a monolithic geometric multigrid solver for fluid structure interactions in ale formulation summary we present a monolithic geometric multigrid solver for fluid structure interaction problems in arbitrary lagrangian eulerian coordinates the coupled dynamics of an incompressible fluid with nonlinear hyperelastic solids gives rise to very large and ill conditioned systems of algebraic equations direct solvers usually are out of question because of memory limitations and standard coupled iterative solvers are seriously affected by the bad condition number of the system matrices the use of partitioned preconditioners in krylov subspace iterations is an option but the convergence will be limited by the outer partitioning our proposed solver is based on a newton linearization of the fully monolithic system of equations discretized by a galerkin finite element method approximation of the linearized systems is based on a monolithic generalized minimal residual method iteration preconditioned by a geometric multigrid solver the special character of fluid structure interactions is accounted for by a partitioned scheme within the multigrid smoother only here fluid and solid field are segregated as dirichlet neumann coupling we demonstrate the efficiency of the multigrid iteration by analyzing 2d and 3d benchmark problems while 2d problems are well manageable with available direct solvers challenging 3d problems highly benefit from the resulting multigrid solver copyright 2015 john wiley sons ltd

construction and calculation of reproducing kernel determined by various linear differential operators we presented a method to construct and calculate the reproducing kernel for the linear differential operator with constant coefficients and a single latent root further we gave the formula for calculation additionally by studying the recurrence relation of the reproducing kernel with arithmetic latent roots we found that the reproducing kernel with a multi knots interpolation constraint can be concisely represented by one with an initial value constraint

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a stabilized finite element method for generalized stationary incompressible flows abstract in this paper we describe a finite element formulation for the numerical solution of the stationary incompressible navier stokes equations including coriolis forces and the permeability of the medium the stabilized method is based on the algebraic version of the sub grid scale approach we first describe this technique for general systems of convection diffusion reaction equations and then we apply it to the linearized flow equations the important point is the design of the matrix of stabilization parameters that the method has this design is based on the identification of the stability problems of the galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix this together with the convergence analysis of the linearized problem leads to a simple expression for the stabilization parameters in the general situation considered in the paper the numerical analysis of the linearized problem also shows that the method has optimal convergence properties

new method based on the hpm and rkhsm for solving forced duffing equations with integral boundary conditions this paper investigates the forced duffing equation with integral boundary conditions its approximate solution is developed by combining the homotopy perturbation method hpm and the reproducing kernel hilbert space method rkhsm hpm is based on the use of the traditional perturbation method and the homotopy technique the hpm can reduce nonlinear problems to some linear problems and generate a rapid convergent series solution in most cases rkhsm is also an analytical technique which can solve powerfully linear boundary value problems therefore the forced duffing equation with integral boundary conditions can be solved using advantages of these two methods two numerical examples are presented to illustrate the strength of the method

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

a new reproducing kernel method for linear nonlocal boundary value problems in the previous works the authors developed the reproducing kernel method rkm for nonlocal boundary value problems a key of the method is the construction of the reproducing kernel rk satisfying the homogenous boundary conditions bcs of the considered problems however it is very difficult to obtain the rk of a reproducing kernel space satisfying nonlocal bcs or nonlinear bcs even if the rk is found its representation is also very complicated compared with the rk without any bcs in this paper we will present a new rkm for linear nonlocal boundary value problems the method can avoid reducing the inhomogeneous bcs to homogeneous bcs and constructing rk satisfying homogeneous nonlocal linear bcs numerical examples are provided to show the effectiveness of the new method

a novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method abstract in this paper a novel method is presented for solving a class of singularly perturbed boundary value problems firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation then the reproducing kernel method is employed to solve the boundary value new problem numerical results show that the present method can provide very accurate analytical approximate solutions

a novel method for nonlinear two point boundary value problems combination of adm and rkm abstract in this paper a novel method is proposed for solving nonlinear two point boundary value problems bvps this method is based on a combination of the adomian decomposition method adm and the reproducing kernel method rkm a major advantage of this method over standard adm is that it can avoid unnecessary computation in determining the unknown parameters the proposed method can be applied to singular and nonsingular bvps numerical results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for solving nonlinear two point boundary value problems

option pricing using a computational method based on reproducing kernel abstract one of the most important subject in financial mathematics is the option pricing the most famous result in this area is black scholes formula for pricing european options this paper is concerned with a method for solving a generalized black scholes equation in a reproducing kernel hilbert space subsequently the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method furthermore the error estimates for obtained approximation in reproducing kernel hilbert space are presented finally a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

