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This article investigates discrete versions of the projection-type and modified projection-type methods for solving Hammerstein integral equations with a weakly singular kernel. The approximating operator utilized is either the orthogonal projection or an interpolatory projection onto a space of piecewise polynomials of degree ≤ r - 1 with respect to a graded partition of [0, 1]. The study reveals that the proposed methods achieve optimal rates of convergence, demonstrating that the numerical quadrature used to estimate the integrals maintains the same rates of convergence as the continuous methods. The theoretical findings are supported by numerical experiments. [ABSTRACT FROM AUTHOR]

Variable-exponent weakly singular Volterra integral equations of the second kind with integral kernels of the form (t - s) - α (t) are considered. In Liang and Stynes (IMA J Numer Anal 19:drad072, 2023) it is shown that a typical solution of such an equation exhibits a weak singularity at the initial time t = 0 , similarly to the case where α (t) is constant. Our paper extends this analysis further by giving a decomposition for the exact solution. To solve the problem numerically, a fractional polynomial collocation method is applied on a graded mesh. The convergence of the collocation solution to the exact solution is analysed rigorously and it is proved that specific choices of the fractional polynomials and mesh grading yield optimal-order convergence of the computed solution. Superconvergence properties of the iterated collocation solution are also analysed. Numerical experiments illustrate the sharpness of our theoretical results. [ABSTRACT FROM AUTHOR]

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures.

A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method's efficacy is supported via various examples. [ABSTRACT FROM AUTHOR]

In the current study, a new numerical algorithm is presented to solve a class of nonlinear fractional integral-differential equations with weakly singular kernels. Cubic hat functions (CHFs) and their properties are introduced for the first time. A new fractional-order operational matrix of integration via CHFs is presented. Utilizing the operational matrices of CHFs, the main problem is transformed into a number of trivariate polynomial equations. Error analysis and the convergence of the proposed method are evaluated, and the convergence rate is addressed. Ultimately, three examples are provided to illustrate the precision and capabilities of this algorithm. The numerical results are presented in some tables and figures. [ABSTRACT FROM AUTHOR]

In this paper, a combination of a time discrete technique on graded meshes with barycentric rational interpolation is employed to solve a class of one- and two-dimensional time-dependent partial integro-differential equations with weakly singular kernels. Equations of this type are also commonly called fractional evolution equations, whose solutions have some singular behaviours near the initial time. This makes that it's difficult to reach an expected convergence result for the numerical treatment concerning time variable on uniform meshes. In the temporal direction, a generalized Crank-Nicolson difference scheme and the product integration formula on graded meshes are used to discretize the differential term and the integral term, respectively. Moreover, it's proved that the time discrete scheme is unconditionally stable and quadratically convergent. Further, a fully discrete numerical approach is structured with spacial directional discretization in terms of barycentric rational interpolation. The results of numerical experiments display the efficiency of the method and confirm the theoretical analyses. In addition, the present scheme can reach second-order convergence in time both for smooth initial value and non-smooth initial value. [ABSTRACT FROM AUTHOR]

In this paper, a novel second-order method based on a change of variable and the symmetrical and repeated quadrature formula is presented for numerical solving second kind Volterra integral equations with non-smooth solutions. Applying the discrete Grönwall inequality with weak singularity, the convergence order  O (N − 2)  in  L ∞  norm is proved, where N refers to the number of time steps. Numerical results are conducted to verify the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

In the current paper, we use backward difference and local discontinuous Galerkin (BD-LDG) methods for temporal and spatial discretization of fourth-order partial integro-differential equations (PIDEs) with memory terms containing weakly singular kernels. This work provides a stability analysis of the proposed method, and at the end, by presenting some numerical experiments, we demonstrate the stability of the resulting scheme and show numerically that the optimal convergence rate is O (h k + 1) in the discrete L 2 norm. [ABSTRACT FROM AUTHOR]

