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The real-world problems that can be resolved by a dynamic mathematical model is an analogous form of a complex integral equation. In this article, we intend to exhibit the solution to a non-homogeneous, nonlinear Volterra integral equation endowed with an arbitrary binary relation utilizing the fixed point solution that is originally proved for a multivalued map and then reduced to a single valued self map. The results proved and the application demonstrated is an added asset to the present literature and can significantly contribute in determining the existence of a solution for a dynamic mathematical model in the form of a Volterra integral equation equipped with an arbitrary binary relation via fixed point findings. Furthermore, the stability of the solution of the Volterra integral equation is verified using the Hyers–Ulam stability concept. [ABSTRACT FROM AUTHOR]

This paper introduces a collocation method for numerically solving Hammerstein generalized integral equations on the whole line. The approach utilizes mapped Gegenbauer functions to transform the Hammerstein integral equation into a system of nonlinear algebraic equations. Additionally, convergence results are obtained using the standard L 2 -norm. Finally, the effectiveness of the proposed method is demonstrated through several numerical experiments, which also serve to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, some new upper and lower bounds on the blow-up time of a certain class of nonlinear Volterra integral equations are presented and compared to previously proposed bounds. In most cases, the new bounds are a significant improvement on previously proposed bounds. We present many new bounds, with varying degrees of accuracy and computational complexity. The derivation of the upper bounds is made possible by a certain covariance inequality from advanced probability theory, which opens the door to many future investigations of blow-up time estimations in many other nonlinear integral equations. [ABSTRACT FROM AUTHOR]

This paper explores a 2D non-linear fractional integral equation of the Riemann–Liouville type. The authors establish the existence of at least one solution for this 2D integral equation in the Banach space of all real continuous functions on [ 0 , c ] × [ 0 , d ] , under some weaker conditions. The main tools used in our considerations are the concept of a measure of noncompactness and Petryshyn's fixed point theorem. Illustrative examples highlight the broad applicability of our findings across a diverse spectrum of integral equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multi-domain hybrid spectral collocation method for the nonlinear Volterra integral equations (VIEs), whose solutions exhibit weak singularities at the endpoint t = 0. In order to efficiently approximate the weakly singular solutions, we divide the interval [ 0 , T ] into N subintervals and use the Gauss points of the generalized log orthogonal functions as the collocation points to approximate the weakly singular integral term in the first subinterval. In the remaining subintervals, we use Legendre and Jacobi Gauss points as the collocation points to approximate the corresponding integral terms. We also provide a rigorous h p -version convergence analysis for the hybrid spectral collocation method under L 2 -norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint. [ABSTRACT FROM AUTHOR]

In this note, we consider a general 2 × 2 system of nonlinear Volterra type integral equations. The modified Newton method (modified NM) is used to reduce the nonlinear problems into 2 × 2 linear system of algebraic integral equations of Volterra type. The latter equation is solved by discretization method. Nystrom method with Gauss–Legendre quadrature is applied for the kernel integrals and Newton forwarded interpolation formula is used for finding values of unknown functions at the selected node points. Existence and uniqueness solution of the problems are proved and accuracy of the quadrature formula together with convergence of the proposed method are obtained. Finally, numerical examples are provided to show the validity and efficiency of the method presented. Numerical results reveal that the proposed methods is efficient and accurate. Comparisons with other methods for the same problem are also presented. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, Galerkin and collocation methods and their iterated versions are discussed to solve the nonlinear Hammerstein type integral equation on the half-line for both convolution and non-convolution kernels using the space of piecewise polynomial subspace. We prove that the approximate and iterated approximate solution in Galerkin and collocation methods converges with the order O (n − r) and O (n − 2 r) respectively in the infinity norm, where n − 1 is the maximum norm of the partition and r denotes the order of the piecewise polynomial employed in the approximation. We improve these results further by considering multi-projection methods. In fact we show that iterated approximate solution in multi-Galerkin, multi-collocation converges with the order O (n − 4 r). Numerical results are presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract In this paper we use spline quasi-interpolating projectors on a bounded interval for the numerical solution of nonlinear integral equations. In particular, we propose a spline quasi-interpolating projection method with high order of convergence and a spline quasi-interpolating collocation method, both in case of smooth kernels and in case of Green's function type ones. We explicitly construct the approximate solutions and we get results related to the convergence orders. Finally, we provide numerical tests, that confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract We use an auxiliary point on the semilocal convergence of Newton's method when the majorant principle of Kantorovich is applied to operators with high order derivatives satisfying a center Lipschitz type condition, so that we extend the classical conditions of these types, that are centered at the starting point of Newton's method, to other points belonging to the domain of definition of the operator involved. This extension provides a modification of the domain of starting points for Newton's method which allows increasing the choice of starting points. We illustrate this study with nonlinear Fredholm integral equations. [ABSTRACT FROM AUTHOR]