comments on a generalization of rosenbrock s theorem by m schlegel in the above paper ibid vol ac 32 p 155 feb 1987 the author presents a generalization of the famous rosenbrock theorem concerning the problem of synthesis of linear state feedback control laws for linear controllable systems the commenters point out that this result was published by them earlier in an alternative form here they give again alternative conditions for solvability of the problem since the original papers published in 1979 and 1985 were in russian in contrast to the conditions given in the above paper their conditions are not redundant and are given in terms of order relation on the lattice of integer partitions

fluid structure interaction by means of variational multiscale reduced order models this is the peer reviewed version of the following article tello a codina r baiges j fluid structure interaction by means of variational multiscale reduced order models int j numer methods eng 2020 121 2601 2625 https doi org 10 1002 nme 6321 which has been published in final form at https onlinelibrary wiley com doi abs 10 1002 nme 6321 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

supg reduced order models for convection dominated convection diffusion reaction equations abstract this paper presents a streamline upwind petrov galerkin supg reduced order model rom based on proper orthogonal decomposition pod this rom is investigated theoretically and numerically for convection dominated convection diffusion reaction problems the supg finite element method was used on realistic meshes for computing the snapshots leading to some noise in the pod data numerical analysis is used to propose the scaling of the stabilization parameter for the supg rom two approaches are used one based on the underlying finite element discretization and the other one based on the pod truncation the resulting supg roms and the standard galerkin rom g rom are studied numerically for many settings the results obtained with the supg roms are more accurate finally one of the choices for the stabilization parameter is recommended

periodic boundary value problems for neutral multi pantograph equations in this paper a new method is proposed for solving nonlinear neutral multi pantograph equations with periodic boundary conditions by combining the advantages of the homotopy perturbed method hpm and the reproducing kernel method rkm the numerical results obtained show that the present method is reliable

reduced order subscales for pod models in this work the reduced order subscales for proper orthogonal decomposition models are presented the basic idea consists in splitting the full order solution into the part which can be captured by the reduced order model and the part which cannot the subscales for which a model is required the proposed model for the subscales is defined as a linear function of the solution of the reduced order model the coefficients of this linear function are obtained by comparing the solution of the full order model with the solution of the reduced order model for the same initial conditions which for convenience are evaluated in the snapshots used to train the original reduced order model the difference between both solutions are the subscales for which a model can be built using a least squares procedure the subscales are then introduced as a correction in the reduced order model resulting in an important improvement in accuracy the enhanced reduced order model is tested in several numerical examples these practical cases show that the use of the subscales leads to more accurate solutions successfully corrects errors introduced by hyper reduction and allows to solve complex flow problems using a reduced number of degrees of freedom

an iterative reproducing kernel method in hilbert space for the multi point boundary value problems abstract in this paper an iterative method is proposed to solve the nonlinear bitsadze samarskii boundary value problems with multi point boundary conditions the algorithm is based on the reproducing kernel hilbert space method we use an iterative scheme to overcome the nonlinearity of the problem the convergence and error estimate of the iterative scheme are established the reproducing kernel hilbert space method is used to generate an approximation of the linearized problem in fact the reproducing kernel hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived in order to show the efficiency and versatility of the proposed method some numerical results are reported the comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method

homotopy perturbation reproducing kernel method for nonlinear systems of second order boundary value problems in this paper based on homotopy perturbation method hpm and reproducing kernel method rkm a new method is presented for solving nonlinear systems of second order boundary value problems bvps hpm is based on the use of traditional perturbation method and homotopy technique the hpm can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases rkm is also an analytical technique which can solve powerfully linear bvps homotopy perturbation reproducing kernel method hp rkm combines advantages of these two methods and therefore can be used to solve efficiently systems of nonlinear bvps three numerical examples are presented to illustrate the strength of the method

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

on existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations the existence and multiplicity of positive solutions are established to periodic boundary value problems for singular nonlinear second order ordinary differential equations the arguments are based only upon the positivity of the green s functions and the krasnosel skii fixed point theorem as an example a periodic boundary value problem is also considered which comes from the theory of nonlinear elasticity

fluid structure interaction with large structural displacements the calculation of viscous flows inside deformable structures faces major difficulties when enforcing the kinematic compatibility at the fluid structure interface and updating the geometry of the domain this paper proposes to overcome these problems by considering the fluid and the structure as a unique continuous medium to be studied in a fixed reference configuration the resulting problem is then split into a fluid and a structural part through an additive decomposition of the space of kinematically admissible test functions this approach treats the structure in a fully lagrangian way and uses an associated arbitrary lagrangian eulerian ale formulation for the fluid such a strategy has three advantages which are detailed in the present paper it can be discretised in time by implicit stable energy conserving time integration schemes and solved by simple iterative uncoupled algorithms the fluid and the structural problems can be approximated by independent finite element spaces the most recent geometrically exact nonlinear shell models can be used for the structures the full strategy has been implemented into an industrial cfd code results obtained in the simulation of industrial hydraulic shock absorbers are presented at the end of the paper

