerkin spectral technique for the numerical solution of multi term time fractional partial differential equations the proposed methods are based on a finite difference scheme in time the time fractional derivatives which have been described in caputo s sense are approximated by a scheme of order o 3 1 2 and the space derivative is discretized with a fourth order compact finite difference procedure and galerkin spectral method we prove the unconditional stability of the compact procedure by coefficient matrix property the l convergence of the compact finite difference method has been proved by the energy method also we obtain an error estimate for galerkin spectral method numerical results are provided to verify the accuracy and efficiency of the proposed schemes

a unified petrov galerkin spectral method for fractional pdes existing numerical methods for fractional pdes suffer from low accuracy and inefficiency in dealing with three dimensional problems or with long time integrations we develop a unified and spectrally accurate petrov galerkin pg spectral method for a weak formulation of the general linear fractional partial differential equations fpdes of the form 0dt2 u j 1dcj ajdxj2 ju u f where 2 2 j 0 1 j 0 1 in a 1 d1 d dimensional space time domain subject to dirichlet initial and boundary conditions we perform the stability analysis in 1 d and the corresponding convergence study of the scheme in multi d the unified pg spectral method applies to the entire family of linear hyperbolichyperbolic parabolicparabolic and ellipticelliptic like equations we develop the pg method based on a new spectral theory for fractional sturm liouville problems fslps recently introduced in zayernouri and karniadakis 2013 specifically we employ the eigenfunctions of the fslp of first kind fslp i called jacobi poly fractonomials as temporal spatial bases next we construct a different space for test functions from poly fractonomial eigenfunctions of the fslp of second kind fslp ii besides the high order spatial accuracy of the pg method we demonstrate its efficiency and spectral accuracy in time integration schemes for solving time dependent fpdes as well rather than employing algebraically accurate traditional methods especially when 2 12 1 finally we formulate a general fast linear solver based on the eigenpairs of the corresponding temporal and spatial mass matrices with respect to the stiffness matrices which reduces the computational cost drastically we demonstrate that this framework can reduce to hyperbolic fpdes such as time and space fractional advection tsfa parabolic fpdes such as time and space fractional diffusion tsfd model and elliptic fpdes such as fractional helmholtz poisson equations with the same ease and cost several numerical tests confirm the efficiency and spectral convergence of the unified pg spectral method for the aforementioned families of fpdes moreover we demonstrate the computational efficiency of the new approach in higher dimensions e g 1 3 1 5 and 1 9 dimensional problems

variational and numerical analysis of a quasistatic viscoelastic problem with normal compliance friction and damage we consider a model for quasistatic frictional contact between a viscoelastic body and a foundation the material constitutive relation is assumed to be nonlinear the mechanical damage of the material caused by excessive stress or strain is described by the damage function the evolution of which is determined by a parabolic inclusion the contact is modeled with the normal compliance condition and the associated version of coulomb s law of dry friction we derive a variational formulation for the problem and prove the existence of its unique weak solution we then study a fully discrete scheme for the numerical solutions of the problem and obtain error estimates on the approximate solutions

computing the eigenvalues of the generalized sturm liouville problems based on the lie group sl 2 r for the generalized sturm liouville problems we can construct an sl 2 r lie group shooting method to find eigenvalues by using the closure property of the lie group a one step lie group transformation between the boundary values at two ends of the considered interval is established hence we can theoretically derive an analytical characteristic equation to determine the eigenvalues for the generalized sturm liouville problems because the closed form formulas are derived to calculate the unknown left boundary values in terms of l the present method provides an easy numerical implementation and has a cheap computational cost numerical examples are examined to show that the present sl 2 r lie group shooting method is effective

dynamics of a network based sis epidemic model with nonmonotone incidence rate abstract this paper studies the dynamics of a network based sis epidemic model with nonmonotone incidence rate this type of nonlinear incidence can be used to describe the psychological effect of certain diseases spread in a contact network at high infective levels we first find a threshold value for the transmission rate this value completely determines the dynamics of the model and interestingly the threshold is not dependent on the functional form of the nonlinear incidence rate furthermore if the transmission rate is less than or equal to the threshold value the disease will die out otherwise it will be permanent numerical experiments are given to illustrate the theoretical results we also consider the effect of the nonlinear incidence on the epidemic dynamics

