stability of a general hiv 1 reaction diffusion model with multiple delays and immune response abstract the aim of this paper is to study a general hiv 1 reaction diffusion model with space dependent diffusion coefficients and antibody immune response the model contains four discrete time delays and three categories of infected cells latently infected cells short lived productively infected cells and long lived productively infected cells the effect of highly active antiretroviral therapies haart used to treat hiv 1 is incorporated into the model the existence non negativity and boundedness of solutions are studied in order to verify the well posedness of the model all possible equilibria of the model are evaluated and the threshold parameters needed for their existence and stability are investigated the global asymptotic stability of the corresponding equilibria are proved by building suitable lyapunov functionals the numerical simulations are conducted in order to confirm the theoretical results and show the behavior of solutions in space and time

global dynamics of an hiv 1 infection model with distributed intracellular delays in this paper an hiv 1 infection model with distributed intracellular delays is investigated where the intracellular delays account for the time the target cells are contacted by the virus particles and the time the contacted cells become actively infected meaning that the contacting virions enter cells and the time the virus has penetrated into a cell and the time the new virions are created within the cell and are released from the cell respectively by analyzing the characteristic equations the local stability of an infection free equilibrium and a chronic infection equilibrium of the model is established by using suitable lyapunov functionals and lasalle s invariance principle it is proved that if the basic reproduction ratio is less than unity the infection free equilibrium is globally asymptotically stable and if the basic reproduction ratio is greater than unity the chronic infection equilibrium is globally asymptotically stable

global stability of an hiv 1 infection model with saturation infection and intracellular delay in this paper an hiv 1 infection model with a saturation infection rate and an intracellular delay accounting for the time between viral entry into a target cell and the production of new virus particles is investigated by analyzing the characteristic equations the local stability of an infection free equilibrium and a chronic infection equilibrium of the model is established by using suitable lyapunov functionals and the lasalle invariant principle it is proved that if the basic reproduction ratio is less than unity the infection free equilibrium is globally asymptotically stable if the basic reproduction ratio is greater than unity the chronic infection equilibrium is globally asymptotically stable

lyapunov functionals for delay differential equations model of viral infections we study global properties of a class of delay differential equations model for virus infections with nonlinear transmissions compared with the typical virus infection dynamical model this model has two important and novel features to give a more complex and general infection process a general nonlinear contact rate between target cells and viruses and the removal rate of infected cells are considered and two constant delays are incorporated into the model which describe i the time needed for a newly infected cell to start producing viruses and ii the time needed for a newly produced virus to become infectious mature respectively by the lyapunov direct method and using the technology of constructing lyapunov functionals we establish global asymptotic stability of the infection free equilibrium and the infected equilibrium we also discuss the effects of two delays on global dynamical properties by comparing the results with the stability conditions for the model without delays further we

the numerical solution of linear multi term fractional differential equations systems of equations in this paper we show how the numerical approximation of the solution of a linear multi term fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity we begin by showing how our method applies to a simple class of problems and we give a convergence result we solve the bagley torvik equation as an example we show how the method can be applied to a general linear multi term equation and give two further examples

technical communique a note on fractional order derivatives of periodic functions in this paper it is shown that the fractional order derivatives of a periodic function with a specific period cannot be a periodic function with the same period the fractional order derivative considered here can be obtained based on each of the well known definitions grunwald letnikov definition riemann liouville definition and caputo definition this concluded point confirms the result of a recently published work proving the non existence of periodic solutions in a class of fractional order models also based on this point it can be easily proved the absence of periodic responses in a wider class of fractional order models finally some examples are presented to show the applicability of the paper achievements in the solution analysis of fractional order systems

global existence and asymptotic behavior of classical solutions to a fractional logistic keller segel system abstract in this paper we consider a fractional parabolic elliptic keller segel system with a logistic source on r n first we establish the regularity of weak solutions of the fractional parabolic equation using blow up arguments combined with liouville type theorems next by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions moreover the global existence and boundedness of classical solutions for given initial data are obtained under some conditions finally we show the asymptotic behavior of the global solutions with strictly positive initial data

boundedness enforced by mildly saturated conversion in a chemotaxis may nowak model for virus infection abstract we study the system u t u u v u f u w v t v v f u w w t w w v which models the virus dynamics in an early stage of an hiv infection in a smooth bounded domain r n n n for a parameter 0 and a given function f c 1 0 satisfying f 0 f 0 0 and f s k f s for all s 1 some k f 0 and r we prove that whenever 2 n solutions to exist globally and are bounded the proof mainly relies on smoothing estimates for the neumann heat semigroup and in the case 1 on a functional inequality furthermore we provide some indication why the exponent 2 n could be essentially optimal