a numerical algorithm for nonlinear multi point boundary value problems in this paper an algorithm is presented for solving second order nonlinear multi point boundary value problems bvps the method is based on an iterative technique and the reproducing kernel method rkm two numerical examples are provided to show the reliability and efficiency of the present method

numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel hilbert space in this paper approximate solutions to a class of fractional differential equations with delay are presented by using a semi analytical approach in hilbert function space further the uniqueness of the solution is proved in the space of real valued continuous functions as well as the existence of the solution is proved in hilbert function space we also prove convergence and perform an analysis error for the proposed approach sophisticated delay differential equations of fractional order are considered as test examples numerical results illustrate the efficiency of the proposed approach computationally

the exact solution of a class of volterra integral equation with weakly singular kernel abstract in this paper the weakly singular volterra integral equations with an infinite set of solutions are investigated among the set of solutions only one particular solution is smooth and all others are singular at the origin the numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation the aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory applying weighted integral we provide a new definition for reproducing kernel space and obtain reproducing kernel function using the good properties of reproducing kernel function the only smooth solution is exactly expressed in the form of series the n term approximate solution is obtained by truncating the series meanwhile we prove that the derivative of approximation converges to the derivative of exact solution uniformly the final numerical examples compared with other methods show that the method is efficient

solving the generalized regularized long wave equation on the basis of a reproducing kernel space on the basis of a reproducing kernel space an iterative algorithm for solving the generalized regularized long wave equation is presented the analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u n is constructed by truncating the series to n terms the convergence of u n to the analytical solution is also proved results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

fast divergence conforming reduced basis methods for steady navier stokes flow abstract reduced basis methods rb methods or rbms form one of the most promising techniques to deliver numerical solutions of parametrized pdes in real time with reasonable accuracy for incompressible flow problems rbms based on lbb stable velocity pressure spaces do not generally inherit the stability of the underlying high fidelity model and instead additional stabilization techniques must be introduced one way of bypassing the loss of lbb stability in the rbm is to inflate the velocity space with supremizer modes this however deteriorates the performance of the rbm in the performance critical online stage as additional dofs must be introduced to retain stability while these dofs do not effectively contribute to accuracy of the rb approximation in this work we consider a velocity only rb approximation exploiting a solenoidal velocity basis the solenoidal reduced basis emerges directly from the high fidelity velocity solutions in the offline stage by means of piola transforms the solenoidality of the velocity space is retained under geometric transformations making the proposed rb method suitable also for the investigation of geometric parameters to ensure exact solenoidality of the high fidelity velocity solutions that constitute the rb we consider approximations based on divergence conforming compatible b splines we show that the velocity only rb method leads to a significant improvement in computational efficiency in the online stage and that the pressure solution can be recovered a posteriori at negligible extra cost we illustrate the solenoidal rb approach by modeling steady two dimensional navier stokes flow around a naca0015 airfoil at various angles of attack

the exact solution for solving a class nonlinear operator equations in the reproducing kernel space in this paper the author will introduce the sobolev space w 1 2 w n firstly and then give the reproducing properties of the space w 1 2 w n as these results of those a class of the nonlinear operator equation n i 1 m i j 1 a i ju f is transformed to the n dimensional linear operator equation au f finally we also give the exact solution of this nonlinear operator equation

solving singular nonlinear second order periodic boundary value problems in the reproducing kernel space abstract in this paper we present a new method for solving a singular nonlinear second order periodic boundary value problem its analytical solution is represented in the form of series in the reproducing kernel space in the mean time the n term approximation u n x to the analytical solution u x is obtained and is proved to converge to the analytical solution some numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

a new method for solving a class of mixed boundary value problems with singular coefficient abstract in 1 2 3 4 5 6 7 it is very difficult to deal with initial boundary value conditions in this paper we give a new method to deal with boundary value conditions the main contribution of this paper is to put mixed boundary value conditions into reproducing kernel hilbert space the numerical examples are studied to demonstrate the accuracy of the present method results obtained by the method indicate the method is simple and effective

fluid structure partitioned procedures based on robin transmission conditions in this article we design new partitioned procedures for fluid structure interaction problems based on robin type transmission conditions the choice of the coefficient in the robin conditions is justified via simplified models the strategy is effective whenever an incompressible fluid interacts with a relatively thin membrane as in hemodynamics applications we analyze theoretically the new iterative procedures on a model problem which represents a simplified blood vessel system in particular the robin neumann scheme exhibits enhanced convergence properties with respect to the existing partitioned procedures the theoretical results are checked using numerical experimentation

