Nonlinear Volterra Integral Equations Research Articles

Efficient Collocation Algorithm for Solving Three-Dimensional Linear and Nonlinear Volterra Integral Equations

Efficient Collocation Algorithm for Solving Three-Dimensional Linear and Nonlinear Volterra Integral Equations

Meshless method and shifted Chebyshev polynomials for solving a class of VIEs of the third kind

The main goal of this study is to solve a class of linear and nonlinear Volterra integral equations (VIEs) of the third kind. The proposed technique estimates the solution of these equations using a technique based on MLS approximation and shifted Chebyshev polynomials. The approach is independent of the geometry of the domain and does not require any background meshes; therefore, it can be considered a meshless approach. The new method is effective and more flexible for many types of VIEs of the third kind, and its algorithm can be simply implemented on computers. The construction of a new technique for the proposed equations has been introduced. The convergence analysis is also studied. The convergence precision of the new approach is checked on classes of VIEs of the third kind, and the results validate the theoretical error estimates. Finally, numerical tests are provided to illustrate the accuracy and reliability of this approach.

Meshless Technique Based on Moving Least Squares Approximation for Numerical Solutions of Linear and Nonlinear Third-Kind VIEs

The principal objective of this paper is to propose a numerical scheme to solve linear and nonlinear third-kind Volterra integral equations (VIEs). The method approximates the solution using the collocation method based on moving least squares (MLS) approximation. The suggested scheme is meshless, since it needs no background mesh or cell structures, and is thus independent of the geometry of the domain. The approach reduces the solution of third-kind VIEs to the solution of systems of algebraic equations. All integrals appearing in the approach are calculated approximately by the Gauss–Legendre integration formula. The convergence analysis of the proposed scheme has been conducted, and its performance has been tested through a series of numerical examples, demonstrating its effectiveness and applicability in comparison to the existing methods.

Computational Analysis of a Novel Iterative Scheme with an Application

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory.

Numerical solution for a generalized form of nonlinear cordial Volterra integral equations using quasilinearization and Legendre-collocation methods

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported.

([formula omitted])-contractive and ([formula omitted])-contractive mapping based fixed point theorems in fuzzy bipolar metric spaces and application to nonlinear Volterra integral equations

In this paper, we introduce some novel concepts within the realm of fuzzy bipolar metric spaces, namely (αη)-contractive type covariant mappings and contravariant mappings, and (βχ)-contractive type covariant mappings. We establish some fixed point theorems that demonstrate both the existence and uniqueness of fixed points for (αη)-contractive type covariant mappings and contravariant mappings, and for (βχ)-contractive type covariant mappings in complete fuzzy bipolar metric spaces utilizing the triangular property. Additionally, to substantiate the findings, some illustrative examples and consequential outcomes are presented. Furthermore, the proven results serve to extend, generalize, and enhance the corresponding outcomes documented in existing literature. A practical application of these findings in the context of non-linear Volterra integral equations is demonstrated, solidifying and reinforcing the credibility of the established results. Overall, this paper contributes to the understanding of fixed point theory in the context of fuzzy bipolar metric spaces and highlights the significance of (αη)-contractive mappings and (βχ)-contractive mappings in this domain.

Homotopy analysis method for solving fuzzy Nonlinear Volterra integral equation of the second kind

Homotopy analysis method for solving fuzzy Nonlinear Volterra integral equation of the second kind

Numerical solution of fuzzy stochastic Volterra integral equations with constant delay

This paper considers the nonlinear fuzzy stochastic Volterra integral equations with constant delay, which are general and include many fuzzy stochastic integral and differential equations discussed in literature. Since Doob's martingale inequality is no longer applicable to such equations, a new maximum inequality is obtained. Combining with the Picard approximation method, the existence and uniqueness of solutions to nonlinear fuzzy stochastic Volterra integral equations with constant delay are given. Moreover we prove that the solution behaves stably in the case of small changes of initial values, kernels and nonlinearities. We further develop a Euler-Maruyama (EM) scheme and prove the strong convergence of the scheme. It is shown that the strong convergence order of the EM method is 0.5 under Lipschitz condition. Moreover, the strong superconvergence order is 1 if further, the kernel h(t,s) of the stochastic term satisfies h(t,t)=0. Numerical examples demonstrated that the numerical results are consistent with the theoretical research conclusions. Furthermore, the application model of the fuzzy stochastic Volterra integral equation with constant delay in population dynamics is considered, and the exact solution of the numerical example is given in explicit form.

On the regularity of solutions to a class of nonlinear Volterra integral equations with singularities

We study the smoothness properties of solutions to nonlinear Volterra integral equations of the second kind on a bounded interval [0,b]. The kernel of the integral operator of the underlying equation may have a diagonal singularity and a boundary singularity. Information about them is given through certain estimates. To characterize the regularity of solutions of such equations we show that the solution belongs to an appropriately weighted space of smooth functions on (0,b], with possible singularities of the derivatives of the solution at the left endpoint of the interval [0,b].

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

<p>The paper introduces an iterative method for solving nonlinear Volterra integral equations and analyzes its convergence, stability, and application through examples. It expresses the general nonlinear Volterra integral equation as a series and decomposes the nonlinear operator to derive a recursive formula for the proposed iterative method. The method ensures absolute and uniform convergence, with stability analysis conducted to ensure bounded errors in the presence of perturbations. Convergence analysis utilizes the Lipschitz condition, demonstrating the uniform convergence of the solution series. Illustrative examples, including power nonlinearity and trigonometric functions, validate the stability and convergence of the method. Through graphical representations, convergence analyses for specific integral equations demonstrate the method’s effectiveness and applicability in solving diverse nonlinear integral equations. Overall, the paper contributes a robust iterative method with insights into its stability and convergence properties, supported by practical examples.</p>

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