This paper is dedicated to solving a class of nonlinear fractional quadratic integral equations of variable order. The block-by-block method based on the Gauss–Lobatto quadrature technique has been developed to solve such integral equations with smooth and weakly singular kernels. In this approach, several values of the unknown functions are obtained through numerical integration without need to any starting value for beginning. The analysis of convergence of the presented method is proved and a rate of convergence is found. Some examples are solved to demonstrate the accuracy of the established approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified Euler–Maruyama (EM) method for Riemann–Liouville stochastic fractional integro-differential equations with weakly singular kernels, and then analyse the strong convergence of the proposed EM method. Specifically, we transform the considered stochastic fractional integro-differential equation into its equivalent form of stochastic Volterra integral equations and derive the corresponding modified EM method. Then, the strong convergence 1 − α order of the proposed method is established, where α is the order of Riemann–Liouville fractional derivative with 0 < α < 1. Finally, numerical experiments are demonstrated to support our theoretical results. [ABSTRACT FROM AUTHOR]

in this work, we present an advanced interpolation method via the Vandermonde matrix for solving weakly singular Volterra integral equations of the second kind. The optimal rules for the node distributions of the two kernel variables were created to guarantee that the kernel's singularity was isolated. The unknown function is interpolated using three matrices: one of which is the monomial matrix, based on the Vandermonde matrix and Chebyshev nodes; the second is the known square Vandermyde matrix, and the third is the unknown coefficient matrix. The singular kernel is interpolated twice and transformed into a double-interpolated non-singular function through five matrices, two of which are monomials. A linear algebraic system can be obtained without using the collocation points by inserting the interpolated unknown function on the left and right sides of the integral equation. The solution of the obtained system yields the unknown coefficients matrix and thereby finds the interpolated solution. The obtained results from solving six examples are faster to converge to the exact ones using the lowest degree of interpolants and are better than those achieved by the other indicated method, which confirms the novelty and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

The discontinuous Galerkin (DG) method is employed to solve second-kind Volterra integral equations (VIEs) with weakly singular kernels. It is proved that the quadrature DG (QDG) method obtained from the DG method by approximating the inner products by suitable numerical quadrature formulas, is equivalent to the piecewise discontinuous polynomial collocation method. The convergence theory is established for VIEs for both uniform and graded meshes. Some numerical experiments are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers the initial value problem of general nonlinear stochastic fractional integro-differential equations with weakly singular kernels. Our effort is devoted to establishing some fine estimates to include all the cases of Abel-type singular kernels. Firstly, the existence, uniqueness and continuous dependence on the initial value of the true solution under local Lipschitz condition and linear growth condition are derived in detail. Secondly, the Euler–Maruyama method is developed for solving numerically the equation, and then its strong convergence is proven under the same conditions as the well-posedness. Moreover, we obtain the accurate convergence rate of this method under global Lipschitz condition and linear growth condition. In particular, the Euler–Maruyama method can reach strong first-order superconvergence when α = 1 α = 1. Finally, several numerical tests are reported for verification of the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, Galerkin and iterated Galerkin methods are applied to approximate the linear second kind Volterra integral equations with weakly singular algebraic kernels using Legendre polynomial basis functions. We discuss the convergence results in both L ² and infinity norms in two cases: when the exact solution is sufficiently smooth and non-smooth. We also apply Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods and derive the super convergence rates. Numerical results are given to verify the theoretical results [ABSTRACT FROM AUTHOR]

In the current paper, a space–time spectral‐Galerkin method is presented for the one‐ and two‐dimensional (1D & 2D) fourth‐order time‐fractional partial integro‐differential equation (TFPIDE) with a weakly singular kernel. In temporal direction, a Petrov‐Galerkin approach is used for discretization. Indeed, eigenfunctions of the first and second kind fractional Sturm‐Liouville problem (FSLP), which called Jacobi polyfractonomials, are used as temporal basis for the trial and test spaces, respectively. Also, spatial discretization is based on the Galerkin approximation with a combination of Legendre polynomials as basis. Fully discrete scheme, give rises to obtaining approximate solution of desired problem via solving a system of linear algebraic equations. Spectral accuracy and efficiency of the considered method are numerically demonstrated by some test problems with smooth and non‐smooth exact solutions in one‐ and two‐dimensional cases. [ABSTRACT FROM AUTHOR]