global stability of endemic equilibrium for a siqrs epidemic model on complex networks abstract we study the global stability of endemic equilibrium of a siqrs epidemic model on complex networks this model was proposed by li et al 2014 the local asymptotic stability of disease free equilibrium and the permanence of the disease were proved by them by use of a combination of lyapunov function method and a monotone iterative technique we prove that the endemic equilibrium is globally asymptotically stable numerical simulations indicate that quarantine strategy is an effective measure in preventing and controlling epidemic spreading

on the stability and convergence of the time fractional variable order telegraph equation in this work we have generalized the time fractional telegraph equation using the concept of derivative of fractional variable order the generalized equation is called time fractional variable order telegraph equation this new equation was solved numerically via the crank nicholson scheme stability and convergence of the numerical solution were presented in details numerical simulations of the approximate solution of the time fractional variable order telegraph equation were presented for different values of the grid point

on the new fractional derivative and application to nonlinear fisher s reaction diffusion equation recently caputo and fabrizio introduced a new derivative with fractional order in this paper we presented useful tools about the new derivative and applied it to the nonlinear fisher s reaction diffusion equation we presented the solution of the modified equation using the notion of iterative method using the theory of fixed point we presented the stability of the used method some numerical simulations were presented for different values of fractional order

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 direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

stability analysis of an sis epidemic model with feedback mechanism on networks abstract in this paper an epidemic model with feedback mechanism on networks is investigated we obtain the basic reproductive number r 0 and analyze the stability behaviors of disease free equilibrium e 0 and endemic equilibrium e when r 0 1 the disease free equilibrium is globally asymptotically stable and when r 0 1 the endemic equilibrium is asymptotically stable although the feedback mechanism cannot change the basic reproductive number r 0 as we prove it should be indicated that it can weaken the spreading of diseases and reduce the endemic level finally the results of the stability and the effectiveness of the feedback mechanism are illustrated by some numerical simulations

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

center stable manifold for planar fractional damped equations in this paper we discuss the existence of a center stable manifold for planar fractional damped equations by constructing a suitable lyapunovperron operator via giving asymptotic behavior of mittagleffler function we obtain an interesting center stable manifold theorem finally an example is provided to illustrate the result

fractional difference finite element approximations for the time space fractional telegraph equation abstract in this paper we study the numerical solution of the time space fractional order fractional for simplicity telegraph equation which can be used in signal analysis for transmission and propagation of electrical signals also the modeling of the reaction diffusion and the random walk of suspension flows and so on the semi discrete and fully discrete numerical approximations are both analyzed where the galerkin finite element method for the spatial riemann liouville fractional derivative with order 1 1 2 and the finite difference schemes for the temporal caputo derivatives with orders 1 2 1 and 2 are analyzed respectively results on the existence and uniqueness of the solution the numerical stability and the error estimates are displayed in details numerical examples are included to confirm the theoretical analysis

a note on global stability of the virose equilibrium for network based computer viruses epidemics abstract it has been recently conjectured in yang et al 2013 27 that the unique virose equilibrium of a network based model is globally asymptotically stable in the feasible region the aim of this paper is to investigate the global stability of the virose equilibrium by using the direct lyapunov method and appropriate grouping techniques for lyapunov function derivatives some sufficient conditions are established which improves the relevant results moreover we provide a new brief proof for global stability of the virus free equilibrium by using comparison method and discuss quantitatively the effect of various immunization schemes including uniform immunization targeted immunization and high risk immunization on the viruses spread which suggests promising strategy to halt viruses spread in computer networks

asymptotic imaging of perfectly conducting cracks in this paper we consider cracks with dirichlet boundary conditions we first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack based on this formula we design a noniterative approach for locating a collection of small cracks in order to do so we construct a response matrix from the boundary measurements the location and the length of the crack are estimated respectively from the projection onto the noise space and the first significant singular value of the response matrix indeed the direction of the crack is estimated from the second singular vector we then consider an extended crack with dirichlet boundary conditions we rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack to reconstruct an extended crack from many boundary measurements we develop two methods for obtaining a good guess several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave

global uniqueness for an inverse boundary value problem arising in elasticity we prove that we can determine the lame parameters of an elastic isotropic inhomogeneous medium in dimensionsn 3 by making measurements of the displacements and corresponding stresses at the boundary of the medium

semirigid inclusions in elastic bodies mechanical interplay and optimal control abstract the paper concerns an analysis of an equilibrium problem for 2d elastic body with two semirigid inclusions it is assumed that inclusions have a joint point and we investigate a junction problem for these inclusions the existence of solutions is proved and different equivalent formulations of the problem are proposed we investigate a convergence to infinity of a rigidity parameter of the semirigid inclusion it is proved that in the limit we obtain an equilibrium problem for the elastic body with a rigid inclusion and a semirigid one a parameter identification problem is investigated in particular the existence of a solution to a suitable optimal control problem is proved

analytic and approximate solutions of the space and time fractional telegraph equations the adomian decomposition method is used to obtain analytic and approximate solutions of the space and time fractional telegraph equations the space and time fractional derivatives are considered in the caputo sense the analytic solutions are calculated in the form of series with easily computable terms some examples are given the results reveal that the adomian method is very effective and convenient

structural optimization using sensitivity analysis and a level set method in the context of structural optimization we propose a new numerical method based on a combination of the classical shape derivative and of the level set method for front propagation we implement this method in two and three space dimensions for a model of linear or nonlinear elasticity we consider various objective functions with weight and perimeter constraints the shape derivative is computed by an adjoint method the cost of our numerical algorithm is moderate since the shape is captured on a fixed eulerian mesh although this method is not specifically designed for topology optimization it can easily handle topology changes however the resulting optimal shape is strongly dependent on the initial guess

global stability of a network based sirs epidemic model with nonmonotone incidence rate abstract this paper studies the dynamics of a network based sirs epidemic model with vaccination and a nonmonotone incidence rate this type of nonlinear incidence can be used to describe the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infective individuals on heterogeneous networks is getting larger using the analytical method epidemic threshold r 0 is obtained when r 0 is less than one we prove the disease free equilibrium is globally asymptotically stable and the disease dies out while r 0 is greater than one there exists a unique endemic equilibrium by constructing a suitable lyapunov function we also prove the endemic equilibrium is globally asymptotically stable if the inhibitory factor is sufficiently large numerical experiments are also given to support the theoretical results it is shown both theoretically and numerically a larger can accelerate the extinction of the disease and reduce the level of disease

high precision identification of an object optimality conditions based concept of imaging a class of inverse problems for the identification of an unknown geometric object from given measurements is considered a concept for object imaging based on optimality conditions and level sets is introduced which provides high resolution properties of the identification problem and stability to discretization and noise errors as a specific case the identification of the center of a test object of arbitrary shape and unknown boundary conditions from d boundary measurements in d spatial dimensions in the context of the helmholtz equation is described in detail for analysis and numerical realization methods from topology optimization generalized singular perturbations endowed with variational techniques and a petrov galerkin enrichment within generalized fem are used

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

damageable contact between an elastic body and a rigid foundation in this work the contact problem between an elastic body and a rigid obstacle is studied including the development of material damage which results from internal compression or tension the variational problem is formulated as a first kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field the existence of a unique local weak solution is stated then a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an euler scheme to discretize the time derivatives error estimates are derived on the approximate solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions finally three two dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme

numerical approximation of higher order time fractional telegraph equation by using a combination of a geometric approach and method of line we propose a simple and accurate numerical scheme for solving the time fractional telegraph tft equation within caputo type fractional derivative a fictitious coordinate is imposed onto the problem in order to transform the dependent variable u x t into a new variable with an extra dimension in the new space with the added fictitious dimension a combination of method of line and group preserving scheme gps is proposed to find the approximate solutions this method preserves the geometric structure of the problem power and accuracy of this method has been illustrated through some examples of tft equation