two dimensional shifted legendre polynomials operational matrix method for solving the two dimensional integral equations of fractional order abstract this work approximates the unknown functions based on the two dimensional shifted legendre polynomials operational matrix method 2d slpom for the numerical solution of two dimensional fractional integral equations the present method reduces these equations to a system of algebraic equations and then this system will be solved numerically by newton s method moreover an estimation of the error bound for this algorithm will be shown by preparing some theorems some examples are presented to demonstrate the validity and applicability of the proposed method with respect to the two dimensional block pulse functions method 2d bpfs and two dimensional bernstein polynomials operational matrix method 2d bpom

superresolution reconstruction through object modeling and parameter estimation a method based on object modeling and parameter estimation is proposed to achieve superresolution reconstruction an efficient method for solving for the model parameters is given that uses linear prediction theory and linear least squares fitting reconstruction results from simulated and real magnetic resonance data are also presented to demonstrate its capability for gibbs ringing reduction and resolution enhancement

barycentric interpolation collocation methods for solving linear and nonlinear high dimensional fredholm integral equations abstract in this article two barycentric interpolation collocation methods are proposed for solving linear and nonlinear high dimensional f r e d h o l m integral equations of the second kind the approaches respectively utilize the modified weighted l a g r a n g e functions and the novel rational functions as the interpolation basis functions they are effective schemes for evaluating the multidimensional undetermined function through the numerical strategies and some composite quadrature formulas the linear and nonlinear f r e d h o l m integral equations are transformed into the corresponding linear and nonlinear algebraic equations further we prove that the discrete collocation methods are equivalent to the n y s t r o m quadrature methods then the convergence analysis is established by the collectively compact theory moreover the error estimation of the approximate solution and the exact solution are also provided numerical examples are presented to illustrate the capability and efficiency of the techniques by compared with the classic l a g r a n g e interpolation collocation method and other methodologies

a new method based on haar wavelet for the numerical solution of two dimensional nonlinear integral equations a new numerical method based on haar wavelet is proposed for two dimensional nonlinear fredholm volterra and volterra fredholm integral equations of first and second kind the proposed method is an extension of the haar wavelet method aziz and siraj ul islam 2013 siraj ul islam et al 2013 and siraj ul islam et al 2014 from one dimensional nonlinear integral equations fredholm and volterra to two dimensional nonlinear integral equations fredholm volterra and volterra fredholm the main characteristic of the method is that unlike several other methods it does not involve numerical integration which results in an improved accuracy of the method in order to show the effectiveness of the method it is applied to several benchmark problems the numerical results are compared with other methods existing in the recent literature

mri artifact cancellation due to rigid motion in the imaging plane a post processing technique has been developed to suppress the magnetic resonance imaging mri artifact arising from object planar rigid motion in two dimensional fourier transform 2 dft mri rotational and translational motions of the target during magnetic resonance magnetic resonance mr scan respectively impose nonuniform sampling and a phase error an the collected mri signal the artifact correction method introduced considers the following three conditions 1 for planar rigid motion with known parameters a reconstruction algorithm based on bilinear interpolation and the super position method is employed to remove the mri artifact 2 for planar rigid motion with known rotation angle and unknown translational motion including an unknown rotation center first a super position bilinear interpolation algorithm is used to eliminate artifact due to rotation about the center of the imaging plane following which a phase correction algorithm is applied to reduce the remaining phase error of the mri signal and 3 to estimate unknown parameters of a rigid motion a minimum energy method is proposed which utilizes the fact that planar rigid motion increases the measured energy of an ideal mr image outside the boundary of the imaging object by using this property all unknown parameters of a typical rigid motion are accurately estimated in the presence of noise to confirm the feasibility of employing the proposed method in a clinical setting the technique was used to reduce unknown rigid motion artifact arising from the head movements of two volunteers

on solving systems of multi pantograph equations via spectral tau method abstract the current manuscript focuses on solving systems of multi pantograph equations the spectral tau method is applied for solving systems of multi pantograph equations with shifted jacobi polynomials as basis functions the convergence analysis of the proposed technique is also investigated we introduced the numerical solutions of some test problems and compared the obtained numerical solutions of such problems with those given using different numerical methods

second kind integral equations for the classical potential theory on open surfaces ii a second kind integral equation formulation is presented for the dirichlet problem for the laplace equation in two dimensions with the boundary conditions specified on a collection of open curves the performance of the obtained apparatus is illustrated with several numerical examples the formulation is a simplification of the equation previously constructed by the authors

numerical solution of integro differential equations of high order by wavelet basis its algorithm and convergence analysis abstract this paper presents for the first time numerical solutions for this particular type of integro differential equations according to equations which will be introduced suitable wavelet galerkin method is provided using wavelet basis in the space c r l 2 r 0 that c r is the holder space of exponent this approach has two advantages first the wavelets basis are arbitrary it means that any differentiable wavelets basis can be used second the desired orders for this equation are the reasons for involving a wide variety of these types of equations the algorithm and convergence analysis of this scheme are described numerical examples plots and tablets of errors confirm the applicability and the validity of the proposed method

numerical solution of three dimensional volterra fredholm integral equations of the first and second kinds based on bernstein s approximation abstract a new and efficient method is presented for solving three dimensional volterra fredholm integral equations of the second kind 3d vfiek2 first kind 3d vfiek1 and even singular type of these equations here we discuss three variable bernstein polynomials and their properties this method has several advantages in reducing computational burden with good degree of accuracy furthermore we obtain an error bound for this method finally this method is applied to five examples to illustrate the accuracy and implementation of the method and this method is compared to already present methods numerical results show that the new method provides more efficient results in comparison with other methods

about a numerical method of successive interpolations for functional hammerstein integral equations a new and robust numerical method for hammerstein functional integral equations is proposed the convergence and the numerical stability of the method are mathematically proved and tested on some examples

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

projection and iterated projection methods for nonliear integral equations consider the nonlinear operator equation x mathcal k x with mathcal k a completely continuous mapping of a domain in the banach space mathcal x into mathcal x and let x denote an isolated fixed point of mathcal k let mathcal k n n geqq 1 denote a sequence of finite dimensional approximating subspaces and let p n be a projection of mathcal x onto mathcal x n the projection method for solving x mathcal k x given by x n p n mathcal k x n and the iterated projection solution is defined as tilde x n mathcal k x n we analyze the convergence of x n and tilde x n to x giving a general analysis that includes both the galerkin and collocation methods a more detailed analysis is then given for a large class of urysohn integral operators in one variable showing the superconvergence of tilde x n to x

restarted adomian method for system of nonlinear volterra integral equations in this paper we generalize restarted adomian method as a new method based on standard adomian method for solving system of nonlinear volterra integral equations

shells structure in uncorrelated scale free networks abstract we study structure of shells in uncorrelated scale free networks using probabilistic arguments we obtain explicit expression for the distance distribution i e average number of nodes at the l th shell for different ranges of the degree exponent to overcome the analytical difficulties when 2 3 we show that the heterogeneous network can be approximated by a disassortative ordered network and the average degree of neighbors of a node must depends on shells we also deduce the mean distance between nodes l as the distance at which distance distribution is maximum taking number of nodes large we retrieve the known scaling forms for the different ranges of mainly the small world and the ultra small world behaviors very good accordance with simulations is also found the expressions of l involve all the network s parameters and can be used as good approximations of mean distance in real world problems

solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method in this paper we will give some results for developing the two dimensional differential transform tddt for double integrals then the tddt method will be developed for solving a class of two dimensional linear and nonlinear volterra integral equations we also give some examples to demonstrate the accuracy of the method

hybrid function method and convergence analysis for two dimensional nonlinear integral equations in the current paper an efficient numerical method based on two dimensional hybrid of block pulse functions and legendre polynomials is developed to approximate the solutions of two dimensional nonlinear fredholm volterra and volterrafredholm integral equations of the second kind the main idea of the presented method is based upon some of the important benefits of the hybrid functions such as high accuracy wide applicability and adjustability of the orders of block pulse functions and legendre polynomials to achieve highly accurate numerical solutions by using the numerical integration and collocation method two dimensional nonlinear integral equations are reduced to a system of nonlinear algebraic equations the focus of this paper is to obtain an error estimate and to show the convergence analysis for the numerical approach under the l2 norm numerical results are presented and compared with the results from other existing methods to illustrate the efficiency and accuracy of the proposed method

extrapolation of nystrom solution for two dimensional nonlinear fredholm integral equations abstract in this paper we analyze the existence of asymptotic error expansion of the nystrom solution for two dimensional nonlinear fredholm integral equations of the second kind we show that the nystrom solution admits an error expansion in powers of the step size h and the step size k for a special choice of the numerical quadrature the leading terms in the error expansion for the nystrom solution contain only even powers of h and k beginning with terms h2p and k2q these expansions are useful for the application of richardson extrapolation and for obtaining sharper error bounds numerical examples show that how richardson extrapolation gives a remarkable increase of precision in addition to faster convergence

a computational method for solving a class of two dimensional volterra integral equations the use of the geometric series theorem together with schauder bases in certain banach spaces allows us to design a numerical algorithm in order to solve an important type of two dimensional linear integral equations a study of the convergence and error is presented here all the calculations can be easily implemented and the efficiency of this method will be shown with numerical results

rationalized haar functions method for solving fredholm and volterra integral equations this paper presents a computational technique for fredholm integral equation of the second kind and volterra integral equation of the second kind the method is based upon haar functions approximation properties of rationalized haar functions are first presented the operational matrix of integration together with product operational matrix and newton cotes nodes are utilized to reduce the computation of integral equations into some algebraic equations the method is computationally attractive and applications are demonstrated through illustrative examples