This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0. A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we study the discrete version of Legendre spectral and iterated Legendre spectral techniques to solve the second kind Hammerstein type weakly singular integral equations. To obtain the convergence analysis, we use the appropriate numerical quadrature rule and obtain the order O (n − r + 1) in discrete Legendre spectral method. If the quadrature rule is minimal, i.e. the number of quadrature nodes and the dimension of the approximating subspace are same, then the optimal rate is obtained O (n − r) in iterated form of discrete Legendre spectral collocation method in L 2 norm and uniform norm. Numerical aspects are given to verify the hypothetical results. [ABSTRACT FROM AUTHOR]

The main purpose of this work is to develop a spectrally accurate collocation method for solving weakly singular integral equations of the second kind with nonsmooth solutions in high dimensions. The proposed spectral collocation method is based on a multivariate Jacobi approximation in the frequency space. The essential idea is to adopt a smoothing transformation for the spectral collocation method to circumvent the curse of singularity at the beginning of time. As such, the singularity of the numerical approximation can be tailored to that of the singular solutions. A rigorous convergence analysis is provided and confirmed by numerical tests with nonsmooth solutions in two dimensions. The results in this paper seem to be the first spectral approach with a theoretical justification for high-dimensional nonlinear weakly singular Volterra type equations with nonsmooth solutions. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to establish a fractional-order operational matrix method based on Euler wavelets for solving linear Volterra–Fredholm integro-differential equations with weakly singular kernel. First, the one-dimensional Euler wavelet is introduced, and then by using it, the operational matrix of fractional integration is constructed. Using the operational matrix of integration of fractional order, the weakly singular linear fractional Volterra–Fredholm integro-differential equations are reduced to a system of algebraic equations. The convergence of the proposed method has been analyzed and the numerical convergence rate for the presented scheme is established. Also, the error analysis of the proposed technique has been investigated. Furthermore, some numerical examples are solved to illustrate the applicability and efficiency of the proposed approach. Finally, a comparison of absolute error values obtained by different wavelet methods has been presented to clarify the error analysis of the proposed method. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the stochastic Volterra integral equations (SVIEs) with double weakly singular kernels. Our primary objective is to examine the well-posedness of the proposed equation. Specifically, we explore the presence of existence, uniqueness, boundedness, and the continuous dependence of the exact solution on the initial data. Additionally, we develop a stochastic θ -scheme as a numerical solution for the equation and demonstrate that the convergence rate of the scheme is influenced by the kernel parameters. To validate the accuracy and reliability of our theoretical findings, we present two numerical examples. Notably, one of these examples concentrates on estimating the price of a European call option using the Heston stochastic volatility model with a singular kernel. Our results, when compared to the corresponding findings by Li et al. [3] , not only relax the integrable limitations of singular kernels but also establish a precise convergence order. In addition, we propose an improved scheme, based on the efficient sum-of-exponentials (SOE) approximation, to address the low computational efficiency of the stochastic θ -scheme. The results confirm that our approach aligns significantly with the expected physical interpretations. • We introduce the novel double singular stochastic Volterra integral equation (SVIE). • We investigate the well-posedness of SVIEs with double weakly singular kernels. • Additionally, the boundedness of the p th moment of the proposed equation is achieved. • The proposed scheme, which is built upon the SOE approximation, effectively reduces the computational cost and storage requirements to O (N log ⁡ N) and O (log ⁡ N) when T ≫ 1 , respectively. When T ≈ 1 , the computational cost and storage requirements are further reduced to O (N log 2 ⁡ N) and O (log 2 ⁡ N) , respectively. • In addition, through our numerical experiments, we have substantiated that the computational efficiency of the proposed scheme, utilizing the SOE approximation, outperforms conventional methods such as the Euler method and the Galerkin approximation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the superconvergence of the "interpolated" collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution u h , two different interpolation postprocessing approximations of higher accuracy: I 2 h 2 m - 1 u h based on the collocation points and I 2 h m u h based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods. [ABSTRACT FROM AUTHOR]