This paper presents a method for solving the two-dimensional nonlinear Volterra–Fredholm integral equation. The main idea of the method is to use the Lagrange interpolation function to approximate the unknown solution and the Legendre–Gauss quadrature formula to approximate the integral. The advantage of the method is that it requires relatively few collocation points to obtain a relatively small error and does not require the calculation of integrals. Under certain sufficient conditions, the existence and uniqueness of the original equation are given. In addition, the existence and uniqueness of the solutions of the discrete equations are given using the theory of compact operators. The convergence analysis and error estimates of the method are also derived. Finally, several numerical examples are used to demonstrate its efficiency and accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we apply the multi-Galerkin method with Kumar–Sloan technique to improve the superconvergence rates and find the numerical solution of non-linear Fredholm–Hammerstein integral equations for both smooth as well as the weakly singular kernels. Considering piecewise polynomial as basis function of the approximating subspace, we derive the improved superconvergence rates for multi-Galerkin method based on Kumar–Sloan approximations which are exactly the same superconvergence rates for the approximate solution as in the case of iterated multi-Galerkin method. We have shown improved superconvergence results without the need for iterated multi-Galerkin method. Theoretical results are verified using numerical examples. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper we prove existence of solution for infinite system of nonlinear integral equations in the Banach spaces ℓ p , p > 1 with the help of a technique associated with measure of noncompactness and generalized Meir–Keeler fixed point theorem. We also provide some illustrative examples in support of our existence theorems. Finally, we introduce an iteration algorithm constructed by modified homotopy perturbation method to solve the above problem with high accuracy. [ABSTRACT FROM AUTHOR]

In this paper, the main objective is to describe an approximate scheme for solving nonlinear quadratic integral equations (NQIEs) of fractional order. The method is based on piecewise linear functions that are called hat functions (HFs). By applying the HFs approximation, their properties and operational matrices, nonlinear equation becomes to a nonlinear system of equations which can be solved by using simple arithmetic methods. Also, the error analysis of HFs method and convergence analysis of proposed model are given. At the end, two nonlinear examples are presented to illustrate the validity and applicability of the explained approach. [ABSTRACT FROM AUTHOR]

In the current paper, an equivalence lemma is proposed which converts a second order implicit form of nonlinear fuzzy Volterra integro-differential equation to four different kinds of nonlinear integral equations based on kind of GH-differentiability of the first and second derivatives. Finally, some examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The classical collocation methods based on piecewise polynomials have been studied for delay Volterra integral equations of the second-kind in Brunner (2004). These collocation methods have uniform order m for any choice of the collocation parameters and can achieve local superconvergence in the grid points by choosing the suitable collocation parameters. In this paper with the aim of increasing the order of classical collocation methods, we use a general class of multistep methods based on Hermite collocation methods and prove that this numerical method has uniform order 2 m + 2 r for r previous time steps and m collocation points. Some numerical examples are given to show the validity of the presented method and to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