short memory principle and a predictor corrector approach for fractional differential equations fractional differential equations are increasingly used to model problems in acoustics and thermal systems rheology and modelling of materials and mechanical systems signal processing and systems identification control and robotics and other areas of application this paper further analyses the underlying structure of fractional differential equations from a new point of view we apprehend the short memory principle of fractional calculus and farther apply a adams type predictor corrector approach for the numerical solution of fractional differential equation and the detailed error analysis is presented combining the short memory principle and the predictor corrector approach we gain a good numerical approximation of the true solution of fractional differential equation at reasonable computational cost a numerical example is provided and compared with the exact analytical solution for illustrating the effectiveness of the short memory principle

splitting methods for the nonlocal fowler equation we consider a nonlocal scalar conservation law proposed by andrew c fowler to describe the dynamics of dunes and we develop a numerical procedure based on splitting methods to approximate its solutions we begin by proving the convergence of the well known lie formula which is an approximation of the exact solution of order one in time we next use the split step fourier method to approximate the continuous problem using the fast fourier transform and the finite difference method our numerical experiments confirm the theoretical results

a numerical scheme for the solution of multi order fractional differential equations we present an algorithm for the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 0 combined with suitable initial conditions the fractional derivatives are described in the caputo sense the algorithm is based on adomian s decomposition approach and the solutions are calculated in the form of a convergent series with easily computable components several numerical examples are presented

a generalized hbv model with diffusion and two delays an hepatitis b virus hbv model with spatial diffusion general incidence rate and time delays subject to homogeneous neumann boundary conditions is formulated the stability of the disease free equilibrium and the chronic infection equilibrium is analyzed by using the linearization method and by constructing appropriate lyapunov functionals furthermore several diffusive hbv models existing in the literature are generalized and extended

boundedness of solutions to a virus infection model with saturated chemotaxis abstract we show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous neumann boundary conditions more precisely the system considered is u t u u 1 u v u w u v t v u w v w t w w v with 0 and solvability and boundedness of the solution are shown under the condition that 2 3 if n 1 1 2 n 2 6 n 4 if 2 n 4 n 4 if n 5

numerical solution of nonlinear fractional differential equations by spline collocation methods an initial value problem for nonlinear fractional differential equations is considered using an integral equation reformulation of the initial value problem some regularity properties of the exact solution are derived on the basis of these properties the numerical solution of initial value problems by piecewise polynomial collocation methods is discussed in particular the attainable order of convergence of proposed algorithms is studied and a global superconvergence effect for a special choice of collocation points is established theoretical results are verified by means of numerical examples

on the convergence of spline collocation methods for solving fractional differential equations in the first part of this paper we study the regularity properties of solutions of initial value problems of linear multi term fractional differential equations we then use these results in the convergence analysis of a polynomial spline collocation method for solving such problems numerically using an integral equation reformulation and special non uniform grids global convergence estimates are derived from these estimates it follows that the method has a rapid convergence if we use suitable nonuniform grids and the nodes of the composite gaussian quadrature formulas as collocation points theoretical results are verified by some numerical examples

numerical algorithms for caputo fractional order differential equations abstractthe initial value problems ivps of caputo fractional order differential equations are very important in control systems modelling and simulation a series of numerical algorithms are proposed in the paper in solving systematically various kinds of caputo equations for linear caputo equations the divergent problems of the existing taylor auxiliary function are pointed out and two effective algorithms are presented to transform the nonzero ivps into zero ones where the closed form solutions are available furthermore algorithms for nonlinear caputo equation are also presented aiming at finding numerical solutions to all kinds of caputo equations error analysis for the proposed algorithms is provided and numerical examples are presented to illustrate the accuracy and effectiveness of the algorithms

anomalous diffusion without scale invariance asymptotic behaviour of a new class of anomalous diffusion equations for subdiffusive transport defined in terms of generalized distributed fractional order time derivatives is considered the effect of slowly varying factors on the scaling function of asymptotic solutions is demonstrated the origin of slowly varying scaling factors in the ctrw models is discussed

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

weakly singular gronwall inequalities and applications to fractional differential equations abstract we obtain some new gronwall type inequalities which are applicable to some weakly singular volterra integral equations similar to the ones first studied by d henry the main interest is that we consider cases with a double singularity and we obtain explicit l bounds rather than l 1 bounds furthermore our bounds involve the exponential function and not the mittag leffler function as in some previous works we give applications to some volterra integral equations with a doubly singular kernel that arise from caputo fractional differential equations where as opposed to previous papers we have a singularity in the nonlinearity

stability preservation analysis for frequency based methods in numerical simulation of fractional order systems in this paper the frequency domain based numerical methods for simulation of fractional order systems are studied in the sense of stability preservation first the stability boundary curve is exactly determined for these methods then this boundary is analyzed and compared with an accurate ideal boundary in different frequency ranges also the critical regions in which the stability does not preserve are determined finally the analytical achievements are confirmed via some numerical illustrations