In this article, we apply projection methods and their iterated versions to approximate the solution of system of Fredholm–Hammerstein integral equations with both smooth and weakly singular kernels of algebraic and logarithmic type using the piecewise polynomial basis functions. We show that the iterated Galerkin approximate solution converges to the exact solution faster than the Galerkin approximate solution for both smooth and weakly singular algebraic and logarithmic type kernels. We improve these results further in iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, Jacobi spectral Galerkin method and its iterated version are developed for second kind weakly singular Volterra–Hammerstein integral equations. We transform the domain of integration from [ 0 , t ] to [ − 1 , 1 ] by using variable and function transformations so that the theory of Jacobi polynomials can be applied and obtain the superconvergence results. Further, we enhance these superconvergence results in iterated Jacobi spectral multi-Galerkin method. We verify the theoretical results by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, new numerical method based on Haar wavelet is proposed for the approximate solution of second and fourth order partial integro-differential equations with weakly singular kernels. The proposed method is based on collocation technique. The performance of the proposed method is tested by applying it to various benchmark problems from the literature. The numerical results establish the accuracy as well as efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time t = 0 , only 1 - α global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for m = 1 at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Galerkin and iterated Galerkin methods to approximate the solution of the second kind nonlinear Volterra integral equations of Hammerstein type with the weakly singular kernel of algebraic type, using piecewise polynomial basis functions based on graded mesh. We prove that the Galerkin method converges with the order of convergence O (n - m) , whereas iterated Galerkin method converges with the order O (n - 2 m) in uniform norm. We also prove that in iterated multi-Galerkin method the order of convergence is O (n - 3 m) . Numerical results are provided to justify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers Lévy noise driven nonlinear stochastic Volterra integral equations with doubly weakly singular kernels, whose singular points include both s = 0 and s = t. The existence and uniqueness theorem of the true solution as well as the strong convergence rate of the Euler–Maruyama (EM) method are developed via establishing some fine estimates. Compared with the corresponding results by Wang (Statist Probab Lett 78: 1062–1071, 2008) and Zhang (J Differential Equations 244: 2226–2250, 2008), our results generalize the Gaussian noise case to the Lévy noise case, relax the integrable limitations of singular kernels and establish an accurate convergence order. Moreover, based on the efficient sum-of-exponentials approximation, a fast EM method is presented to improve the low computational efficiency of the EM method. Specifically, when T ≫ 1, the computational complexity O(N2) and the storage O(N) of the EM method are reduced to O (N log N) and O (log N) respectively; while T ≈ 1, they are reduced to O (N log 2 N) and O (log 2 N) respectively, where T and N denote the terminal time and the total number of time steps respectively. Finally, the numerical example supports the theoretical results and explains the priority of the fast EM method. [ABSTRACT FROM AUTHOR]

In the present work, a localized Laplace transform based numerical scheme is constructed for fractional partial integro-differential equations with weakly singular kernels. By application of the Laplace transform the time variable is eliminated and the spatial derivatives are approximated by RBF which transformed the problem into a linear system of equations. The resultant system lead to differentiation matrices which are sparse contrary to large global collocation matrices. The inverse Laplace transform is then computed numerically using contour integration to recover the solution. The efficiency and accuracy of the method is demonstrated by several numerical experiments. The results obtained by the present method are compared with analytical solutions and other numerical methods. [ABSTRACT FROM AUTHOR]

Collocation solutions by globally continuous piecewise polynomials to second-kind Volterra integral equations (VIEs) with smooth kernels are uniformly convergent only for certain sets of collocation points. In this paper we establish the analogous convergence theory for VIEs with weakly singular kernels, for both uniform and graded meshes. [ABSTRACT FROM AUTHOR]

The Jacobi spectral collocation method is being proposed to solve Fredholm-Hammerstein integral equations with the weakly singular kernel and smooth solutions. By using the appropriate quadrature rule, the integral operator is approximated by the discrete operator, which gives rise to the Jacobi collocation method for Fredholm-Hammerstein integral equations with the weakly singular kernel. The convergence analysis for the approximated solution with the exact solution is being discussed for both weighted -norm and infinity-norm. Numerical examples are presented to validate the theoretical estimate. [ABSTRACT FROM AUTHOR]

