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In this article, an accurate computational approaches based on Lagrange basis and Jacobi-Gauss collocation method is suggested to solve a class of nonlinear stochastic Itô-Volterra integral equations (SIVIEs). Since the exact solutions of this kind of equations are not still available, so finding an accurate approximate solutions has attracted the interest of many scholars. In the proposed methods, using Lagrange polynomials and zeros of Jacobi polynomials, the considered system of linear and nonlinear stochastic Volterra integral equations is reduced to linear and nonlinear systems of algebraic equations. Solving the resulting algebraic systems by Newton's methods, approximate solutions of the stochastic Volterra integral equations are constructed. Theoretical study is given to validate the error and convergence analysis of these methods; the spectral rate of convergence for the proposed method is established in the L ∞ -norm. Several related numerical examples with different simulations of Brownian motion are given to prove the suitability and accuracy of our methods. The numerical experiments of the proposed methods are compared with the results of other numerical techniques. [ABSTRACT FROM AUTHOR]

In this paper, a combination approach based on Bernoulli polynomials and Gauss-Jacobi quadrature formula is developed to solve the system of nonlinear variable-order fractional Volterra integral equations (V-O-FVIEs). For this, we extend the constant coefficient in the Gauss-Jacobi formula to the variable coefficient and used it in our method. The method converts the system of V-O-FVIEs into the corresponding nonlinear system of algebraic equations. In addition, we use Gronwall inequality and the collectively compact theory to prove the existence and uniqueness of the solution of the original equation and the approximate equation, respectively. The convergence analysis and the error estimation of proposed method are discussed. Finally, some numerical examples illustrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

Volterra integral equation of the second kind with weakly singular kernel usually exhibits singular behavior at the origin, which deteriorates the accuracy of standard numerical methods. This paper develops a singularity separation Chebyshev collocation method to solve this kind of Volterra integral equation by splitting the interval into a singular subinterval and a regular one. In the singular subinterval, the general psi-series expansion for the solution about the origin or its Padé approximation is used to approximate the solution. In the regular subinterval, the Chebyshev collocation method is used to discretize the equation. The details of the implementation are also discussed. Specifically, a stable and fast recurrence procedure is derived to evaluate the singular weight integrals involving Chebyshev polynomials analytically. The convergence of the method is proved. We further extend the method to the nonlinear Volterra integral equation by using the Newton method. Three numerical examples are provided to show that the singularity separation Chebyshev collocation method in this paper can effectively solve linear and nonlinear weakly singular Volterra integral equations with high precision. [ABSTRACT FROM AUTHOR]

In this research, two systems of nonlinear Volterra integral equation and Volterra integro‐differential equation were considered. New results in sense of Ulam stabilities in relation to these two systems were proved on a finite interval. The proof of the results on the Ulam stabilities of that classes of the equations are based on the nonlinear alternative related to Banach's contraction principle. The outcomes of this research give new contribution to the theory of Ulam stabilities. [ABSTRACT FROM AUTHOR]

The blow-up phenomenon may occur in nonlinear integral equations and differential equations, which has important significance for the simulation of some practical problems. This paper is devoted to predicting the blow-up time for convolution Volterra-Hammerstein integral equations using the series expansion of the solution about the origin. First, the finite-term series expansion for the solution about the origin is obtained via Picard iteration and symbolic computation, which has high accuracy near the origin. Second, the Padé approximation of the series expansion is made to extend the convergence region of the series. The blow-up time of the solution is estimated by calculating the smallest positive root of the denominator of the rational function. Third, an integral transform is performed to the series expansion to convert the branch point to a pole of order one at the blow-up time, so the predicted accuracy of the blow-up time is improved remarkably. Numerical examples illustrate that the proposed method is very efficient for estimating the blow-up time of Volterra integral equations and some kinds of differential equations. [ABSTRACT FROM AUTHOR]

In this article, we consider Lucas series for N = 2, 5, 6 in numerical solutions of linear and nonlinear Volterra integral equations of the second kind. The series is used to transform the equation into a system of nonlinear algebraic equations, and the unknown parameters are determined. The application of this method has shown that the Lucas series is a powerful and active candidate that can be used to approximate the solution of linear and nonlinear Volterra integral equations of the second kind. The method gave a good performance, as can be seen from the four sample problems considered. The numerical results reveal computational efficiency of the method and it is also seen to be highly accurate and converge to the exact solution in some cases. Approximate and exact solutions are plotted to further confirm the accuracy of the method. [ABSTRACT FROM AUTHOR]

