solving boundary value problems for delay differential equations by a fixed point method a general linear boundary value problem for a nonlinear system of delay differential equations dde in short is reduced to a fixed point problem v a v with a properly chosen generally nonlinear operator a the unknown fixed point v is approximated by piecewise linear function v h defined by its values v i v h t i at grid points t i i 0 1 n where n is a given positive integer and h max 1 i n t i t i 1 under suitable assumptions the existence of a fixed point of a is equivalent to existence of so called e h approximate fixed points of v h a v h which can be found by minimization of l 2 n norm of residuum v h a v h with respect to the variables v i these e h approximate fixed points are used for obtaining approximate solutions of the original boundary value problem for a system of dde numerical experiments with the boundary value problem for a system of delay differential equations of population dynamics as well as with two periodic boundary value problems one for the periodic distributed delay lotka volterra competition system and the second one modeling two coupled identical neurons with time delayed connections show an efficiency of this kind of approach

optimal estimation from limited noisy data abstract we discuss the following problem let a b f j dy f j e j 1 j n g3 j e p c jp given f j e j 1 j n estimate f x the idea is as follows the estimate is sought in the form a b f j dy f j j 1 j n j p c jp given f j j 1 j n estimate f x the idea is as follows the estimate is sought in the form f n x j 1 n j x f j j the problem is to find j x such that 1 e j 0 then tf n f 0 n 2 if g3 j e p c jp then d f n f 2 where 0 is a certain number we find j 1 j n optimal in a certain sense namely if e j 0 then j are found from the requirements a j 1 n j x a b j dy 1 a b l 1 n j x j y 2 x y 2 dy min note that a j 1 n j x j y n x y in an optimal way if e j 0 then j are found from a and c 2 if this problem is soluble and if not one increases 2 so that this problem becomes soluble

spline collocation method for integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions to initial or boundary value problems of fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms

inverse problem of non homogeneous residual stress identification in thin plates abstract vibration of solid bodies with residual stresses has been attracting attention of researchers from different countries for a long time problems of residual stress analysis have its applications in fields of building mechanical engineering aircraft construction biomechanics manufacturing of composite and functionally gradient materials the most common model of residual stresses or prestresses is the homogeneous prestress state model however in fact the prestress state is often non homogeneous under natural conditions one of the most powerful nondestructive methods of reconstruction of non homogeneous prestress state is the acoustical method in the present paper the direct problem formulations for 3d bodies and thin plates with non homogeneous prestress fields are described a formulation of the inverse problem on a reconstruction of non homogeneous prestress state is given on the basis of acoustical method the problem is reduced to the iterative process at each step of the latter the direct problem and the integral fredholm equation of the first kind are solved two ways of obtaining operator equations of the inverse problem are presented for two oscillation regimes in plane and out of plane plate vibration modes numerical results of solving the inverse problem on a reconstruction of the uniaxial prestress function in case of out of plane vibration for a thin rectangular plate are presented features and characteristics of the solutions obtained are revealed the most auspicious conditions for a better quality of the identification procedure are pointed out

stability of piecewise polynomial collocation for computing periodic solutions of delay differential equations we prove numerical stability of a class of piecewise polynomial collocation methods on nonuniform meshes for computing asymptotically stable and unstable periodic solutions of the linear delay differential equation dot y t a t y t b t y t tau f t by a periodic boundary value approach this equation arises e g in the study of the numerical stability of collocation methods for computing periodic solutions of nonlinear delay equations we obtain convergence results for the standard collocation algorithm and for two variants in particular estimates of the difference between the collocation solution and the true solution are derived for the standard collocation scheme the convergence results are unconditional that is they do not require mesh ratio restrictions numerical results that support the theoretical findings are also given

inversion of limited angle tomographic data abstract a new method is given for analytical inversion of the radon transform given limited angle data numerical aspects of this inversion problem are discussed

multidimensional inverse problems and completeness of the products of solutions to pde a method is given for proving uniqueness theorems for some inverse problems the method is based on a result on completeness of the products of solutions to pde as an example the following uniqueness theorems are proved 1 the scattering amplitude a k known for all s2 and a fixed k 0 determines the compactly supported q x l2 d uniquely 2 the surface data u x y k known for all x y p x x3 0 and a fixed k 0 determine the compactly supported x l2 d d r 3 x x3 0 is arbitrarily small determine aj x j 1 2 uniquely here 2u k2u k2a1x a2 x u x y in r3 a1 l2 d a2 h2 d the same conclusion holds if the surface data are known at two distinct frequencies 4 the surface data u x y k known for all x y p and all k 0 determine v x and h k uniquely here 2 k2 k2 x u x y h k x l2 d h k is fourier transform of a wavelet of compact support 5 the conductivity x w2 2 d x c 0 is uniquely determined by the measurements of u and un on d here n is the outward normal to d d r3 is a bounded domain with a smooth boundary d x u 0 in d 6 necessary and sufficient conditions are given for a function a k s2 k 0 is fixed to be the scattering amplitude corresponding to a local potential from a certain class