In this article, we consider the multi-Galerkin method for solving the Fredholm–Hammerstein integral equations with weakly singular kernels, using piecewise polynomial bases. We show that multi-Galerkin method has order of convergence for the algebraic kernel, whereas for logarithmic kernel, it converges with the order in uniform norm, where h is the norm of the partition and r is the smoothness of the solution. We also discuss the iterated multi-Galerkin method. We prove that iterated multi-Galerkin method has order of convergence for the algebraic kernel and for logarithmic kernel, it has order of convergence of order in uniform norm. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the Galerkin method to approximate the solution of Fredholm–Hammerstein integral equations of second kind with weakly singular kernels, using Legendre polynomial bases. We prove that for both the algebraic and logarithmic kernels, the Legendre Galerkin method has order of convergence O (n − r) , whereas the iterated Legendre Galerkin method converges with the order O (n − r − α + 1 2 ) for the algebraic kernel, and order O (log n n − r − 1 2 ) for logarithmic kernel in both L 2 -norm and infinity norm, where n is the highest degree of the Legendre polynomial employed in the approximation and r is the smoothness of the solution. We also propose the Legendre multi-Galerkin and iterated Legendre multi-Galerkin methods. We prove that iterated Legendre multi-Galerkin method has order of convergence O ((1 + c l o g n) n − r − 2 α + 1 2 ) for the algebraic kernel, and order of convergence O ( (log n) 2 (1 + c l o g n) n − r − 3 2 ) for logarithmic kernel in both L 2 -norm and infinity norm. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Current-time responses having a singularity at time t = 0 are typical of some transient electroanalytical experiments (such as e.g. the potential step chronoamperometry). A reliable simulation of such singular responses cannot be performed by conventional simulation techniques, as they fail to provide accurate results close to the singularity. A recent extension [L. K. Bieniasz, J. Comput. Appl. Math. 323 (2017) 136] of the adaptive Huber method for solving Volterra integral equations (IEs) is adopted for the simulation of singular responses at planar and spherical electrodes, in cases when one-dimensional, semi-infinite diffusion transport is coupled with (pseudo-)first order homogeneous reactions. Highly accurate approximants are elaborated, for computing certain integrals of the kernel functions occurring in the related IEs, and needed in this simulation approach. Formerly published approximants to the coefficients of the adaptive Huber method are also improved. The resulting algorithm is tested on four examples of IEs describing chronoamperometry for E rev C irr , EC rev E , C rev E and E rev C' irr reaction mechanisms. Singular transients are simulated automatically with a prescribed accuracy, also for t arbitrarily close to the singularity. [ABSTRACT FROM AUTHOR]

In this paper, several methods for approximating the solution of Hammerstein equations with weakly singular kernels are considered. The paper is motivated by the results reported in papers [7, 12]. The orders of convergence of the proposed methods and those of superconvergence of the iterated methods are analyzed. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In the present paper, a modification of hat functions (MHFs) has been considered for solving a class of nonlinear fractional integro-differential equations with weakly singular kernels, numerically. The fractional order operational matrix of integration is introduced. We provide an error estimation for the approximation of a function by a series of MHFs. To suggest a numerical method, the main problem is converted to an equivalent Volterra integral equation of the second kind and operational matrices of MHFs are used to reduce the problem to the solution of bivariate polynomial equations. Finally, illustrative examples are provided to confirm the accuracy and validity of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a piecewise fuzzy polynomial interpolation method is proposed to approximate the solutions of fuzzy delay integral equations with weakly singular kernels. Firstly, the existence and uniqueness of the solutions of these fuzzy delay integral equations are proved by using the Gronwall inequality and an iterative method. Then, the fuzzy weakly singular delay integral equations are discretized into related algebraic equations by using piecewise fuzzy polynomial interpolation and the fuzzy Gauss-Jacobi quadrature formula. Moreover, the convergence of the numerical method is investigated and an error estimate is provided. Finally, the accuracy and efficiency of the proposed method are verified by some numerical experiments. [ABSTRACT FROM AUTHOR]

