Methods For Volterra Integral Equations Research Articles

A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha }$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

Stability analysis of ef Gaussian direct quadrature methods for Volterra integral equations

The present paper analyzes the numerical stability of exponentially fitted Gaussian direct quadrature methods for Volterra integral equations. By applying the exponential fitting technique, the coefficients of these methods are specially tuned on the problem. The analysis of numerical stability of these methods with respect to the classical basic test equation is carried out, which takes into account the dependence of the methods both on the stepsize and on the problem parameters. Several stability regions are shown and some numerical experiments confirm theoretical expectations.

On the convergence rate of collocation methods for Volterra integral equations with weakly singular oscillatory trigonometric kernels

This paper presents efficient collocation methods for linear Volterra integral equations with weakly singular highly oscillatory kernels. The numerical steepest descent method and generalized Gauss–Laguerre rule are utilized to calculate the weakly singular oscillatory integrals, which is the main challenge of the problem. Moreover, we derive the corresponding error estimation formula in terms of the frequency and the step length. This formula reflects, to some extent, the global convergence of the method. However, numerical examples show that the method is very effective and verify the correctness of the theoretical results.

A bridge on Lomnitz type creep laws via generalized fractional calculus

This paper is dedicated to achieving the unification of the scattered Lomnitz type creep laws. Thereupon, a novel generalized fractional calculus, that is, Katugampola-like fractional calculus is well-defined and employed as an effective tool, along with some fundamental qualities as by-products. By virtue of the hereditary theory of viscoelasticity and Volterra integral equation method, a compatible stress-strain relation with memory effects and time-varying viscosity established results in a unified Lomnitz creep law. Then, a comparative analysis on the creep function and relaxation function inherited from the unified Lomnitz creep law is exhibited via reasoning and numerical simulation. Besides, physical interpretation of such unified Lomnitz creep law is also provided with the aid of the modified spring-dashpot model of Maxwell type. Results carried out in this paper might provide a full characterization of the creep of igneous rocks to meet some special theoretical and experimental requirements.

Development of the Approximating Functions Method for Problems in a Planar Waveguide with Constant Polarization

Article introduces an extension of the approximating functions method, a particular case of the finite element method (FEM) with interpolating functions in the form of Lagrange polynomials of a special form, to solve electrodynamics problems in a planar waveguide with constant polarization in the spatial-temporal domain using the Volterra integral equation method. The main goal of the article is to expand the area of ​​applicability of this method to three-dimensional problems in a planar waveguide with constant polarization, as well as to obtain general interpolation expressions in analytical form, which will be used to construct a system of nonlinear equations for solving specific problems.

Efficient collocation methods for Volterra integral equations with highly oscillatory kernel

In this paper, we construct efficient collocation methods to deal with linear Volterra integral equations with highly oscillatory kernel. To discretize the oscillatory integrals in the collocation equation, a Filon type method is adopted. Based on some lemmas, we analyze the asymptotic property of the solution and obtain the convergence of the method, which is related to both the frequency ω and step length h. The proposed method does not only have the same order at the collocation points as the classical collocation method, but also may enjoy an asymptotic order which could reach two in some cases. Some numerical examples are presented at last to show the theoretical results and the efficiency of the method.

THE NEW MODIFICATION OF THE APPROXIMATING FUNCTIONS METHOD FOR CLOUD COMPUTING

The approach for building cloud-ready fault-tolerant calculations by approximating functions method, which is an analytical-numerical part of Volterra integral equation method for solving 1D+T nonlinear electromagnetic problems, is presented. The solving process of the original algorithm of the method is modified: it is broken down into the sequential chain of stages with a fixed number of sequential or parallel steps, each of which is built in a fault-tolerant manner and saves execution results in fault-tolerant storage for high availability. This economizes RAM and other computer resources and does not damage the calculated results in the case of a failure, and allows stopping and starting the calculations easily after manual or accidental shutdown. Also, the proposed algorithm has self-healing and data deduplication for cases of corrupted saved results. The presented approach is universal and does not depend on the type of medium or the initial signal. Also, it does not violate the natural description of non-stationary and nonlinear features, the unified definition of the inner and outer problems, as well as the inclusion of the initial and boundary conditions in the same equation as the original approximating functions method. The developed approach stress-tested on the known problems, stability checked and errors compared.