discrete galerkin method for fredholm integro differential equations with weakly singular kernels approximations to a solution and its derivatives of a boundary value problem of an nth order linear fredholm integro differential equation with weakly singular or other nonsmooth kernels are determined these approximations are piecewise polynomial functions on special graded grids for their finding a discrete galerkin method and an integral equation reformulation of the boundary value problem are used optimal global convergence estimates are derived and an improvement of the convergence rate of the method for a special choice of parameters is obtained to illustrate the theoretical results a collection of numerical results of a test problem is presented

periodic solutions for delay differential equations model of plankton allelopathy abstract in this paper we propose a modified delay differential equation model of the growth of two species of plankton having competitive and allelopathic effects on each other by using the continuation theorem of coincidence degree theory a set of easily verifiable sufficient conditions are obtained for the existence of positive periodic solutions for this model

numerical solution of some inverse scattering problems of geophysics some results are reported of the numerical solution of the inverse problem which consists of finding an inhomogeneity v x in the 3d region r 3 x x3 0 under the surface of the earth from the knowledge of the acoustic field u x y k at low frequencies k for all positions of the source y and the receiver x on the surface p x x3 0 of the earth the inhomogeneity v x in the refraction coefficient is related to the data of u x y k on p by an integral equation of the first kind this inverse problem is extremely ill conditioned the numerical computations in this paper show how to successfully obtain v x which was a priori assumed to be a known constant in an a priori unknown region b and zero outside b an ad hoc b searching procedure together with a regularization method have been used to solve the resulting ill condioned linear system for the discretized problem a subsequent direct comparison finds b more accurately this type of computation can be used to detect piecewise constant inhomogeneity in a medium for example to detect a hole or a crack in a rock or in a construction element

periodic solution for a two species nonautonomous competition lotka volterra patch system with time delay abstract by using the continuation theorem of coincidence degree theory the existence of a positive periodic solution for a two species nonautonomous competition lotka volterra patch system with time delay x 1 t x 1 t r 1 t a 1 t x 1 t b 1 t y t d 1 t x 2 t x 1 t x 2 t x 2 t r 2 t a 2 t x 2 t d 2 t x 1 t x 2 t y t y t r 3 t a 3 t x 1 t b 3 t y t t 0 k s y t s ds is established where ri t ai t i 1 2 3 di t i 1 2 bi t i 1 3 and t are all positive periodic continuous functions with period w 0 is a nonnegative constant and k s is a continuous nonnegative function on 0

a new stable inverse method for identification of the elastic constants of a three dimensional generally anisotropic solid abstract this article presents a new approach for inverse identification of all elastic constants of a 3d generally anisotropic solid with arbitrary geometry via measured strain data to eradicate the nonlinear inequality constraints posed on the elastic constants the problem is first transformed to an unconstrained one by the cholesky factorization theorem the cost function is defined by the tikhonov regularization method and the inverse problem is solved using the damped gauss newton technique where a meshless method is employed for the direct and sensitivity analyses to demonstrate the effectiveness of the proposed approach several examples are presented in the end where all experimental data are numerically simulated analyses of these examples show that all twenty one elastic constants of an example material can be correctly identified even when measurement errors are relatively large and initial guesses are far from exact values

a smoothed inverse eigenstrain method for reconstruction of the regularized residual fields abstract a smoothed inverse eigenstrain method is developed for reconstruction of residual field from limited strain measurements a framework for appropriate choice of shape functions based on the prior knowledge of expected residual distribution is presented which results in stabilized numerical behavior the analytical method is successfully applied to three case studies where residual stresses are introduced by inelastic beam bending laser forming and shot peening the well rehearsed advantage of the proposed eigenstrain based formulation is that it not only minimizes the deviation of measurements from its approximations but also will result in an inverse solution satisfying a full range of continuum mechanics requirements the smoothed inverse eigenstrain approach allows suppressing fluctuations that are contrary to the physics of the problem furthermore a comprehensive discussion is performed on regularity of the asymptotic solution in the tikhonov scheme and the regularization parameter is then exactly determined utilizing morozov discrepancy principle gradient iterative regularization method is also examined and shown to have an excellent convergence to the tikhonov morozov regularization results