Kernels obtained from the heat equation arise in several modeling contexts, like some double porosity models, or viscous flows in networks of thin tubes. These kernels are weakly singular at initial time. An accurate approximation must therefore take this singularity into account. In this paper we obtain an asymptotic expansion for small times, which we use to build a numerical scheme for approximating the kernels. Convergence of the scheme and relevance of a correction through the asymptotics are proven both analytically and numerically. Finally, we show that this approximation applies to the model on the graph studied by the authors in Canon et al. [4] (2021). • Small time asymptotic expansion of weakly singular kernels • Convergent numerical approximation of a diffusion equation with incompatible initial and boundary data • Application to simplified models for viscous fluids in networks in relation to blood vessels flows and microfluidics [ABSTRACT FROM AUTHOR]

We apply Jacobi spectral collocation approximation to a two-dimensional nonlinear weakly singular Volterra integral equation with smooth solutions. Under reasonable assumptions on the nonlinearity, we carry out complete convergence analysis of the numerical approximation in the L∞-norm and weighted L2-norm. The provided numerical examples show that the proposed spectral method enjoys spectral accuracy. [ABSTRACT FROM AUTHOR]

Numerical methods for solving nonlinear systems of weakly singular Volterra integral equations (VIEs) possessing weakly singular solutions appear almost nonexistent in the literature, except for a few treatments of single first kind Abel equations. To reduce this gap, an extension is presented, of the adaptive Huber method designed for VIEs with singular kernels such as K ( t , τ ) = ( t − τ ) − 1 / 2 and K ( t , τ ) = exp [ − λ ( t − τ ) ] ( t − τ ) − 1 / 2 (where λ ≥ 0 ) and a variety of nonsingular kernels. The method was thus far restricted to bounded solutions having at least two derivatives. Under a number of assumptions specified, the extension applies to solutions U μ ( t ) that can be written as sums of singular components c μ t − 1 / 2 (with unknown coefficients c μ ), and nonsingular components U ¯ μ ( t ) . In the solution process, factor t − 1 / 2 is handled analytically, whereas c μ and U ¯ μ ( t ) are determined numerically. Computational experiments reveal that the extended method determines singular solutions equally well as the unextended method determined nonsingular solutions. The method is intended primarily for a class of VIEs encountered in electroanalytical chemistry, but it can also be of interest to other application areas. [ABSTRACT FROM AUTHOR]

In a number of transient electroanalytical experiments (such as, in particular, the potential step chronoamperometry) one expects current-time responses having a singularity at time t = 0. A reliable simulation of such singular responses is a difficult task. Conventional simulation techniques fail to provide accurate results close to the singularity. A recent extension [L. K. Bieniasz, J. Comput. Appl. Math. 323 (2017) 136] of the adaptive Huber method for solving Volterra integral equations (IEs) is shown to overcome this difficulty in cases when the current behaves like t −1/2 for t → 0. The method requires an availability of highly accurate approximants for computing certain integrals of the kernel functions occurring in the IEs. Relevant approximants are elaborated for several most important kernel functions specific of one-dimensional diffusion, and one kernel function for two-dimensional diffusion. The resulting algorithm is tested on two examples of single IEs describing chronoamperometry and cyclic voltammetry for a single reversible charge transfer, and one example of an IE system describing chronoamperometry for an E rev E rev mechanism. Singular transients are simulated automatically with a prescribed accuracy, even for t arbitrarily close to the singularity. [ABSTRACT FROM AUTHOR]

In this paper, we present an hp-version Legendre-Jacobi spectral collocation method for the nonlinear Volterra integro-differential equations with weakly singular kernels. We derive hp-version error bounds of the collocation method under the $$H^1$$ -norm for the Volterra integro-differential equations with smooth solutions on arbitrary meshes and singular solutions on quasi-uniform meshes. Numerical experiments demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this article, we apply Jacobi spectral Galerkin, multi-Galerkin methods and their iterated versions to approximate the system of Volterra integral equations for smooth as well as weakly singular kernels and obtain the superconvergence results. Before doing this, first, we develop the regularity properties of the system of linear Volterra integral equations. We show that the Jacobi spectral iterated Galerkin approximation yields better converge rates over Jacobi spectral Galerkin method. We make the improvement of the superconvergence rates further for both smooth and weakly singular kernels in Jacobi spectral iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results. [ABSTRACT FROM AUTHOR]