The New Modification of the Approximating Functions Method for Cloud Computing

Abstract The approach for building cloud-ready fault-tolerant calculations by approximating functions method, which is an analytical-numerical part of Volterra integral equation method for solving 1D+T nonlinear electromagnetic problems, is presented. The solving process of the original algorithm of the method is modified: it is broken down into the sequential chain of stages with a fixed number of sequential or parallel steps, each of which is built in a fault-tolerant manner and saves execution results in fault-tolerant storage for high availability. This economizes RAM and other computer resources and does not damage the calculated results in the case of a failure, and allows stopping and starting the calculations easily after manual or accidental shutdown. Also, the proposed algorithm has self-healing and data deduplication for cases of corrupted saved results. The presented approach is universal and does not depend on the type of medium or the initial signal. Also, it does not violate the natural description of non-stationary and nonlinear features, the unified definition of the inner and outer problems, as well as the inclusion of the initial and boundary conditions in the same equation as the original approximating functions method. The developed approach stress-tested on the known problems, stability checked and errors compared.

A GLMs-based difference-quadrature scheme for Volterra integro-differential equations

In this paper, a class of general linear methods is combined with a special quadrature rule inherited from natural Runge–Kutta methods for Volterra integral equations to design a more stable and efficient algorithm. The constructed methods of orders 1,2, and 3 are A0–stable and the method of order 4 is A0(α)–stable. The capability of the proposed methods in solving stiff and nonstiff problems is shown by some numerical experiments.

Complete monotonicity-preserving numerical methods for time fractional ODEs

The time fractional ODEs are equivalent to convolutional Volterra integral equations with completely monotone kernels. We therefore introduce the concept of complete monotonicity-preserving ($\mathcal{CM}$-preserving) numerical methods for fractional ODEs, in which the discrete convolutional kernels inherit the $\mathcal{CM}$ property as the continuous equations. We prove that $\mathcal{CM}$-preserving schemes are at least $A(\pi/2)$ stable and can preserve the monotonicity of solutions to scalar nonlinear autonomous fractional ODEs, both of which are novel. Significantly, by improving a result of Li and Liu (Quart. Appl. Math., 76(1):189-198, 2018), we show that the $\mathcal{L}$1 scheme is $\mathcal{CM}$-preserving, so that the $\mathcal{L}$1 scheme is at least $A(\pi/2)$ stable, which is an improvement on stability analysis for $\mathcal{L}$1 scheme given in Jin, Lazarov and Zhou (IMA J. Numer. Analy. 36:197-221, 2016). The good signs of the coefficients for such class of schemes ensure the discrete fractional comparison principles, and allow us to establish the convergence in a unified framework when applied to time fractional sub-diffusion equations and fractional ODEs. The main tools in the analysis are a characterization of convolution inverses for completely monotone sequences and a characterization of completely monotone sequences using Pick functions due to Liu and Pego (Trans. Amer. Math. Soc. 368(12):8499-8518, 2016). The results for fractional ODEs are extended to $\mathcal{CM}$-preserving numerical methods for Volterra integral equations with general completely monotone kernels. Numerical examples are presented to illustrate the main theoretical results.

Highly stable multistep Runge–Kutta methods for Volterra integral equations

In this paper, we investigate highly stable multistep Runge–Kutta methods for Volterra integral equations. First, the order conditions for order p and stage order $$q=p$$ are presented, and a convergence theorem is given. The numerical stability conditions for the basic and convolution test equations are derived. Then, the methods with one or two stages are studied in detail. Some A-stable and $$V_0$$ -stable m-stage methods with order $$p>m$$ are obtained. For one-stage methods, we also construct $$A_0$$ -stable and -stable methods of orders 3 and 4. Finally, numerical experiments are given to confirm the theoretical results.

Implementation of general linear methods for Volterra integral equations

General linear methods (GLMs) are a large family of methods which have been already introduced for the numerical solution of Volterra integral equations of the second kind. In this paper, various issues relevant to the implementation of these methods in a variable stepsize environment are discussed, and the corresponding code is developed.

Mapped spectral collocation methods for Volterra integral equations with noncompact kernels

This paper is devoted to solve weakly singular Volterra integral equations with noncompact kernels, which differ from the well-known case of Abel-type equations. We consider using the mapped Laguerre spectral method to deal with this type of equations. The construction and analysis of log orthogonal functions collocation method are presented in this paper and some numerical examples are included to show the efficiency of the proposed method.

Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels

This article deals with the implementation of Jacobi spectral and pseudospectral Galerkin approaches for solving weakly singular Volterra integral equations with noncompact kernels. The computational domain of the main problem and intervals of the involved integrals are transformed into [-1,1] by some appropriate change of variables. Therefore, one can apply the Jacobi Galerkin method and impose the inner product conditions on the residual of the considered equation. Also, Jacobi Gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the Jacobi weight function in the pseudospectral Galerkin method. For both cases of spectral and pseudospectral Galerkin schemes, rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in L∞ and weighted L2 norms. Several numerical examples are also given to confirm the theoretical prediction.

Multistep Collocation Methods for Volterra Integral Equations with Weakly Singular Kernels

Multistep Collocation Methods for Volterra Integral Equations with Weakly Singular Kernels

Barycentric rational collocation methods for Volterra integral equations with weakly singular kernels

In this paper, we investigate barycentric rational collocation methods for weakly singular Volterra integral equations. Since the exact solution usually has a weak singularity at the initial point, a smoothing transformation is first used to convert the original equation to a new equation whose exact solution is smooth enough. Then, by the collocation approach based on barycentric rational interpolation, a discrete numerical scheme is constructed, and a general convergence result is established. Finally, the theoretical results are illustrated by some numerical examples.

The numerical solution of fractional differential equations using the Volterra integral equation method based on thin plate splines

Finding the approximate solution of differential equations, including non-integer order derivatives, is one of the most important problems in numerical fractional calculus. The main idea of the current paper is to obtain a numerical scheme for solving fractional differential equations of the second order. To handle the method, we first convert these types of differential equations to linear fractional Volterra integral equations of the second kind. Afterward, the solutions of the mentioned Volterra integral equations are estimated using the discrete collocation method together with thin plate splines as a type of free-shape parameter radial basis functions. Since the scheme does not need any background meshes, it can be recognized as a meshless method. The proposed approach has a simple and computationally attractive algorithm. Error analysis is also studied for the presented method. Finally, the reliability and efficiency of the new technique are tested over several fractional differential equations and obtained results confirm the theoretical error estimates.

Stability Issues for Selected Stochastic Evolutionary Problems: A Review

We review some recent contributions of the authors regarding the numerical approximation of stochastic problems, mostly based on stochastic differential equations modeling random damped oscillators and stochastic Volterra integral equations. The paper focuses on the analysis of selected stability issues, i.e., the preservation of the long-term character of stochastic oscillators over discretized dynamics and the analysis of mean-square and asymptotic stability properties of ϑ -methods for Volterra integral equations.

On fast multipole methods for Volterra integral equations with highly oscillatory kernels

This paper explores the fast multipole methods (FMMs) to accelerate the approximation for weakly singular Volterra integral equations with highly oscillatory trigonometric kernels. By constructing the fast translation path, the FMM is utilized to speed up the iterative method, which reduces the complexity from O ( N 2 ) to O ( N ) . Especially, we use the collocation method to discretize the Volterra integral equation with constants and linear elements respectively, then apply the GMRES to solve the dense and non-symmetric linear system. In addition, the highly oscillatory integrals derived from the algorithm are calculated effectively by the steepest descent method. The proposed method shows that the numerical solutions become more accurate as the frequency increases. Both of the optimal convergence rates of truncation and the error bounds analysis are represented in the end.

Superconvergence results of Legendre spectral projection methods for Volterra integral equations of second kind

In this paper, Legendre spectral projection methods are applied for the Volterra integral equations of second kind with a smooth kernel. We prove that the approximate solutions of the Legendre Galerkin and Legendre collocation methods converge to the exact solution with the order $${\mathcal {O}}(n^{-r})$$ in $$L^2$$ -norm and order $${\mathcal {O}}(n^{-r+\frac{1}{2}})$$ in infinity norm, and the iterated Legendre Galerkin solution converges with the order $${\mathcal {O}}(n^{-2r})$$ in both $$L^2$$ -norm and infinity norm, whereas the iterated Legendre collocation solution converges with the order $${\mathcal {O}}(n^{-r })$$ in both $$L^2$$ -norm and infinity norm, n being the highest degree of Legendre polynomials employed in the approximation and r being the smoothness of the kernels. We have also considered multi-Galerkin method and its iterated version, and prove that the iterated multi-Galerkin solution converges with the order $${\mathcal {O}}(n^{-3r})$$ in both infinity and $$L^2$$ norm. Numerical examples are given to illustrate the theoretical results.