a spectral tau algorithm based on jacobi operational matrix for numerical solution of time fractional diffusion wave equations in this paper an efficient and accurate spectral numerical method is presented for solving second fourth order fractional diffusion wave equations and fractional wave equations with damping the proposed method is based on jacobi tau spectral procedure together with the jacobi operational matrix for fractional integrals described in the riemann liouville sense the main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations in the unknown expansion coefficients of the sought for spectral approximations the validity and effectiveness of the method are demonstrated by solving five numerical examples numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier an efficient numerical scheme for time fractional diffusion wave equations is proposed a time space jacobi tau approximation is developed for such equations several tau spectral methods can be achieved as special cases the validity and applicability of the proposed method are demonstrated accurate numerical results are obtained by selecting limited collocation nodes

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

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

a numerical scheme for a class of nonlinear fredholm integral equations of the second kind in this paper an iterative approach for obtaining approximate solutions for a class of nonlinear fredholm integral equations of the second kind is proposed the approach contains two steps at the first one we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation solution of the discretized form converges to the exact solution of the problem following that in the next step solution of the discretized form is approximated by an iterative approach we finally on some examples show the efficiency of the proposed approach

statistical analysis of articulation points in configuration model networks an articulation point ap in a network is a node whose deletion would split the network component on which it resides into two or more components aps are vulnerable spots that play an important role in network collapse processes which may result from node failures attacks or epidemics therefore the abundance and properties of aps affect the resilience of the network to these collapse scenarios we present analytical results for the statistical properties of aps in configuration model networks in order to quantify their abundance we calculate the probability p i in rm ap that a random node i in a configuration model network with p k k is an ap we also obtain the conditional probability p i in rm ap k that a random node of degree k is an ap and find that high degree nodes are more likely to be aps than low degree nodes using bayes theorem we obtain the conditional degree distribution p k k rm ap over the set of aps and compare it to p k k we propose a new centrality measure based on aps each node can be characterized by its articulation rank r which is the number of components that would be added to the network upon deletion of that node for nodes which are not aps the articulation rank is r 0 while for aps r ge 1 we obtain a closed form expression for the distribution of articulation ranks p r r configuration model networks often exhibit a coexistence between a giant component and finite components to examine the distinct properties of aps on the giant and on the finite components we calculate the probabilities presented above separately for the giant and the finite components we apply these results to ensembles of configuration model networks with a poisson exponential and power law degree distributions the implications of these results are discussed in the context of common attack scenarios and network dismantling processes

an efficient algorithm for mr image reconstruction without low spatial frequencies it is demonstrated that if the image to be reconstructed is known to have some zero valued pixels the dynamic ranges can be better used by disregarding the largest signals and using signal restoration methods low frequency and high frequency signals are related using the knowledge that some pixels are zero by a set of linear equations in which the number of equations is equal to the number of zero pixels and the number of unknowns is equal to the number of low frequency signal samples rejected an improved fourier transform ft magnetic resonance mr imaging method based on a least square error lse technique and an efficient algorithm for signal restoration when the low frequency components are discarded are presented in this method the regions of support in both the image domain and the frequency domain can have arbitrary shapes and all zero pixels in the image domain can be taken into account the algorithm has been tested on simulated and experimental data with acceptable 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

study on existence of solutions for some nonlinear functional integral equations abstract using the technique of measures of noncompactness in banach algebra we employ abstract fixed point theorems such as darbo s theorem to study the existence solution in banach algebra c 0 a for some functional integral equations which include many key integral and functional equations that arise in nonlinear analysis

adaptive numerical approach based upon chebyshev operational vector for nonlinear volterra integral equations and its convergence analysis abstract in the current paper we present a direct numerical scheme to approximate a second kind nonlinear volterra integral equations nvies the scheme is based upon shifted chebyshev polynomials and its operational matrices which eventually leads to the sparsity of the coefficients matrix of obtained system the main idea of the proposed approach is based on a useful property of chebyshev polynomials that yields to construct a new operational vector this vector eliminates any requirement of using projection methods and also enhances the accuracy vs other methods applied projection methods the constructive technique and the convergence analysis of this approach under the l w 2 norm are also described numerical experiments and comparisons confirm the applicability and the validity of the presented scheme