In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlined the resulting algebraic system in a practical manner. Instead of dealing with a system of 2m × 2m equations, we have simplified the problem to finding the solution for a set of 2 × 2 nonlinear equations. Here, m represents the number of block pulse functions, which is typically a large positive integer. This approach significantly reduces computational costs, especially for large values of m. We provide proofs for both the existence and uniqueness of the solution for this system. Additionally, we investigate two types of stability: Ulam-Hyers stability and generalized Ulam-Hyers stability. We have evaluated the effectiveness of the proposed method by applying it to three different problems and comparing our results with existing literature. The outcomes of these tests highlight the efficiency and efficacy of our proposed method. [ABSTRACT FROM AUTHOR]

A numerical method to solve the nonlinear Volterra integral equations of the first kind with discontinuous kernels is proposed. Usage of operational matrices for this kind of equation is a cost-efficient scheme. Shifted Legendre polynomials are applied for solving Volterra integral equations with discontinuous kernels by converting the equation to a system of nonlinear algebraic equations. The convergence analysis is given for the approximated solution and numerical examples are demonstrated to denote the precision of the proposed method. [ABSTRACT FROM AUTHOR]

We present the inverse problem of successive determination of the two unknowns that are a coefficient characterizing the properties of a medium with weakly horizontal inhomogeneity and the kernel of an integral operator describing the memory of the medium. The direct initial-boundary value problem involves the zero data and the Neumann boundary condition. The trace of the Fourier image of a solution to the direct problem on the boundary of the medium serves as additional information. Studying inverse problems, we assume that the unknown coefficient is expanded into an asymptotic series in powers of a small parameter. Also, we construct some method for finding the coefficient that accounts for the memory of the environment to within an error of order . At the first stage, we determine a solution to the direct problem in the zero approximation and the kernel of the integral operator, while the inverse problem reduces to an equivalent problem of solving the system of nonlinear Volterra integral equations of the second kind. At the second stage, we consider the kernel given and recover a solution to the direct problem in the first approximation and the unknown coefficient. In this case, the solution to the equivalent inverse problem agrees with a solution to the linear system of Volterra integral equations of the second kind. We prove some theorems on the unique local solvability of the inverse problems and present the results of numerical calculations of the kernel and the coefficient. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the solution of non-linear Volterra integral equation of the second kind by using the Rishi transform. The solutions of two numerical problems in compact form are determined by applying the Rishi transform, which suggests that the Rishi transform can be used as a tool for solving these and the other types of related real world problems across various disciplines. [ABSTRACT FROM AUTHOR]

We study a class of integral equations of convolution type on the whole line with a monotone and odd nonlinearity. We prove constructive existence and absence theorems for nonnegative (nontrivial) and bounded solutions. We study the asymptotic behavior of the constructed solution at . We also prove the uniqueness of the solution in the class of nonnegative (nonzero) and bounded functions and present specific examples of this class of equations that can be applied in various fields of mathematical physics. [ABSTRACT FROM AUTHOR]

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Our method opens the door to many expansions in the problems of monotone variational inequalities, image restoration, convex optimization, and split convex feasibility. Moreover, some experimental examples were conducted to gauge the usefulness and efficiency of the technique compared with the iterative methods in the literature. Finally, the proposed approach is applied to solve the nonlinear Volterra integral equation with a delay. [ABSTRACT FROM AUTHOR]

A family of first- and second-order accurate noniterative numerical methods is constructed for solving systems of nonlinear Volterra integral equations of the second kind. The methods are examined for A-, L-, and P-stability. The conclusions are illustrated by numerical results obtained for test equations with stiff and oscillating components. [ABSTRACT FROM AUTHOR]

In this article, we solve the second type of nonlinear Volterra picture fuzzy integral equation (NVPFIE) using an accelerated form of the Adomian decomposition method (ADM). Based on (α , δ , β) -cut, we convert the NVPFIE to the nonlinear Volterra integral equations in a crisp form. An accelerated version of the ADM is used to solve this transformed system, which is based on a new formula for the Adomian polynomial. The sufficient condition that guarantees a unique solution is obtained using this new Adomian polynomial, error estimates are given, and the convergence of the series solution is proven. Numerical cases are discussed to illustrate the effectiveness of this approach. [ABSTRACT FROM AUTHOR]