fractional linear multistep methods for abel volterra integral equations of the second kind fractional powers of linear multistep methods are suggested for the numerical solution of weakly singular volterra integral equations the proposed methods are convergent of the order of the underlying multistep method also in the generic case of solutions which are not smooth at the origin the stability properties stability region a stability a a stability are closely related to those of the underlying linear multistep method

a constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation abstract this paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations an algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed several illustrative examples with linear and nonlinear fractional differential equations are given to demonstrate the effectiveness of the method

maximum principle and numerical method for the multi term time space riesz caputo fractional differential equations the maximum principle for the space and time space fractional partial differential equations is still an open problem in this paper we consider a multi term time space riesz caputo fractional differential equations over an open bounded domain a maximum principle for the equation is proved the uniqueness and continuous dependence of the solution are derived using a fractional predictor corrector method combining the l1 and l2 discrete schemes we present a numerical method for the specified equation two examples are given to illustrate the obtained results

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

global stability for reaction diffusion equations in biology the aim of this work is to study the global stability for some diffusion equations in biology by constructing lyapunov functionals these lyapunov functionals are obtained from those for ordinary differential equations several examples from virology and epidemiology are given to illustrate our method

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

hiv dynamics modeling data analysis and optimal treatment protocols we present an overview of some concepts and methodologies we believe useful in modeling hiv pathogenesis after a brief discussion of motivation for and previous efforts in the development of mathematical models for progression of hiv infection and treatment we discuss mathematical and statistical ideas relevant to structured treatment interruptions sti among these are model development and validation procedures including parameter estimation data reduction and representation and optimal control relative to sti results from initial attempts in each of these areas by an interdisciplinary team of applied mathematicians statisticians and clinicians are presented

asymptotic properties of fractional delay differential equations abstract in this paper we study the asymptotic properties of d dimensional linear fractional differential equations with time delay we present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse laplace transform method and prove polynomial decay of stable solutions two examples illustrate the obtained analytical results

solving a multi order fractional differential equation using adomian decomposition abstract an algorithm has been developed to convert the multi order fractional differential equation d y t f t y t d 1 y t d n y t y k 0 c k k 0 m where m d denotes caputo fractional derivative of order into a system of fractional differential equations further adomian decomposition method has been employed to solve the system of fractional differential equations some illustrative examples are presented

the synchronization of fractional order rossler hyperchaotic systems abstract the synchronization of fractional order hyperchaotic systems is studied using the rossler system as an example based on the laplace transformation theory sufficient conditions for global synchronization of the systems are given analytically also the variational iteration method is implemented to give the approximate solution for the fractional order error system of the two identical hyperchaotic systems which is in good agreement with the approximate solution using the classical laplace transformation method numerical methods and simulations on the master slave systems are presented to verify the results obtained

the geometry on smooth toroidal compactifications of siegel varieties we study smooth toroidal compactifications of siegel varieties thoroughly from the view points of hodge theory and kahler einstein metric we observe that any cusp of a siegel space can be identified as a set of certain weight one polarized mixed hodge structures we then study the in finity boundary divisors of toroidal compactifications and obtain a global volume form formula of an arbitrary smooth siegel variety ag g 1 with a smooth toroidal compactification ag such that d ag ag is normal crossing we use this volume form formula to show that the unique group invariant kahler einstein metric on ag endows some restraint combinatorial conditions for all smooth toroidal compactifications of ag again using the volume form formula we study the asymptotic behavior of logarithmical canonical line bundle on any smooth toroidal compactification of ag carefully and we obtain that the logarithmical canonical bundle degenerate sharply even though it is big and numerically effective

global stability and periodic solution of the viral dynamics it is well known that the mathematical models provide very important information for the research of human immunodeficiency virus type 1 and hepatitis c virus hcv however the infection rate of almost all mathematical models is linear the linearity shows the simple interaction between the t cells and the viral particles in this paper we consider the classical mathematical model with saturation response of the infection rate by stability analysis we obtain sufficient conditions on the parameters for the global stability of the infected steady state and the infection free steady state we also obtain the conditions for the existence of an orbitally asymptotically stable periodic solution numerical simulations are presented to illustrate the results

stability analysis of a virus dynamics model with general incidence rate and two delays the aim of this work is to study the dynamical behavior of a virus dynamics model with general incidence rate and two delays the first delay represents the time from the virus entry to the production of new viruses and the second delay corresponds to the time necessary for a newly produced virus to become infectious lyapunov functionals are constructed and lasalle invariance principle for delay differential equations is used to establish the global asymptotic stability of the disease free and the chronic infection equilibria the results obtained show that the global dynamics are completely determined by the value of a certain threshold parameter called the basic reproduction number r 0 and under some assumptions on the general incidence function our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection these results can be applied to a variety of possible incidence functions that could be used in virus dynamics model as well as epidemic models