maxentropic reconstruction of fourier and laplace transforms in this note the first of a series we present a maxentropic procedure to recover a function from the knowledge of a finite number of values of a linear integral transform and apply it to the case of the fourier or the laplace transforms also we consider the case in which the values of the transform are known to be polluted by noise

piecewise polynomial collocation for fredholm integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of initial or boundary value problems of linear fredholm integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of convergence of numerical solutions for all values of the nonuniformity parameter of the underlying grid

the collocation method in the numerical solution of boundary value problems for neutral functional differential equations part ii differential equations with deviating arguments we consider the numerical solution of boundary value problems for neutral differential equations with deviating arguments by the collocation method which includes the finite element method and the spectral element method by using the results of part i siam j numer anal 53 2015 pp 2771 2793 of this paper we obtain concrete convergence theorems for this particular type of functional differential equations

the iterated galerkin method for linear integro differential equations abstract the iterated galerkin method of i h sloan which yields superconvergence for the approximate solution of integral equations is generalized to get similar results for the larger class of integro differential equations two approaches are investigated to obtain superconvergence in different norms moreover some additional aspects are discussed

on a boundary value problem for delay differential equations of population dynamics and chebyshev approximation in this paper a boundary value problem for delay differential equations of population dynamics is considered we obtain approximate solutions by using chebyshev polynomial series and newton raphson s procedure and give the error estimation the method of the error estimation has been obtained in an existence theorem proved by a part of the authors we carry out some numerical experiments by a computer language matlab

numerical analysis of singularly perturbed delay differential equations with layer behavior in this paper we present a numerical method to solve boundary value problems for singularly perturbed differential difference equations with negative shift in recent papers the term negative shift has been used for delay siam j appl math 54 1994 249 siam j appl math 42 1982 502 siam j appl math 45 1985 687 siam j appl math 45 1985 708 siam j appl math 54 1994 273 similar bvps are associated with expected first exit time problem of the membrane potential in models for neuron and in variational problem in control theory the stability and convergence analysis of the method is discussed also the effect of small shift on the boundary layer solution in both the cases i e boundary layer on the left side as well as the right side is discussed by considering numerical experiments several numerical examples are solved using the presented method compared the computed result with exact solution and plotted the graphs of the solution of the problems

numerical recovery of the layered medium from the surface data abstract in this paper an inverse problem is formulated and its numerical solution is given the problem is of interest for example in geophysics numerical results show that the algorithm is efficient for noisy data practical recommendations are given the problem consists of finding n z the refraction coefficient of the medium given the surface data u x k x p x x3 0 and all k 0 k0 where k0 0 is an arbitrary small wave number the governing equation is 2 k2n z u x x x1 x2 z analytical solution to this inverse problem is given a numerical implementation is suggested and the results of the numerical experiments are reported

finite element boundary value integration of wheeler feynman electrodynamics the electromagnetic two body problem is solved as a boundary value problem associated to an action functional we show that the functional is frechet differentiable and that its conditions for criticality are the mixed type neutral differential delay equations with state dependent delay of wheeler feynman electrodynamics we construct a finite element method that finds c 1 smooth solutions when suitable past and future positions of the particles are given as boundary data the numerical trajectories satisfy a variational problem defined in a finite dimensional hermite functional space of c 1 piecewise polynomials the numerical variational problem is solved using a combination of newton s method intercalated with boundary adjustments to ensure that the velocity of the solution is continuous with the boundary data we recover the known circular orbits and compute several other novel trajectories of the wheeler feynman electrodynamics we also discuss the local convexity of the functional close to the new found trajectories and the possibility of solutions with less regularity

a new approach to numerical solution of fixed point problems and its application to delay differential equations in this paper we consider a certain approximation of fixed points of a continuous operator a mapping the metric space into itself by means of finite dimensional e h fixed points of a these finite dimensional functions are obtained from functions defined on discrete space grid points related to a parameter h 0 by applying suitably chosen extension operators p h a theorem specifying necessary and sufficient conditions for existence of fixed points of a in terms of e h fixed points of a is given a corollary which follows the theorem yields an approximate method for a fixed point problem and determines conditions for its convergence an example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples

basis functions for residual stresses abstract we consider arbitrary preexisting residual stress states in arbitrarily shaped unloaded bodies these stresses must be self equilibrating and traction free common treatments of the topic tend to focus on either the mechanical origins of the stress or methods of stress measurement at certain locations here we take the stress field as given and consider the problem of approximating any such stress field in a given body as a linear combination of predetermined fields which can serve as a basis we consider planar stress states in detail and introduce an extremization problem that leads to a linear eigenvalue problem eigenfunctions of that problem form an orthonormal basis for all possible residual stress states of sufficient smoothness in numerical examples convergence of the approximating stress fields is demonstrated in the l2 norm for continuous stress fields as well as for a stress field with a simple discontinuity finally we outline the extension of our theory to three dimensional bodies and states of stress our approach can be used to describe arbitrary preexisting residual stress states in arbitrarily shaped bodies using basis functions that are determined by the body geometry alone

on an inverse problem for inhomogeneous thermoelastic rod abstract in recent years different fields of engineering have been increasingly incorporating functionally graded materials with variable physical properties that significantly improve a quality of elements of designs the efficiency of practical application of thermoelastic inhomogeneous materials depends on knowledge of exact laws of heterogeneity and to define them it is necessary to solve coefficient inverse problems of thermoelasticity in the present research a scheme of solving the inverse problem for an inhomogeneous thermoelastic rod is presented two statements of the inverse problem are considered in the laplace transform space and in the actual space the direct problem solving is reduced to a system of the fredholm integral equations of the 2nd kind in the laplace transform space and an inversion of the solutions obtained on the basis of the theory of residues the inverse problem solving is reduced to an iterative procedure at its each step it is necessary to solve the fredholm integral equation of the 1st kind to solve it the tikhonov method is used specific examples of a reconstruction of variable characteristics required are given

monotone methods for periodic solutions of second order scalar functional differential equations an iterative method is presented which starting from a lower or from an upper periodic solution provides a monotone sequence converging to a periodic solution of 1 with some restrictions on the growth off the method extends to functional differential equations of type 1 two numerical examples with an a posteriori error analysis are given

numerical bifurcation analysis of delay differential equations numerical methods for the bifurcation analysis of delay differential equations ddes have only recently received much attention partially because the theory of ddes smoothness boundedness stability of solutions is more complicated and less established than the corresponding theory of ordinary differential equations as a consequence no established software packages exist at present for the bifurcation analysis of ddes we outline existing numerical methods for the computation and stability analysis of steady state solutions and periodic solutions of systems of ddes with several constant delays

the numerical solution of forward backward differential equations decomposition and related issues this paper focuses on the decomposition by numerical methods of solutions to mixed type functional differential equations mfdes into sums of forward solutions and backward solutions we consider equations of the form x t ax t bx t 1 cx t 1 and develop a numerical approach using a central difference approximation which leads to the desired decomposition and propagation of the solution we include illustrative examples to demonstrate the success of our method along with an indication of its current limitations

interpolation correction for collocation solutions of fredholm integro differential equations in this paper we discuss the collocation method for a large class of fredholm linear integro differential equations it will be shown that when a certain higher order interpolation operation is added to the collocation solution of this equation the new approximations will under suitable assumptions admit a multiterm error expansion in even powers of the step size h based on this expansion ideal multilevel correction results of this collocation solution are obtained

configurational balance laws for incompatibility in stress space conservation laws have been recently obtained by requiring that a positive definite functional of the stress gradient the eulerlagrange equations of which are the beltramimichell compatibility con

on uniqueness of an inverse problem for a 1 d wave equation from transmission data this paper concerns the uniqueness of a coefficient inverse problem for a one dimensional wave equation we allow the unknown coefficient under consideration to be discontinuous this type of inverse problem arises in seismology corresponding to the case in which the medium to be detected has discontinuous interfaces by virtue of spectra theory of the sturm liouville problem and asymptotics of the green function we prove that the discontinuous coefficient can be uniquely recovered from the source function f in l 1 0 infty and the response function g furthermore lemma 6 in this paper corrects some mistakes in t kimura and t suzuki siam j appl math 53 1993 pp 1747 1761 about the asymptotics of the green function this correction means that some related parts in the aforementioned article also need to be rectified correspondingly