Lyapunov functions/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it is therefore further explored in this paper. In this manuscript, we utilize Lyapunov functionals combined with Laplace transform to qualitatively analyze the solutions of the integral equation In addition, we extend our method to nonlinear integral equations, integral equations with infinite delay, and integral equations with several kernels. We mention that Laplace transform has been used to solve integral equations of convolution types but has never been applied directly to integral equations that are not of the convolution type. In addition, our method allows us to find the upper estimates, and our necessary conditions are easy to verify. [ABSTRACT FROM AUTHOR]

A class of nonlinear singular integral equations on the whole line is considered. This class of equations for specific representations of nonlinearity and kernel arises in dynamic theory of p-adic strings, in mathematical theory of the pandemic spread within the framework of modified Diekmann-Kapper model, in the theory of radiative transfer in nonhomogeneous medium and kinetic theory of gases. In this paper we will investigate and reveal new qualitative properties of the solution of the above mentioned equations. Relying on these properties as well as some geometric inequalities for convex downward functions we will prove the uniqueness of the solution in the class of bounded functions whose infimum is a positive number. At the end we will present examples of such equations, satisfying all restrictions of the proven statements. [ABSTRACT FROM AUTHOR]

In this study we consider linear and nonlinear Volterra integral equations (VIEs) of the second kind with discontinuous kernel. A novel iterative method using floating point arithmetic (FPA) is presented to solve the problem. Also a convergence theorem and error analysis of the method are presented. The main novelty of this study is to validate the results using the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method which is based on stochastic arithmetic (SA). Moreover instead of applying usual mathematical softwares we use the CADNA (Control of Accuracy and Debugging for Numerical Applications) library which must be done on Linux operating system using C, C++ or Fortran codes. Applying the method and the library we will be able to find the optimal results such as optimal error, optimal iteration of the method and optimal approximation. Proving the main theorem of the CESTAC method we show the equality between the number of common significant digits (NCSDs) of two successive iterations with the exact and approximate solutions. Thus it can help us to apply a new termination criterion instead of absolute error to show the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

The well known results of the theory of classical gaseous polytropes are presented in the framework of an integral approach where the standard Lane-Emden differential equation for a spherically symmetric gravitating mass is examined via its equivalent in the form of a nonlinear integral Volterra equation of the 2nd kind. It is shown that the inverse Laplace transform of the Lane-Emden equation for polytropes with an index of n=5 (Schuster model) is a recurrence relation for Bessel functions of the first kind. The invariance of the nonlinear integral Volterra equation with respect to homological transformations is shown, as well as the possibility of obtaining singular solutions under certain conditions. It is also shown that for whole integral and half integral polytrope indices, this equation is equivalent to a multidimensional integral equation, and finding the expansion of the Emden function in a power series of the dimensionless distance ξ from the center of the polytrope is equivalent to finding the Neumann series and the iterated nuclei in the Fredholm theory. Approximations of the Emden functions in closed form and their applicability to various astrophysical objects will be presented and discussed in the second part of this paper. Polytropes of other geometries and dimensionalities are not considered here. [ABSTRACT FROM AUTHOR]

In this paper, a spectral method is implemented to solve a nonlinear multidimensional Volterra integral equation. Under some condition, the spectral convergence analysis of the solution u$$ u $$ is investigated in both L∞$$ {L}^{\infty } $$ norm and Lωα,β2$$ {L}_{\omega^{\alpha, \beta}}^2 $$ norm. The results of a numerical example confirm the theoretical prediction. [ABSTRACT FROM AUTHOR]

In this paper, using Bianchini-Grandolfi gauge functions, we define a new notion of BG-contraction mapping and proximal BG-contraction mapping. After that, we prove some best proximity point results. Hence, we generalize and improve results existing in the literature such as [12]. Also, we provide nontrivial example to demonstrate the effectiveness of our result. Finally, we give an existence and uniqueness result for the solution of nonlinear Freldhom and Volterra integral equations via our result. [ABSTRACT FROM AUTHOR]