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 Volterra–Fredholm 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 L 2 -norm. Numerical results are presented and compared with the results from other existing methods to illustrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Two-dimensional First Boubaker polynomials (2D-FBPs) have been formulated and developed as the set of basis for the expansion of bivariate functions. These polynomials are an extension of one-dimensional (1D) FBPs that have previously been used to solve 1D nonlinear integral equations. The dual operational matrix, four operational matrices for integration and operational matrix of product for 2D-FBPs are presented in this study newly. The explicit formula for the operational matrix of product is based on the operational matrix of product by 2D Taylor polynomials (TPs), which has been determined too. Moreover, 2D-FBPs and their operational matrices are used to approximate solution of four types of 2D nonlinear integral equations, that are 2D nonlinear Fredholm, Volterra, Volterra–Fredholm and the system of Volterra integral equations. Using Newton–Cotes collocation nodes, our approach leads to the solution of the set of nonlinear algebraic equations and can be solved uniquely by an iteration method. Since the proposed approach does not require any integration, so all the computations can be easily performed. An explicit and simpler upper bound is obtained for the error vector function of the operational matrix of integration P 2 D b. In addition, an explicit and simpler upper bound is presented for each bivariate function that expand by 2D-FBPs, as well as the convergence analysis has been proved. Six test problems are given to verify the validity and applicability of the presented methods. The numerical results are compared with exciting methods in literature. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to provide a numerical approach for two-dimensional Volterra integral equations (2D-VIEs). An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution v ∈ S p − 1 , p − 1 (− 1) (Π N , M) for approximating the solution of 2D-VIEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the accuracy of the presented method. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of a solution for the system of integro-differential equations. By using a certain measure of noncompactness through the generalized Darbo fixed point theorem, we deduce conditions that guarantee the existence of a solution for the system. The advantage of the Darbo fixed point theorem is it helps us to impose weaker conditions in equations, because the contraction condition reduces to continuous and self-mapping conditions. To illustrate our results, we provide multiple examples and solve them by using the artificial small parameter method, with tables and figures that demonstrate the exactitude of the results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence and uniqueness of the common solution for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations using the common fixed point results equipped with illustrative examples. Some common fixed point results satisfying the generalized contraction condition involving w -distance and weak altering distance functions are proved. Then, an example is provided to support the usability of our result along with numerical experiments. [ABSTRACT FROM AUTHOR]

In this work, we use Newton-type iterative schemes to obtain a domain of existence of solution, approximate the solution of Chandrasekhar H -equations and deal with the case of nonlinear integral equations with non-separable kernels. A change of variable in the Chandrasekhar H -equation allows us to apply a previous study by describing nonlinear integral equations of Hammerstein-type with non-separable kernel. We use the Bernstein polynomials for approximating the non-separable kernel and then we apply a semilocal converge study done previously to the Chandrasekhar H -equation. Moreover, we apply Newton-type iterative schemes for some specific Chandrasekhar H -equations to approximate the H -function solution and compare our results with others obtained previously. [ABSTRACT FROM AUTHOR]

This work is devoted to solve integral equations formulated in terms of the kernel functions and Nemytskii operators. This type of equations appear in different applied problems such as electrostatics and radiative heat transfer problems. We deal with both cases separable and non-separable kernels by setting the theoretical semilocal convergence results for an adequate iterative scheme that can be useful for approximating the solution of the infinite dimensional problem. We pay special attention to non-separable kernels avoiding the solution given in previous works where the original nonlinear integral equation has been approximated by means of an equation with separable kernel. However, in this case, we introduce an approximation of the derivative operator that it is needed for applying the iterative scheme considered. Moreover, we study the localization and separation of possible solutions of nonlinear integral equation by means of a result of semilocal convergence for the iterative scheme considered. The theoretical results obtained have been tested with some applied problems showing competitive results. [ABSTRACT FROM AUTHOR]

In this paper, the efficient methods are proposed for solving nonlinear Fredholm integral equations with a weakly singular kernel. By using Sinc approximation with the single exponential (SE) and double exponential (DE) transformations, the methods are developed. Sinc approximation has considerable advantages. Approximation by Sinc functions handles singularities in the problem and such approximations yield rapidly convergent schemes for solving the problem. Furthermore, convergence of proposed methods is discussed by preparing the theorems which show exponential convergence and guarantee the applicability of those. Finally, some numerical examples are presented to illustrate efficiency and accuracy of the numerical schemes. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a novel technique is being formulated for the numerical solution of integral equations (IEs) as well as integro-differential equations (IDEs) of first and higher orders. The present approach is an improved form of the Haar wavelet methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). The proposed modifications resulted in computational efficiency and simple applicability of the earlier methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013). In addition to this, the new approach is being extended from IDEs of first order to IDEs of higher orders with initial- and boundary-conditions. Unlike the methods (Aziz and Siraj-ul-Islam, 2013, Siraj-ul-Islam et al., 2013) (where the kernel function is being approximated by two-dimensional Haar wavelet), the kernel function in the present case is being approximated by one-dimensional Haar wavelet. The modified approach is easily extendable to higher order IDEs. Numerical examples are being included to show the accuracy and efficiency of the new method. [Copyright &y& Elsevier]