mixed type functional differential equations a numerical approach the equations considered in this paper are mixed type functional equations sometimes known as forward backward differential equations that take the form u t au t bu t 1 cu t 1 t i r we consider basic existence and uniqueness properties when i t 1 t 2 and we seek solutions u c t 1 1 t 2 1 that satisfy u t w 1 t t t 1 1 t 1 u t w 2 t t t 2 t 2 1 for prescribed functions w 1 w 2 absolutely continuous respectively on t 1 1 t 1 t 2 t 2 1 with arbitrary boundary conditions specified in this way the problem turns out to be ill posed and so existence and uniqueness questions have an important role to play in developing numerical schemes we consider with t 1 t 2 n numerical approximations of a solution when it exists the numerical methods that we consider are linear q methods and we investigate computationally their effectiveness through some illustrative examples

dynamic behaviors of a delay differential equation model of plankton allelopathy in this paper we consider a modified delay differential equation model of the growth of n species of plankton having competitive and allelopathic effects on each other we first obtain the sufficient conditions which guarantee the permanence of the system as a corollary for periodic case we obtain a set of delay dependent condition which ensures the existence of at least one positive periodic solution of the system after that by means of a suitable lyapunov functional sufficient conditions are derived for the global attractivity of the system for the two dimensional case under some suitable assumptions we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation examples show the feasibility of the main results

the collocation method in the numerical solution of boundary value problems for neutral functional differential equations part i convergence results we consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method the collocation method can be applied in two versions the finite element method and the spectral element method we give convergence results for the collocation method deduced by the convergence theory developed in s maset numer math 2015 pp 1 31 for a general discretization of an abstract reformulation of the problems such convergence results are then applied in part ii s maset siam j numer anal 53 2015 pp 2794 2821 of this paper to boundary values problems for a particular type of neutral functional differential equations namely differential equations with deviating arguments

periodic solutions for delay lotka volterra competition systems abstract by using the continuation theorem of coincidence degree theory sufficient and realistic conditions are obtained for the existence of positive periodic solutions for the periodic distributed delay lotka volterra competition system du i t dt u i t r i t a ii t u i t j 1 j i n a ij t 0 t ij k ij s u j t s ds i 1 2 n and the periodic state dependent delay lotka volterra competition system du i t dt u i t r i t a ii t u i t j 1 j i n a ij t u j t j t u 1 t u n t i 1 2 n where ri aii 0 aij 0 j i i j 1 2 n are continuous periodic functions tij 0 j i i j 1 2 n kij c tij 0 0 0 tijkij s ds 1 j i i j 1 2 n i c rn 1 r and i i 1 2 n are periodic with respect to their first arguments respectively

reconstruction of inhomogeneous properties of orthotropic viscoelastic layer abstract the consecutive scheme of reconstructing the functions characterizing the instantaneous and long termed modules for inhomogeneous orthotropic viscoelastic layer whose properties continuously vary through a thickness is proposed the identification problem is solved on the basis of the additional information on the integral characteristics of displacement fields measured on top border of a layer iterative process of reconstruction of six unknown functions is formulated to reconstruct the rest six functions systems of fredholm s integral equations of the first kind with smooth kernels are received computational experiment on a reconstruction of various laws of inhomogeneity is conducted a comparative analysis of the results obtained is carried out

an abstract framework in the numerical solution of boundary value problems for neutral functional differential equations we consider the numerical solution of boundary value problems for general neutral functional differential equations the problems are restated in an abstract form and then a general discretization of the abstract form is introduced and a convergence analysis of this discretization is developed

collocation methods for the computation of periodic solutions of delay differential equations in this paper we investigate collocation methods for the computation of periodic solutions of autonomous delay differential equations ddes periodic solutions are found by solving a periodic two point boundary value problem which is an infinite dimensional problem for ddes in contrast to the case of ordinary differential equations we investigate three collocation methods based on piecewise polynomials we discuss computational issues and show numerical orders of convergence using an extensive number of tests we compare our numerical results with known theoretical convergence results for initial value problems for ddes in particular we show how superconvergence at the mesh points can be lost or recovered depending on the dde model under consideration and on the choice of collocation discretization we end with a brief discussion of adaptive mesh selection

conservation laws of linear elasticity in stress formulations in this paper we present new conservation laws of linear elasticity which have been discovered these newly discovered conservation laws are expressed solely in terms of the cauchy stress tensor and they are genuine non trivial conservation laws that are intrinsically different from the displacement conservation laws previously known they represent the variational symmetry conditions of combined beltrami michell compatibility equations and the equilibrium equations to derive these conservation laws noether s theorem is extended to partial differential equations of a tensorial field with general boundary conditions by applying the tensorial version of noether s theorem to pobedrja s stress formulation of threedimensional elasticity a class of new conservation laws in terms of stresses has been obtained