The objective of this paper was to present a new inverse problem statement and numerical method for the Volterra integral equations with piecewise continuous kernels. For such Volterra integral equations of the first kind, it is assumed that kernel discontinuity curves are the desired ones, but the rest of the information is known. The resulting integral equation is nonlinear with respect to discontinuity curves which correspond to integration bounds. A direct method of discretization with a posteriori verification of calculations is proposed. The family of quadrature rules is employed for approximation purposes. It is shown that the arithmetic complexity of the proposed numerical method is O (N 3) . The method has first-order convergence. A generalization of the method is also proposed for the case of an arbitrary number of discontinuity curves. The illustrative examples are included to demonstrate the efficiency and accuracy of proposed solver. [ABSTRACT FROM AUTHOR]

A numerical method based on the inverse Sumudu transform and the Bernstein polynomials operational matrix of integration is developed. The derived method is implemented in solving linear differential, integral and integro-differential equations. Also, a procedure for overcoming nonlinearity is developed and implemented to solve nonlinear Volterra integral equations. The approximate results are compared with the exact solutions and an existing method. Error estimation shows that the proposed method has elevated level of accuracy for just a few terms of the polynomial. [ABSTRACT FROM AUTHOR]

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor's theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples. [ABSTRACT FROM AUTHOR]

The problems of Engineering and Science can easily represent by developing their mathematical models in the terms of integral equations. Various analytical and numerical methods are available that can be used for solving integral equations of different kinds. In this paper, authors have considered recently developed integral transform "Rishi Transform" for obtaining the exact solution of non-linear Volterra integral equation of first kind (NLVIEFK). Four numerical problems have considered for demonstrating the complete procedure of determining the exact solution. Results of these problems depict that Rishi transform is very effective integral transform and it provides the exact solution of NLVIEFK without doing complicated calculation work. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of reconstructing the historical distribution for a nonlinear diffusion equation, in which the diffusion is driven by not only a nonlocal operator but also a locally one. The problem naturally arises in many real-world applications including the biological population dynamic where a population competes for the resources and diffuses by a combination of classical and nonlocal dispersal processes. By using the Banach fixed point theorem and some appropriate estimates, we first construct an example to show the ill-posedness of the problem. Next, we propose a filter regularization method written in form of a nonlinear Volterra integral equation, in combination with the new technique recently developed calling the globally Lipschitz approximation. Finally, several numerical tests, with a combination of the finite difference schemes and the fast Fourier transform (FFT) algorithm, are also presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This article considers the nonlinear functional Volterra integral equation, the Urysohn type class of nonlinear integral equations. In an approach based on the Picard iterative method, the solution's existence and uniqueness are demonstrated. The trapezoidal and Euler discretization methods are used for the approximation of a numerical solution, and a nonlinear algebraic system of equations is obtained. The first and second order of convergence for the Euler and the trapezoidal methods respectively to the solution is manifested by employing the Gronwall inequality and its discrete form. A new Gronwall inequality is devised in order to prove the trapezoidal method's convergence. Finally, some numerical examples are provided which attest to the application, effectiveness, and reliability of the methods. [ABSTRACT FROM AUTHOR]

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]

The paper is devoted to investigate the existence of solutions of infinite systems of nonlinear quadratic integral equations of Volterra type. We look for solutions of those infinite systems of integral equations which are defined, continuous and bounded on the real half-axis and take real values. In our considerations the basic role will be played by the technique of measures of noncompactness defined in the space of functions defined, continuous and bounded on the real half-axis with values in the sequence space l ∞ consisting of real sequences equipped with the standard supremum norm. We are going to prove an existence result which generalizes all recently obtained results for infinite systems of integral equations. An example illustrating the obtained result will be included. [ABSTRACT FROM AUTHOR]

This paper proposes a discrete collocation method based on the fuzzy Lagrange interpolation and the fuzzy Gauss-Legendre quadrature formula for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations. Firstly, the existence and uniqueness of the solution of the original equation are proved by using a two-dimensional Gronwall inequality and an iterative method. Secondly, the discrete collocation method is utilized to convert the linear and nonlinear fuzzy Volterra integral equations into the corresponding linear and nonlinear systems of algebraic equations. Finally, the error estimation and convergence analysis of the proposed numerical method are discussed in terms of the uniform modulus of continuity. Some numerical experiments illustrate that the scheme possesses a higher accuracy and yields a better approximation when compared with other known methods. [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]

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]

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

Edge theoretic extended contractions are introduced and coincidence point theorems and common fixed-point theorems are proved for such contraction mappings in a metric space endowed with a graph. As further applications, we have proved the existence of a solution of a nonlinear integral equation of Volterra type and given a suitable example in support of our result. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions. [ABSTRACT FROM AUTHOR]