application of reproducing kernel method to third order three point boundary value problems abstract this paper investigates the analytical approximate solutions of third order three point boundary value problems using reproducing kernel method the solution obtained by using the method takes the form of a convergent series with easily computable components however the reproducing kernel method can not be used directly to solve third order three point boundary value problems since there is no method of obtaining reproducing kernel satisfying three point boundary conditions this paper presents a method for solving reproducing kernel satisfying three point boundary conditions so that reproducing kernel method can be used to solve third order three point boundary value problems results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three point boundary value problems

optimal control of production rate in a failure prone manufacturing system we address the problem of controlling the production rate of a failure prone manufacturing system so as to minimize the discounted inventory cost where certain cost rates are specified for both positive and negative inventories and there is a constant demand rate for the commodity produced the underlying theoretical problem is the optimal control of a continuous time system with jump markov disturbances with an infinite horizon discounted cost criterion we use two complementary approaches first proceeding informally and using a combination of stochastic coupling linear system arguments stable and unstable eigenspaces renewal theory parametric optimization etc we arrive at a conjecture for the optimal policy then we address the previously ignored mathematical difficulties associated with differential equations with discontinuous right hand sides singularity of the optimal control problem smoothness and validity of the dynamic programming equation etc to give a rigorous proof of optimality of the conjectured policy it is hoped that both approaches will find uses in other such problems also we obtain the complete solution and show that the optimal solution is simply characterized by a certain critical number which we call the optimal inventory level if the current inventory level exceeds the optimal one should not produce at all if less one should produce at the maximum rate while if exactly equal one should produce exactly enough to meet demand we also give a simple explicit formula for the optimal inventory level

existence of two positive solutions of a singular nonlinear periodic boundary value problem abstract we consider the singular periodic boundary value problem for the second order scalar differential equation x f t x x under the assumption that f has the repulsive singularity at x 0 we present sufficient conditions for the existence of at least two different positive solutions of the problem we use the lower and upper solutions method here

production control of a manufacturing system with multiple machine states the production control of a single product manufacturing system with arbitrary number of machine states failure modes is discussed the objective is to find a production policy that would meet the demand for the product with minimum average inventory or backlog cost the optimal production policy has a special structure and is called a hedging point policy if the hedging points are known the optimal production rate is readily specified assuming a set of tentative hedging points the simple structure of the optimal policy is utilized to find the steady state probability distribution of the surplus inventory or backlog once this function is determined the average surplus cost is easily calculated in terms of the values of the hedging points the average cost is then minimized to find the optimum hedging points

artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flows this note investigates the so called arti cial added mass e ect which is responsible for devastating instabilities within sequentially staggered fluid structure interaction fsi simulations where incompressible fluids are considered a discrete representation of the added mass operatorma is given and instability conditions are evaluated for different temporal discretisation schemes it is proven that for every sequentially staggered scheme and given spatial discretisation of a problem a mass ratio between fluid and structural mass density can be found at which the coupled system becomes unstable the analysis is quite general and does not depend upon the particular spatial discretisation schemes used however here special attention is put on stabilised finite elements employed on the fluid partition numerical investigations further highlight the results

a domain decomposition strategy for reduced order models application to the incompressible navier stokes equations abstract in this work a domain decomposition strategy for non linear hyper reduced order models is presented the basic idea consists of restricting the reduced order basis functions to the nodes belonging to each of the subdomains into which the physical domain is partitioned an extension of the proposed domain decomposition strategy to a hybrid full order reduced order model is then described the general domain decomposition approach is particularized for the reduced order finite element approximation of the incompressible navier stokes equations with hyper reduction when solving the reduced incompressible navier stokes equations instabilities in the form of large gradients of the recovered reduced order unknown at the subdomain interfaces may appear which is the motivation for the design of additional stability terms giving rise to penalty matrices numerical examples illustrate the behavior of the proposed method for the simulation of the reduced order systems showing the capability of the approach to adapt to configurations which are not present in the original snapshot set

periodic boundary value problems for nonlinear higher order ordinary differential equations we study a class of boundary value problems for linear and nonlinear ordinary differential equations of higher order we study the solution set and present a maximum principle as a consequence we show the validity of the monotone method and obtain the extremal solutions between the upper and lower solutions

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

element boundary terms in reduced order models for flow problems domain decomposition and adaptive coarse mesh hyper reduction abstract in this paper we present a finite element based reduced order model and in particular we consider two aspects related to the introduction of inter element boundary terms in the formulation the first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non matching meshes and discontinuous physical properties the second is a coarse mesh hyper reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms as the finite element full order model the reduced order model is based on the variational multi scale framework with sub grid scales defined not only in the element interiors but also on the inter element boundaries we present some examples of application using the incompressible navier stokes equations and the boussinesq approximation

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

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic