Nonlinear Volterra Integral Equations Research Articles

On the existence of solutions of non-linear 2D Volterra integral equations in a Banach Space

On the existence of solutions of non-linear 2D Volterra integral equations in a Banach Space

Solving Parametric Volterra Integral Equation from Distributed-Order Rough Heston Model and Option Pricing

The rough Heston model has recently attracted the attention of many financial practitioners and researchers because it maintains the basic structure of the classic Heston model and has an advantage in describing the microstructure foundation of the market. In this paper, we study the distributed-order rough Heston model with an exponential tempered factor, and from the characteristic function of log-price in this model, we obtain a nonlinear parametric Volterra integral equation. Finally, the Fourier-cosine methods are combined with the Adams methods to price the option.

Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations

Boubaker polynomials and their applications for solving fractional two-dimensional nonlinear partial integro-differential Volterra integral equations

A sharp error estimate of Euler‐Maruyama method for stochastic Volterra integral equations

In this paper, the Euler‐Maruyama method is used to solve a class of nonlinear stochastic Volterra integral equations (SVIEs), which can be derived from stochastic fractional substantial integro‐differential equations (SFSIDEs). The existence and uniqueness of the solution are proved for the nonlinear SVIEs under the non‐Lipschitz condition. Moreover, the sharp convergence estimate with if if and O(h) if , respectively, is established for the nonlinear SVIEs, which can be revised as supplemented theory results of our recent work (J Comput Appl Math 356 (2019) 377–390). In particular, it also closes a gap that was left on the convergence analysis in (J Comput Appl Math 317 (2017) 447–457). Finally, numerical experiments are given to illustrate the effectiveness of the presented method.

Construction of solutions of integral equations with Stieltjes functionals and bifurcation parameters

The nonlinear Volterra integral equations with loads on the desired solution are studied. Loads are given using the Stieltjes integrals. The equations contain a parameter, for any value of which the equation has a trivial solution. The necessary and sufficient conditions on the values of the parameter are derived in the neighborhood where theequation has nontrivial real solutions.

Numerical solution of two-dimensional nonlinear Volterra integral equations of the first kind using operational matrices of hybrid functions

In this paper, operational matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are employed to approximate the solutions of two-dimensional nonlinear Volterra integral equations of the first kind (2DNVIEF). To this aim, we first convert 2DNVIEF to the two-dimensional linear Volterra integral equations of the second kind (2DLVIES). Then the operational matrices of integration, together with the product operational matrix, can be applied to reduce the 2DLVIES to a system of algebraic equations. The convergence of the proposed method is studied, and an error estimation under the -norm is provided. Finally, some numerical examples are prepared to clarify the accuracy and efficiency of the numerical method especially in dealing with functions that are not continuous in the whole interval.

An existence theorem for nonlinear functional Volterra integral equations via Petryshyn's fixed point theorem

<abstract><p>Using the method of Petryshyn's fixed point theorem in Banach algebra, we investigate the existence of solutions for functional integral equations, which involves as specific cases many functional integral equations that appear in different branches of non-linear analysis and their applications. Finally, we recall some particular cases and examples to validate the applicability of our study.</p></abstract>

Exact solution of non-linear volterra integral equation of first kind using Rishi transform

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.

Solving a Class of Two-dimensional Nonlinear Volterra Integral Equations As a Generalized Problem of Moments

It will be shown that solve an equation two-dimensional Volterra nonlinear can be solved numerically applying the techniques of inverse generalized moments problem in two steps writing the Volterra's equation as a Klein-Gordon equation of the form wtt – wxx = H(x, t), which H(x, t) it is unknown. In a first step, H(x, t) is numerically approximate, and in a second step we numerically approximate the solution of Klein-Gordon equation using the H(x, t) previously approximated. The method is illustrated with examples.

Solvability and numerical method for non-linear Volterra integral equations by using Petryshyn’s fixed point theorem

In this paper, utilizing the technique of Petryshyn’s fixed point theorem in Banach algebra, we analyze the existence of solution for functional integral equations, which includes as special cases many functional integral equations that arise in various branches of non-linear analysis and its application. Finally, we introduce the numerical method formed by modified homotopy perturbation approach to resolving the problem with acceptable accuracy.

A meshless local Galerkin integral equation method for solving a type of Darboux problems based on the radial basis functions

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. doi:10.1017/S1446181121000377

Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations

Abstract In this article, we prove some new common fixed point results under the generalized contraction condition using w-distance and weak altering distance functions. Also, the validity of the results is demonstrated by an example along with numerical experiment for approximating the common fixed point. Later, as applications, the unique common solutions for the system of nonlinear Fredholm integral equations, nonlinear Volterra integral equations and nonlinear fractional differential equations of Caputo type are derived.

A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays

In this paper, we develop a hybrid spectral method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel and vanishing delays. Our main strategy is to divide the original interval into subintervals, to employ the shifted generalized log orthogonal functions (GLOFs) as the basis on the first interval, to take the classical shifted Legendre polynomials as the basis on other intervals. We analyze the existence and uniqueness of the numerical scheme, and derive the corresponding error estimates. A series of examples demonstrate the effectiveness of the proposed method.

Common solution to a pair of nonlinear Fredholm and Volterra integral equations and nonlinear fractional differential equations

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.

New spectral element method for Volterra integral equations with weakly singular kernel

In this paper, we put forward a new spectral element method for the nonlinear second-kind Volterra integral equations (VIEs) with weakly singular kernel, which employs shifted Müntz–Jacobi functions and shifted Legendre polynomials as basis functions. This method is capable of approximating the limited regular solution more efficiently. We analyze the existence and uniqueness of the solution to the numerical scheme and derive the hp-version optimal convergence under some reasonable assumptions. A series of numerical examples are presented to demonstrate the efficiency of the new method.

A MESHLESS LOCAL GALERKIN INTEGRAL EQUATION METHOD FOR SOLVING A TYPE OF DARBOUX PROBLEMS BASED ON RADIAL BASIS FUNCTIONS

AbstractThe 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.

Discrete collocation method for solving two-dimensional linear and nonlinear fuzzy Volterra integral equations

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.

Comparison of Two Wavelet Methods for Accurate Solution of Two-Dimensional Nonlinear Volterra Integral Equation

Comparison of Two Wavelet Methods for Accurate Solution of Two-Dimensional Nonlinear Volterra Integral Equation

Existence of solutions of infinite systems of nonlinear Volterra integral equations on a Fréchet space

Existence of solutions of infinite systems of nonlinear Volterra integral equations on a Fréchet space

Numerical solution of two-dimensional weakly singular Volterra integral equations with non-smooth solutions

Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at t=0 even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot’s quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.