We study the valuation of callable barrier reverse convertible contracts written on one or two underlying asset prices. We assume the issuer of the contract can call early redemption at any time during a pre-specified time interval. We identify the optimal redemption policy and show, in the single underlying asset case, it is characterized by a time-dependent boundary. The boundary satisfies a nonlinear integral equation of the Volterra type. When there are two underlying assets, the boundary is a surface depending on one price in addition to time. Valuation formulas and associated integral equations are derived. Numerical experiments are performed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a new and efficient numerical method to approximate the solutions of two‐dimensional nonlinear fractional Fredholm and Volterra integral equations. For this aim, the two‐variable shifted fractional‐order Jacobi polynomials are introduced and their operational matrices of fractional integration and product are derived. These operational matrices and shifted fractional‐order Jacobi collocation method are utilized to reduce the understudy equations to systems of nonlinear algebraic equations. Then, the arising systems can be solved by the Newton method. Discussion on the convergence analysis and error bound of the proposed method is presented. The efficiency, accuracy, and validity of the presented method are demonstrated by its application to three test examples and by comparing our results with the results obtained by existing numerical methods in the literature recently. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution for a class of nonlinear weakly singular Volterra integral equation of the first kind. The existence and uniqueness issue of this nonlinear Volterra integral equations is studied completely. An hp -version collocation method in conjunction with Jacobi polynomials is introduced so as an appropriate numerical solution to be found. We analyze it properly and find an error estimation in L 2 -norm. The efficiency of the method is illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we give a nonlinear F-contraction form of the Sadovskii fixed point theorem and we also investigate the existence of solutions for a functional integral equation of Volterra type. [ABSTRACT FROM AUTHOR]

This work is concerned with the extension of the Jacobi spectral Galerkin method to a class of nonlinear fractional pantograph differential equations. First, the fractional differential equation is converted to a nonlinear Volterra integral equation with weakly singular kernel. Second, we analyze the existence and uniqueness of solutions for the obtained integral equation. Then, the Galerkin method is used for solving the equivalent integral equation. The error estimates for the proposed method are also investigated. Finally, illustrative examples are presented to confirm our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is used for solving component Volterra nonlinear systems by means of the combined Sumudu transform with Adomian decomposition process. We equate the numerical results with the exact solutions to demonstrate the high accuracy of the solution results. The results show that the approach is very straightforward and effective. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to study fractional Brownian motion and its applications to nonlinear stochastic integral equations. Bernstein polynomials have been applied to obtain the numerical results of the nonlinear fractional stochastic integral equations. Design/methodology/approach: Bernstein polynomials have been used to obtain the numerical solutions of nonlinear fractional stochastic integral equations. The fractional stochastic operational matrix based on Bernstein polynomial has been used to discretize the nonlinear fractional stochastic integral equation. Convergence and error analysis of the proposed method have been discussed. Findings: Two illustrated examples have been presented to justify the efficiency and applicability of the proposed method. The corresponding obtained numerical results have been compared with the exact solutions to establish the accuracy and efficiency of the proposed method. Originality/value: To the best of the authors' knowledge, nonlinear stochastic Itô–Volterra integral equation driven by fractional Brownian motion has been for the first time solved by using Bernstein polynomials. The obtained numerical results well establish the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Numerical solutions of nonlinear system of two-dimensional integral equations have been rarely investigated in the literature. In this study, we suggest a numerically practical algorithm to approximate the solutions of nonlinear system of two-dimensional Volterra–Fredholm and Volterra integral equations. This scheme is based on two-dimensional Legendre wavelet to reduce these nonlinear systems of integral equations to a system of nonlinear algebraic equations. The main characteristic of this approach is high accuracy and computational efficiency of performing which are the consequences of Legendre wavelet properties. The main benefit of this basic function is their ability to detect singularities and their efficiency in dealing with non-sufficiently smooth function in comparison with Legendre polynomials and they minimize the error. The convergence analysis and error bound of the proposed Legendre wavelet method is investigated. Numerical examples confirm that the Legendre wavelet collocation method is accurate, reliable for solving nonlinear system of two-dimensional integral equations. [ABSTRACT FROM AUTHOR]

In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement. [ABSTRACT FROM AUTHOR]