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

extrapolation method of iterated collocation solution for two dimensional nonlinear volterra integral equations in this paper we study the numerical solution of two dimensional nonlinear volterra integral equation by collocation and iterated collocation method asymptotic error expansion of iterated collocation solution is obtained using richardson s extrapolation we show that the accuracy of numerical solution in the iterated collocation method can be improved greatly some numerical examples are given to illustrate this theory

differential transform method for the system of two dimensional nonlinear volterra integro differential equations in this paper we develop the differential transform method dtm for solving a class of the system of two dimensional linear and nonlinear volterra integro differential equations of the second kind to this end we give some preliminary results of the differential transform and describe the method of this paper we also give some examples to demonstrate the accuracy of the presented method

numerical solution of a class of mixed two dimensional nonlinear volterra fredholm integral equations using multiquadric radial basis functions an efficient numerical method is proposed for solving the mixed nonlinear volterra fredholm two dimensional integral equations ies using two dimensional radial basis functions rbfs this method is based on interpolation by radial basis functions including multiquadrics mqs using legendre gauss lobatto nodes and weights also a theorem is proved for convergence analysis some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method

comments and corrections on the use of polar sampling theorems in ct in two recent papers an exact polar interpolation formula was used to reconstruct computer tomography ct imagery by a procedure known as direct fourier transform inversion the resulting imagery compared favorably to ct imagery reconstructed by filtered convolution back projection strictly speaking however the interpolation formula was inappropriately used since it is valid only for an odd number of azimuthal samples while in ct one uses an even number of samples when the number of samples n is large say 200 as in ct the error is not noticeable and the appropriateness of the formula has no practical significance however when n is small a large error can result we derive in this paper exact azimuthal interpolation formulas for n even and arbitrary n

numerical solutions of nonlinear two dimensional partial volterra integro differential equations by haar wavelet a non linear kernel comprising a function of partial derivatives of arbitrary order is approximated by theorem 2 in ref 12 after substituting the kernel approximation in the original equation a nonlinear system is obtained using the haar wavelet solving the nonlinear system the nonlinear two dimensional integro differential volterra equation with partial derivatives is converted to a simple equation containing partial derivatives solving this simple equation we can approximate the solution of the nonlinear two dimensional integro differential volterra equation

revealing the microstructure of the giant component in random graph ensembles the microstructure of the giant component of the erd s renyi network and other configuration model networks is analyzed using generating function methods while configuration model networks are uncorrelated the giant component exhibits a degree distribution which is different from the overall degree distribution of the network and includes degree degree correlations of all orders we present exact analytical results for the degree distributions as well as higher order degree degree correlations on the giant components of configuration model networks we show that the degree degree correlations are essential for the integrity of the giant component in the sense that the degree distribution alone cannot guarantee that it will consist of a single connected component to demonstrate the importance and broad applicability of these results we apply them to the study of the distribution of shortest path lengths on the giant component percolation on the giant component and spectra of sparse matrices defined on the giant component we show that by using the degree distribution on the giant component one obtains high quality results for these properties which can be further improved by taking the degree degree correlations into account this suggests that many existing methods currently used for the analysis of the whole network can be adapted in a straightforward fashion to yield results conditioned on the giant component

numerical approaches for systems of volterra fredholm integral equations in this work we introduce two numerical methods for solving systems of volterra fredholm integral equations in the nonlinear case we suggest a fixed point method where the iterations are perturbed in a suitable way according to a schauder basis in the banach space of continuous functions c a b 2 in the linear case we propose a collocation method based on a particular class of approximating functions in both methods convergence analysis and or low computational cost are analysed taking into account the properties of the basis under consideration numerical results confirm the theoretical study

numerical solution of some classes of integral equations using bernstein polynomials this paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using bernstein polynomials as basis the integral equations considered are fredholm integral equations of second kind a simple hypersingular integral equation and a hypersingular integral equation of second kind the method is explained with illustrative examples also the convergence of the method is established rigorously for each class of integral equations considered here

tomographic image reconstruction from incomplete view data by convex projections and direct fourier inversion we consider the problem of reconstructing cat imagery by the direct fourier method dfm when not all view data are available to restore the missing information we use the method of projections onto convex sets pocs pocs is a recursive image restoration technique that finds a solution consistent with the measured data and a priori known constraints in both the space and fourier domain because dfm reconstruction is a frequency domain technique it is ideally matched to pocs restoration when for one reason or another we are forced to generate an image from a less than complete set of view data we design and apply an algorithm prdf which interpolates extrapolates the missing fourier domain information by pocs and reconstructs an image by dfm a simulated human thorax cross section is restored and reconstructed the restorations using pocs are compared with the gerchberg papoulis extrapolation method and shown to be superior applications of prdf to other types of medical imaging modalities are discussed

on the construction of recurrence relations for the expansion and connection coefficients in series of jacobi polynomials formulae expressing explicitly the jacobi coefficients of a general order derivative integral of an infinitely differentiable function in terms of its original expansion coefficients and formulae for the derivatives integrals of jacobi polynomials in terms of jacobi polynomials themselves are stated a formula for the jacobi coefficients of the moments of one single jacobi polynomial of certain degree is proved another formula for the jacobi coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given a simple approach in order to construct and solve recursively for the connection coefficients between jacobi jacobi polynomials is described explicit formulae for these coefficients between ultraspherical and jacobi polynomials are deduced of which the chebyshev polynomials of the first and second kinds and legendre polynomials are important special cases two analytical formulae for the connection coefficients between laguerre jacobi and hermite jacobi are developed