spline collocation for nonlinear fractional boundary value problems abstract we consider a class of boundary value problems for nonlinear fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem some regularity properties of the exact solution are derived based on these properties and spline collocation techniques the numerical solution of boundary value problems by suitable non polynomial approximations is discussed optimal global convergence estimates are derived and a superconvergence result for a special choice of grid and collocation parameters is given theoretical results are tested by two numerical examples

spline collocation methods for linear multi term fractional differential equations abstract some regularity properties of the solution of linear multi term fractional differential equations are derived based on these properties the numerical solution of such equations by piecewise polynomial collocation methods is discussed the results obtained in this paper extend the results of pedas and tamme 2011 15 where we have assumed that in the fractional differential equation the order of the highest derivative of the unknown function is an integer in the present paper we study the attainable order of convergence of spline collocation methods for solving general linear fractional differential equations using caputo form of the fractional derivatives and show how the convergence rate depends on the choice of the grid and collocation points theoretical results are verified by some numerical examples

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

numerical analysis for distributed order differential equations in this paper we present and analyse a numerical method for the solution of a distributed order differential equation of the general form 0 ma r d ru t dr f t where m is a positive real number and where the derivative d r is taken to be a fractional derivative of caputo type of order r we give a convergence theory for our method and conclude with some numerical examples

remarks on fractional derivatives in this paper we further discuss the properties of three kinds of fractional derivatives the grunwald letnikov derivative the riemann liouville derivative and the caputo derivative especially we compare the riemann liouville derivative with the caputo derivative and sequential property of the caputo derivative is also derived which is helpful in translating the higher fractional order differential systems into lower ones besides we also compare the riemann liouville derivative and the caputo derivative with the classical derivative

distributed order equations as boundary value problems there has been a recent increase in interest in the use of distributed order differential equations particularly in the case where the derivatives are given in the caputo sense to model various phenomena recent papers have provided insights into the numerical approximation of the solution and some results on existence and uniqueness have been proved in each case the representation of the solution depends among other parameters on caputo type initial conditions in this paper we discuss the existence and uniqueness of solutions and we propose a numerical method for their approximation in the case where the initial conditions are not known and instead some caputo type conditions are given away from the origin

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

systems based decomposition schemes for the approximate solution of multi term fractional differential equations we give a comparison of the efficiency of three alternative decomposition schemes for the approximate solution of multi term fractional differential equations using the caputo form of the fractional derivative the schemes we compare are based on conversion of the original problem into a system of equations we review alternative approaches and consider how the most appropriate numerical scheme may be chosen to solve a particular equation

fractional integral inequalities and applications fractional integral inequality results when 0 q 1 are developed when the nonlinear term is increasing in u and satisfies a one sided lipschitz condition using the integral inequality result and the computation of the solution of the linear fractional equation of variable coefficients gronwall inequality results are established this yields the results of q 1 as a special case as an application of this the uniqueness and continuous dependence of the solution on the initial parameters of the nonlinear fractional differential equations are established

on shifted jacobi spectral approximations for solving fractional differential equations in this paper a new formula of caputo fractional order derivatives of shifted jacobi polynomials of any degree in terms of shifted jacobi polynomials themselves is proved we discuss a direct solution technique for linear multi order fractional differential equations fdes subject to nonhomogeneous initial conditions using a shifted jacobi tau approximation a quadrature shifted jacobi tau q sjt approximation is introduced for the solution of linear multi order fdes with variable coefficients we also propose a shifted jacobi collocation technique for solving nonlinear multi order fractional initial value problems the advantages of using the proposed techniques are discussed and we compare them with other existing methods we investigate some illustrative examples of fdes including linear and nonlinear terms we demonstrate the high accuracy and the efficiency of the proposed techniques

global properties of delayed hiv dynamics models with differential drug efficacy in cocirculating target cells the objective of this work is to investigate the qualitative behaviors of three hiv dynamical models with two types of cocirculating target cells and intracellular discrete time delay in the two types of target cells the drug efficacy is assumed to be different the models take into account both short lived infected cells and long lived chronically infected cells the incidence rate of infection is given by bilinear and saturation functional response in the first and second model respectively while it is given by a general function in the third model lyapunov functionals are constructed and lasalle s invariance principle is applied to prove the global asymptotic stability of all equilibria of the models in the first two models we have derived the basic reproduction number r0 it has been proven that the disease free equilibrium is globally asymptotically stable gas when r0 1 and the endemic equilibrium is gas when r0 1 for the third model we have derived r0 and established a set of sufficient conditions for global stability of both disease free and endemic equilibria of the model at the end theoretical results have been checked by numerical simulations