Abstract: In this paper, we give an extension of Darbo’s fixed point theorem associated with measures of noncompactness, and present some results on the existence of coupled fixed points for a class of condensing operators in Banach spaces. Moreover, as an application, we study the problem of existence of solutions for a general system of nonlinear integral equations. [Copyright &y& Elsevier]

Abstract: In this study, we present the existence of solutions for some nonlinear integral equations known as Hammerstein type which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo’s theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on the Sinc approximation to solve the equation which forms the basis for the conductor-like screening model for real solvents (COSMO-RS). The approach is based on the recognition of the Hammerstein integral equation for the determination of the chemical potential of a surface segment as a function of screening charge density. Then convergence of this technique is discussed by preparing a theorem which shows an exponential type convergence rate and guarantees the applicability of that. [Copyright &y& Elsevier]

In this paper, we investigate nonlinear functional Volterra–Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods. [ABSTRACT FROM AUTHOR]

The purpose of this study is to construct a new efficient iterative method of successive approximation based on the three-point Simpson quadrature rule for solving three-dimensional nonlinear Fredholm integral equations. We have also provided the convergence and error analysis of the proposed method. Furthermore, we present the numerical stability analysis of the method with respect to the choice of the first iteration. Finally, some numerical experiments illustrate the accuracy of the method. [ABSTRACT FROM AUTHOR]

Abstract: We analyse collocation methods for nonlinear homogeneous Volterra—Hammerstein integral equations with non-Lipschitz nonlinearity. We present different kinds of existence and uniqueness of nontrivial collocation solutions and give conditions for such existence and uniqueness in some cases. Finally, we illustrate these methods with an example of a collocation problem and give some examples of collocation problems that do not fit in the cases studied previously. [Copyright &y& Elsevier]

Abstract: The famous Newton–Kantorovich hypothesis (Kantorovich and Akilov, 1982 , Argyros, 2007 , Argyros and Hilout, 2009 ) has been used for a long time as a sufficient condition for the convergence of Newton’s method to a solution of an equation in connection with the Lipschitz continuity of the Fréchet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton–Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton–Kantorovich theorem (Argyros, 1998 ; ; Ezquerro and Hernández, 2002 ; ; Proinov 2009, 2010 ). Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 ), as well as a two boundary value problem with a Green’s kernel (Argyros, 2007 ) are also provided in this study. [Copyright &y& Elsevier]

Abstract: By establishing a comparison result and using the method of upper and lower solutions and the monotone iterative technique, we investigate the systems of nonlinear mixed integro-differential equations with deviating arguments and mixed boundary value conditions. Sufficient conditions are established for the existence of extremal system of solutions of the given problem. [Copyright &y& Elsevier]

Abstract: Some explicit bounds on solutions to a class of new power nonlinear Volterra–Fredholm type discrete inequalities are established, which can be used as effective tools in the study of certain sum-difference equations. Application examples are also given. [Copyright &y& Elsevier]

Abstract: We study the blowing-up behavior of solutions of a class of nonlinear integral equations of Volterra type that is connected with parabolic partial differential equations with concentrated nonlinearities. We present some analytic results and, in the case of the kernel of Abel-kind with power nonlinearity and fixed initial data, we give a numerical approximation by using one-point collocation methods. By means of the numerical simulations, we give the dependence of the blow-up time from the parameters of the equation. [Copyright &y& Elsevier]

Abstract: The existence and uniqueness solution of the nonlinear integral equation of Hammerstein type with discontinuous kernel are discussed. The normality and continuity of the integral operator are proved. Toeplitz matrix method is used, as a numerical method, to obtain a nonlinear system of algebraic equations. Also, many important theorems related to the existence and uniqueness of the produced algebraic system are derived. Finally, numerical examples, when the kernel takes a logarithmic and Carleman forms, are discussed and the estimate error, in each case, is calculated. [Copyright &y& Elsevier]