numerical solution for three dimensional nonlinear mixed volterra fredholm integral equations via three dimensional block pulse functions we are concerned here with a three dimensional nonlinear mixed volterra fredholm integral equations of the second kind which include many key integral that appear in the theory of nonlinear parabolic boundary value problems the existence of a unique solution will be proved a new numerical method for solving these type of equations will be presented the method is based upon three dimensional block pulse functions approximation in addition convergence analysis of the method is discussed illustrative examples are included to demonstrate the validity and applicability of the technique

applying the modified block pulse functions to solve the three dimensional volterra fredholm integral equations the main aim of this work is to give further studies for the multi dimensional integral equations in this work we solve special types of the three dimensional volterra fredholm linear integral equations of the second kind via the modified block pulse functions some theorems are included to show convergence and advantage of the method we solve some examples to investigate the applicability and simplicity of the method the numerical results confirm that the method is efficient and simple

an iterative method for the extrapolation of band limited functions abstract we derive an iterative algorithm to extrapolate band limited functions using the operators of fourier transform and projection on convex subsets the algorithm allows the use of a priori knowledge in the extrapolation procedure

a numerical technique for solving a class of fractional variational problems this paper presents a numerical method for solving a class of fractional variational problems fvps with multiple dependent variables multi order fractional derivatives and a group of boundary conditions the fractional derivative in the problem is in the caputo sense in the presented method the given optimization problem reduces to a system of algebraic equations using polynomial basis functions an approximate solution for the fvp is achieved by solving the system the choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed we extensively discuss the convergence of the method and finally present illustrative examples to demonstrate validity and applicability of the new technique

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

a new approach for numerical solution of two dimensional nonlinear fredholm integral equations in the most general kind of kernel based on bernstein polynomials and its convergence analysis abstract regarding the efficient previous method for approximation of one dimensional functions integrals through bernstein polynomials in amirfakhrian 2011 this paper presents a development of this method for approximation of two dimensional functions integrals for the first time then by combining this approximation with bernstein collocation method for numerical solution of two dimensional nonlinear fredholm integral equations the kernels double integrals of integral equations will be approximated combination of two dimensional functions numerical integration method with numerical solution of integral equations method in both methods bernstein polynomials were used will result in increase of convergence speed and accuracy of the method the convergence analysis of the method is completely presented finally numerical examples are presented to illustrate the efficiency and superiority of our method in comparing it with other methods

extrapolation method for solving two dimensional volterral integral equations of the second kind abstract in this paper we propose a numerical quadrature method for solving two dimensional linear and nonlinear volterra integral equations firstly we generalize one dimensional quadrature formula to two dimensional case and its corresponding error asymptotic expansion based on the quadrature formula and the error expansion we next construct an iterative scheme and extrapolation algorithm the numerical solution of any point can be calculated by iterative scheme and the error accuracy and convergence order of the numerical solution are further improved by extrapolation algorithm using the extrapolation algorithm we can improve the convergence order from o h 0 2 to o h 0 3 or even o h 0 4 since the numerical solution of each point is obtained by assignment operation and iteration the computational complexity can be greatly reduced finally four numerical examples are given to illustrate the effectiveness of the method

richardson extrapolation of iterated discrete galerkin solution for two dimensional fredholm integral equations in this paper we study the numerical solution of two dimensional fredholm integral equation by discrete galerkin and iterated discrete galerkin method we are able to derive an asymptotic error expansion of the iterated discrete galerkin solution this expansion covers arbitrarily high powers of the discretization parameters if the solution of the integral equation is smooth the expansion gives rise to richardson type extrapolation schemes which rapidly improve the original rate of the convergence numerical experiments confirm our theoretical results

bessel polynomial solutions of high order linear volterra integro differential equations abstract in this study a practical matrix method which is based on collocation points is presented to find approximate solutions of high order linear volterra integro differential equations vides under the mixed conditions in terms of bessel polynomials numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results the results show the efficiency and accuracy of the present work all of the numerical computations have been performed on the computer using a program written in matlab v7 6 0 r2008a

comments on solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method tari et al a tari m y rahimi s shahmorad f talati solving a class of two dimensional linear and nonlinear volterra integral equations by the differential transform method j comput appl math 228 2009 70 76 presented some fundamental properties of tdtm for the kernel functions in two dimensional volterra integral equations here we suggest simple proofs of those fundamental properties by using the basic properties of tdtm furthermore we present some fundamental properties of tdtm for the kernel functions of a quotient type in two dimensional volterra integral equations numerical illustrations are demonstrated to show the effectiveness of the tdtm for solving two dimensional volterra integral equations

existence of solution for some nonlinear two dimensional volterra integral equations via measures of noncompactness in this paper we analyze the existence of solution for two dimensional nonlinear volterra integral equations 2dvie by using the techniques of measures of noncompactness and petryshyn fixed point theorem which contains as particular cases a lot of integral and functional integral equations that arise in nonlinear analysis also some illustrative examples are given

an integral equation describing an asexual population in a changing environment in this paper we establish the existence of a travelling wave solution to an intrinsically nonlinear differential integral equation arising from the mathematical modelling of an asexual reproduction process the techniques used are fixed point theorems and asymptotic analysis the proofs although abstract give a hint on how to obtain the solutions numerically

solving system of volterra fredholm integral equations with bernstein polynomials and hybrid bernstein block pulse functions this work approximates the unknown functions based on the bernstein polynomials and hybrid bernstein block pulse functions in conjunction with the collocation method for the numerical solution of system of fredholm volterra integral equations in both methods the system of integral equations is approximated by the gauss quadrature formula with respect to the legendre weight function the proposed methods reduce the system of integral equations to a system of algebraic equations that can be easily solved by any usual numerical methods moreover the convergence analysis of these algorithms will be shown by preparing some theorems numerical experiments are presented to show the superiority and efficiency of proposed methods in comparison with some other well known methods

on a method for solving a two dimensional nonlinear integral equation of the second kind in this article the existence of at least one solution of a nonlinear integral equation of the second kind is proved the degenerate method is used to obtain a nonlinear algebraic system where the existence of at least one solution of this system is discussed finally computational results with error estimates are obtained using maple software

a sixth order compact finite difference method for non classical vibration analysis of nanobeams including surface stress effects a non classical model for the free vibrations of nanobeams accounting for surface stress effects is developed in this study based on gurtin murdoch elasticity theory the influence of surface stress is incorporated into the euler bernoulli beam theory a compact finite difference method cfdm of sixth order is employed for discretizing the non classical governing differential equation to obtain the natural frequencies of nanobeams subject to different boundary conditions to check the validity of the present numerical solution based on an exact solution an explicit formula for the fundamental frequency of simply supported nanobeams is obtained good agreement between the results of exact and numerical solutions is achieved confirming the validity and accuracy of the present numerical scheme the comparison between the results generated by cfdm with those obtained by the conventional finite difference method fdm further reveals the advantages of the compact method over its classical counterpart the influences of beam thickness surface density surface residual stress surface elastic constants and boundary conditions on the natural frequencies of nanobeams are also investigated it is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness

an optimal technique for constraint based image restoration and reconstruction a new technique for finding an optimal feasible solution to the general image reconstruction and restoration problem is described this method allows the use of prior knowledge of the properties of both the solution and any noise present on the data the problem is formulated as the optimization of a cost function over the intersection of a number of convex constraint sets each set being defined as containing those solutions consistent with a particular constraint a duality theorem is then applied to yield a dual problem in which the unknown image is replaced by a model defined in terms of a finite dimensional parameter vector and the kernels of the integral equations relating the data and solution the dual problem may then be solved for the model parameters using a gradient descent algorithm this method serves as an alternative to the primal constrained optimization and projection onto convex sets pocs algorithms problems in which this new approach is appropriate are discussed an example is given for image reconstruction from noisy projection data applying the dual method results in a fast nonlinear algorithm simulation results demonstrate the superiority of the optimal feasible solution over one obtained using a suboptimal approach

distribution of shortest path lengths in subcritical erdős renyi networks networks that are fragmented into small disconnected components are prevalent in a large variety of systems these include the secure communication networks of commercial enterprises government agencies and illicit organizations as well as networks that suffered multiple failures attacks or epidemics the structural and statistical properties of such networks resemble those of subcritical random networks which consist of finite components whose sizes are nonextensive surprisingly such networks do not exhibit the small world property that is typical in supercritical random networks where the mean distance between pairs of nodes scales logarithmically with the network size unlike supercritical networks whose structure has been studied extensively subcritical networks have attracted relatively little attention a special feature of these networks is that the statistical and geometric properties vary between different components and depend on their sizes and topologies the overall statistics of the network can be obtained by a summation over all the components with suitable weights we use a topological expansion to perform a systematic analysis of the degree distribution and the distribution of shortest path lengths dspl on components of given sizes and topologies in subcritical erd ifmmode mbox h o else h o fi s r enyi er networks from this expansion we obtain an exact analytical expression for the dspl of the entire subcritical network in the asymptotic limit the dspl which accounts for all the pairs of nodes that reside on the same finite component fc is found to follow a geometric distribution of the form p mathrm fc l ensuremath ell ll ensuremath infty 1 ensuremath c c ensuremath ell ensuremath 1 where cl1 is the mean degree using computer simulations we calculate the dspl in subcritical er networks of increasing sizes and confirm the convergence to this asymptotic result we also obtain exact asymptotic results for the mean distance ensuremath langle l ensuremath rangle mathrm fc and for the standard deviation of the dspl ensuremath sigma l mathrm fc and show that the simulation results converge to these asymptotic results using the duality relations between subcritical and supercritical er networks we obtain the dspl on the nongiant components of er networks above the percolation transition

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

low cost numerical solution for three dimensional linear and nonlinear integral equations via three dimensional jacobi polynomials in recent years numerical methods have been introduced to solve two dimensional volterra and fredholm integral equations in this study a numerical scheme is constructed to solve classes of linear and nonlinear three dimensional integral equations volterra fredholm and mixed volterrafredholm this operational approach is proposed to easily and directly solve these equations at low computational costs the scheme is based on the jacobi polynomials on the interval 0 1 where three variable jacobi polynomials are introduced and their operational matrices of integration and product are derived compared to other existing methods for multidimensional problems the jacobi operational method eliminates the time consuming computations and solely employs the one dimensional operational matrix to construct corresponding multidimensional operational matrices the absolute error of the proposed method is almost constant on the studied interval even at higher dimensions confirming the stability of the proposed operational jacobi method required theorems on the convergence of the method are proved in jacobi weighted sobolev space it is established that the error function vanishes as n increases the method is evaluated using several illustrative examples which indicate the proposed method with lesser computational size compared to the blockpulse functions differential transform and degenerate kernel methods

a generalization of discrete gronwall inequality and its application to weakly singular volterra integral equation of the second kind this paper presents a new discrete gronwall inequality using the inequality we prove convergence and error estimate of the numerical solutions of the second weakly singular volterra integral equation where discrete equation is derived by novot s quadrature formula

a new algorithm for integral of trigonometric functions with mechanization in this paper by using the theories of mathematical analysis the authors systematically studied the problem of integrating of trigonometric functions utilizing omnipotence expressions tan x2 t a new integral algorithm for the integral of trigonometric functions was established and a new maple program trigint for the calculation mechanizing for the integral of trigonometric functions was established trigint could evaluate not only the integral of trigonometric functions utilizing omnipotence expresssions correctly but also all the other kinds of functions the results indicated that the algorithm established in this paper had advantages of simple idea excellent property for operation and powerful competence this would be useful for some calculating problems of engineering mathematics

an improved algorithm for rotational motion artifact suppression in mri an improved algorithm for planar rotational motion artifact suppression in standard two dimensional fourier transform magnetic resonance images is presented it is shown that interpolation of acquired view data on the uncorrupted k space create data overlap and void regions the authors present a method of managing overlap data regions using weighted averaging of redundant data the weights are assigned according to a priority ranking based on the minimum distance between the data set and the k space grid points an iterative estimation technique for filling the data void regions using projections onto convex sets pocs is also described the method has been successfully tested using computer simulations

numerical solution of a class of two dimensional nonlinear volterra integral equations using legendre polynomials in this paper a method for finding an approximate solution of a class of two dimensional nonlinear volterra integral equations is discussed the properties of two dimensional shifted legendre functions are presented the operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations some results concerning the error analysis are obtained we also consider the application of the method to the solution of certain partial differential equations illustrative examples are included to demonstrate the validity and applicability of the technique

a tau approach for solution of the space fractional diffusion equation fractional differentials provide more accurate models of systems under consideration in this paper approximation techniques based on the shifted legendre tau idea are presented to solve a class of initial boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain the fractional derivatives are described in the caputo sense the technique is derived by expanding the required approximate solution as the elements of shifted legendre polynomials using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations from the computational point of view the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use

an efficient algorithm for solving volterra integro differential equations based on alpert s multi wavelets galerkin method abstract this paper is concerned with the application of the multi wavelets galerkin method to linear and nonlinear second order volterra integro differential equations using the operational matrices of integration and wavelet transform matrix we reduce the problem to a set of algebraic equations for linear type we use thresholding to reduce the computational effort we demonstrate that our solutions may be efficiently represented in a multi wavelets basis due to flexible vanishing moments property of this type of multi wavelets the l 2 convergence of the scheme for the proposed equation has been investigated a series of numerical tests is provided to demonstrate the validity and applicability of the technique

new approach to solve two dimensional fredholm integral equations abstract in this paper we present an efficient iterative method based on quadrature formula to solve two dimensional nonlinear fredholm integral equations we investigate the convergence analysis of the proposed method the error estimation is given in terms of uniform and partial modulus of continuity in addition we show the numerical stability analysis of the method with respect to the choice of the first iteration finally some numerical experiments confirm the theoretical results and illustrate the accuracy of the method

a meshfree method for solving multidimensional linear fredholm integral equations on the hypercube domains the main purpose of this article is to describe a numerical scheme for solving multidimensional linear fredholm integral equations of the second kind on the hypercube domains the method is based on interpolation by radial basis functions rbfs to approximate the solution of the linear fredholm integral equations error analysis is presented for this method finally several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method

