Nonlinear Volterra Integral Equations Research Articles

An iterative method for robust solutions to nonlinear Volterra integral equations: Stability, convergence, and practical applications

An iterative method for robust solutions to nonlinear Volterra integral equations: Stability, convergence, and practical applications

A Novel Super-Convergent Numerical Method for Solving Nonlinear Volterra Integral Equations Based on B-Splines

A Novel Super-Convergent Numerical Method for Solving Nonlinear Volterra Integral Equations Based on B-Splines

Collocation methods for nonlinear Volterra integral equations with oscillatory kernel

This work is devoted to the numerical solution of second kind nonlinear Volterra integral equations with highly oscillatory kernel. We use a collocation approach by discretizing the oscillatory integrals in the collocation equation using a Filon-type quadrature rule. We investigate the convergence of the numerical method in terms of step length h and frequency ω. As h decreases, the suggested technique converges with order d, while its asymptotic order as the frequency increase, is at least 1 and may reach 2 in some cases. Numerical experiments validate theoretical results.

Adomian Decomposition Method and Block by Block Method for Solving Nonlinear Functional Volterra Integral Equation in Two-dimensions

This work proposes a new defintion of the nonlinear functional Volterra integral equation ib 2D of the second kind with continuous kernel (NT-DFVIE). Furthermore, the work is concerned to study this new equation numerically. The existence of a unique solution of the equation is proved. In addition, the approximate solutions are obtained by two powerful methods Adomian Decomposition Method (ADM) and Block by block method (BBM). The given numerical examples showed the efficiency and accuracy of the introduced methods.

On Ulam stabilities of iterative Fredholm and Volterra integral equations with multiple time-varying delays

In present study, we deal with nonlinear iterative Fredholm and Volterra integral equations (Fredholm and Volterra IEs) including variable time delays. We are interested here in the investigations of the uniqueness of solutions and Ulam type stabilities of that the iterative Fredholm and the Volterra IEs. The proofs of the new outcomes of the study with regard to these concepts are done in the light of the Banach fixed point theorem (Banach FPT) and the Bielecki metric. As for new contributions of the present study, here, first time we develop the relative outcomes that can be found in the literature to certain nonlinear iterative Fredholm and Volterra IEs including several variable time delays. Finally, a concrete example is introduced at the end of the study.

Solving Nonlinear Volterra Integral Equations by Mohanad Decomposition Method

In this research article, we introduce the Mohanad transform-decomposition method, which is a new analytical approach. The basic characteristics and facts of the proposed method are presented and analyzed. This new method is a simple method that combines the Mohanad transform with the decomposition method. This new approach is utilized to handle nonlinear integro-differential equations, the results obtained from this method are expressed in the form of an infinite series that converges rapidly to the exact ones. The maximum absolute error is computed for the proposed examples, and some figures are presented to show the accuracy of the obtained results. All the numerical results and computations in this study are gained by using Mathematica software.

Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions

Numerical solutions of a class of linear and nonlinear Volterra integral equations of the third kind using collocation method based on radial basis functions

New Results on Ulam Stabilities of Nonlinear Integral Equations

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results.

Some new bounds for the blow-up time of solutions for certain nonlinear Volterra integral equations

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.

A comparative study of discretization techniques for augmented Urysohn type nonlinear functional Volterra integral equations and their convergence analysis

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.

Laplace Transform for the Solution of Non-Linear Volterra Integral Equation of Second Kind

Laplace Transform for the Solution of Non-Linear Volterra Integral Equation of Second Kind

A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel

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 hp-version convergence analysis for the hybrid spectral collocation method under L2-norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint.

On the Ulam stabilities of nonlinear integral equations and integro‐differential equations

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.

Convergence and data dependence results of the nonlinear Volterra integral equation by the Picard's three step iteration

<abstract> <p>Picard's three step iteration algorithm was one of the iteration algorithms that was recently shown to be faster than some other iterative algorithms in the existing literature. The purpose of this paper was to study using this iteration algorithm for the solution of nonlinear Volterra integral equations. It was investigated that the sequences obtained from this iteration algorithm converged to the solution of nonlinear Volterra integral equations. Moreover, data dependence was obtained for nonlinear Volterra integral equations. An example was given that confirmed the applicability of the newly proven theorems.</p> </abstract>

Application of Rishi transform to the solution of nonlinear Volterra integral equation of the second kind ∗

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 sug gests 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.

Regularization of the Solution of a Nonlinear Integral Equation of the First Kind

Integral equations, the main branch of mathematics, are widely used in physics, engineering, mechanics, control theory and other fields. Related to the application of integral equations, new fields are developing, such as economics, some sections of biology, etc. The theory of integral equations mainly developed in the late nineteenth-early twentieth century, starting with Vito Volterra (1982, 1986), Eric Ivar Fredholm (2010), David Hilbert, Erhard Schmidt, etc. scientists began to study it. Nevertheless, within the framework of mathematical concepts that existed before the first half of the twentieth century, such problems were considered incorrect because a small change in the given functions led to a greater change in the desired functions. The Volterra equation of the first kind is an integral equation that has an exact solution only in some cases. The limit of integration has been carried out in very small quantities on non-classical linear and nonlinear integral equations with variable limits and the construction of solutions in these works is based on numerical methods. Therefore, for the so-called non-classical Volterra integral equations, it is relevant to determine the conditions that ensure the uniqueness and regularization of their solutions. In this paper, the uniqueness of the solution of the non-classical nonlinear integral Volterra equation of the first kind is resolved. The aim of the study is to solve the non-classical Volterra integral equation of the first kind, that is, to determine the conditions that ensure the uniqueness of the solution of the nonlinear non-classical Volterra integral equation of the first kind. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems.

Optimal Control of Non-linear Volterra Integral Equations with Weakly Singular Kernels Based on Genocchi Polynomials and Collocation Method

AbstractWe consider a problem of finding the best way to control a system, known as an optimal control problem (OCP), governed by non-linear Volterra Integral Equations with Weakly Singular kernels. The equations are based on Genocchi polynomials. Depending on the applicable properties of Genocchi polynomials, the considered OCP is converted to a non-linear programming problem (NLP). This method is speedy and provides a highly accurate solution with great precision using a small number of basis functions. The convergence analysis of the approach is also provided. The accuracy and flawless performance of the proposed technique and verification of the theory are examined with some examples.

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

Numerical approximation of nonlinear stochastic Volterra integral equation based on Walsh function

Existence and stability of solution for a nonlinear Volterra integral equation with binary relation via fixed point results

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.

Investigation of the F* Algorithm on Strong Pseudocontractive Mappings and Its Application

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the F* iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the F* iterative algorithm converges strongly and faster than other current iterative schemes in the literature and extends to the fixed point of a strong pseudocontractive mapping. Finally, under a nonlinear quadratic Volterra integral equation, the application of our findings is shown.