Abstract: In this paper we give necessary and sufficient conditions for blow-up of solutions for a particular class of nonlinear Volterra equations. We also give some examples. [Copyright &y& Elsevier]

Abstract: In this survey paper, the author examines nonlinear Volterra integral equations of the second kind with solutions that blow-up or quench. The focus is on analytical results, although a few words about numerical solutions for such equations are provided. The integral equations arise in the mathematical modeling of thermal processes within a reactive–diffusive medium. The scope of this review is on the published literature between 1997 and 2005, serving as an update to a previous review by the same author. [Copyright &y& Elsevier]

Abstract: A definition of the discrete filled function is given in this paper. Based on the definition, a discrete filled function is proposed. Theoretical properties of the proposed discrete filled function are investigated, and an algorithm for discrete global optimization is developed from the new discrete filled function. The implementation of the algorithms on several test problems is reported with satisfactory numerical results. [Copyright &y& Elsevier]

Abstract: A new technique is used instead of the classical majorant principle to analyze the R-order of convergence of the Newton process when more general conditions than the Kantorovich ones are considered. [Copyright &y& Elsevier]

In the paper there are solved direct and inverse boundary problems and analytical solutions are obtained for optimization problems in the case of some nonlinear integral operators. It is modeled the plane potential flow of an inviscid, incompressible and nonlimited fluid jet, witch encounters a symmetrical, curvilinear obstacle—the deflector of maximal drag. There are derived integral singular equations, for direct and inverse problems and the movement in the auxiliary canonical half-plane is obtained. Next, the optimization problem is solved in an analytical manner. The design of the optimal airfoil is performed and finally, numerical computations concerning the drag coefficient and other geometrical and aerodynamical parameters are carried out. This model corresponds to the Helmholtz impermeable parachute problem. [Copyright &y& Elsevier]

Mixed Volterra–Fredholm integral equations can arise from mathematical modelling of the spatio-temporal development of an epidemic. In this paper a hybrid Legendre block-pulse method is used to provide solutions to a general integral equation of this type. The main idea of this approach is to reduce a Volterra–Fredholm integral equation to a system of algebraic equations. An order convergence analysis is provided. The accuracy of the proposed method is then confirmed by solving some problems, the first being a nonlinear epidemic problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of Multistep Collocation Method (MCM) for solving the non-linear Volterra Functional Integral Equations (VFIEs) including two types of linear and non-linear lag functions. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. In each subinterval, where the solution is smooth enough, we can apply MCM to approximate the solution. Convergence analysis of the method for both types is also analyzed and illustrative five examples are provided to confirm the applicability of this method. [ABSTRACT FROM AUTHOR]

In this article, we establish the existence of solution for two dimensional nonlinear fractional integral equation using fixed point theorem and measure of noncompactness. Applicability of our results is shown by some examples and for validity of the proposed method we make an iterative algorithm by semi-analytic technique that finds a closed form of the solution with an acceptable accuracy. Ability of the proposed method is granted by comparison with another method found in existing literature. [ABSTRACT FROM AUTHOR]

In this article, we consider a class of derivative dependent Fredholm-Hammerstein integral equations i.e., the integral equation, where the nonlinear function inside the integral sign is dependent on derivative and the kernel function is of Green's type. We propose the piecewise polynomial based Galerkin and iterated Galerkin methods to solve these type of derivative dependent Fredholm-Hammerstein integral equations. We discuss the convergence and error analysis of the proposed methods and also obtain the superconvergence results for iterated Galerkin approximations. Some numerical results are given to illustrate this improvement. [ABSTRACT FROM AUTHOR]

An interesting and computationally efficient scheme is proposed for solving a class of two-dimensional nonlinear Volterra–Fredholm integral equations. The collocation method in conjunction with two-dimensional hybrid functions is utilized to convert these types of equations to a nonlinear system of equations that can be solved numerically by the Newton's method. The major advantages of this method are simplicity of performing and favorable convergence rate for an approximate solution. The convergence analysis of the hybrid functions and the error bound of the presented method are discussed in detail. Moreover, some numerical experiments are provided to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]