reconstruction from limited single particle diffraction data via simultaneous determination of state orientation intensity and phase free electron lasers now have the ability to collect x ray diffraction patterns from individual molecules however each sample is delivered at unknown orientation and may be in one of several conformational states each with a different molecular structure hit rates are often low typically around 0 1 limiting the number of useful images that can be collected determining accurate structural information requires classifying and orienting each image accurately assembling them into a 3d diffraction intensity function and determining missing phase information additionally single particles typically scatter very few photons leading to high image noise levels we develop a multitiered iterative phasing algorithm to reconstruct structural information from single particle diffraction data by simultaneously determining the states orientations intensities phases and underlying structure in a single iterative procedure we leverage real space constraints on the structure to help guide optimization and reconstruct underlying structure from very few images with excellent global convergence properties we show that this approach can determine structural resolution beyond what is suggested by standard shannon sampling arguments for ideal images and is also robust to noise

image restoration by the method of convex projections part 2 applications and numerical results the image restoration theory discussed in a previous paper by youla and webb 1 is applied to a simulated image and the results compared with the well known method known as the gerchberg papoulis algorithm the results show that the method of image restoration by projection onto convex sets by providing a convenient technique for utilizing a priori information performs significantly better than the gerchberg papoulis method

two dimensional triangular functions and their applications to nonlinear 2d volterra fredholm integral equations two dimensional orthogonal triangular functions 2d tfs are presented as a new set of basis functions for expanding 2d functions their properties are determined and an operational matrix for integration obtained furthermore 2d tfs are used to approximate solutions of nonlinear two dimensional integral equations by a direct method since this approach does not need integration all calculations can be easily implemented and several advantages in reducing computational burdens arise finally the efficiency of this method will be shown by comparison with some numerical results

multi dimensional spectral tau methods for distributed order fractional diffusion equations abstract the distributed order fractional diffusion equation is a generalization of the standard fractional diffusion equation that can model processes lacking power law scaling over the whole time domain an important application of distributed order diffusions is to model ultraslow diffusion where a plume of particles spreads at a logarithmic rate to broaden the range of applicability of distributed order fractional diffusion models efficient numerical methods are needed to solve the model equation in this work we develop spectral tau schemes to discretize the fractional diffusion equation with distributed order fractional derivative in time and dirichlet boundary conditions the model solution is expanded in multi dimensions in terms of legendre polynomials and the discrete equations are obtained with the tau method numerical examples are provided to highlight the convergence rate and the flexibility of this approach the proposed spectral tau methods yield an exponential rate of convergence when the solution is smooth our results confirm that nonlocal numerical methods are best suited to discretize distributed order fractional differential equations as they naturally take the global behavior of the solution into account

a new algorithm for the solution of nonlinear two dimensional volterra integro differential equations of high order abstract in this manuscript a new numerical approach is developed for approximating the solution of the one dimensional and two dimensional volterra integro differential equations 2d vides of high order new operational matrices of integration based on shifted jacobi polynomials are presented and used in combination with the collocation method to convert the 2d vide into a system of algebraic equations in addition the convergence analysis of the suggested approach is investigated the efficiency of the algorithm is demonstrated by means of several examples and the results are compared with those given using other numerical schemes

using operational matrix of two dimensional bernstein polynomials for solving two dimensional integral equations of fractional order in this paper we first preset two dimensional bernstein polynomials operational matrix 2dbpom for two dimensional fractional integral equations 2dfie then we introduce a computational method based on 2dbpom to reduce the 2dfie to those of solving a system of algebraic equations also we give the error analysis finally this method is applied for solving three examples to illustrate the applicability and accuracy of presented method

numerical solutions of the integral equations homotopy perturbation method and adomian s decomposition method in this paper a homotopy perturbation method is proposed to solve non singular integral equations comparisons are made between adomian s decomposition method and the proposed method it is shown adomian s decomposition method is a homotopy only finally by using homotopy perturbation method a new iterative scheme like adomian s decomposition method is proposed for solving the non singular integral equations of the first kind the results reveal that the proposed method is very effective and simple

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

distance distribution in configuration model networks we present analytical results for the distribution of shortest path lengths between random pairs of nodes in configuration model networks the results which are based on recursion equations are shown to be in good agreement with numerical simulations for networks with degenerate binomial and power law degree distributions the mean mode and variance of the distribution of shortest path lengths are also evaluated these results provide expressions for central measures and dispersion measures of the distribution of shortest path lengths in terms of moments of the degree distribution illuminating the connection between the two distributions

numerical solution of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces based on block pulse functions in this paper the numerical technique based on block pulse functions bpfs has been developed to approximate the solutions of nonlinear volterrafredholmhammerstein integral equations in two dimensional spaces these functions are orthogonal and have compact support on 0 1 the proposed method reduces the integral equations to a system of nonlinear algebraic equations that can be easily solved by any numerical method also the convergence of the proposed approach is discussed furthermore in order to show the accuracy and reliability of the above mentioned algorithm the new approach is verified through some numerical examples a numerical technique based on block pulse functions is proposed two dimensional nonlinear volterrafredholmhammerstein integral equations are firstly numerically solved the proposed method is to transform the volterrafredholm integral equations into a system of algebraic equations the main characteristic behind the algorithm is operator theory