Nonlinear Volterra Integral Equations Research Articles

SOME NEW STABILITY RESULTS OF VOLTERRA INTEGRAL EQUATIONS ON TIME SCALES

In this paper, we generalize two types of Volterra integral equations given on time scales and examine their Hyers-Ulam and Hyers-Ulam-Rassias stabilities. We also prove these stability results for the non-homogeneous nonlinear Volterra integral equation on time scales and provide an example to support these results. Moreover, we show that the general Volterra type integral equation given on time scales has the Hyers-Ulam-Rassias stability. Our results extend and improve some recent developments announced in the current literature.

Quasilinearization-based Legendre collocation method for solving a class of functional Volterra integral equations

In this paper, a combination of the quasilinearization and the Legendre spectral collocation methods is introduced to approximate the solution of the nonlinear functional Volterra integral equations. Throughout this process, the quasilinearization method converts the nonlinear functional Volterra integral equation to a sequence of linear integral equations. Then, in each iteration, the obtained linear integral equation is solved using the Legendre spectral collocation method. After that, a convergence analysis is discussed in detail. Finally, several numerical examples are included to demonstrate the capability and validity of the proposed method.

System of Non-Linear Volterra Integral Equations in a Direct-Sum of Hilbert Spaces

We use the contraction mapping theorem to present the existence and uniqueness of solutions in a short time to a system of non-linear Volterra integral equations in a certain type of direct-sum H[a; b] of a Hilbert space V[a; b]. We extend the local existence and uniqueness of solutions to the global existence and uniqueness of solutions to the proposed problem. Because the kernel function is a transcendental function in H[a; b] on the interval [a; b], the results are novel and very important in numerical approximation.

Some best proximity point results and an application to nonlinear integral equations on partial metric spaces

In this paper, using Bianchini-Grandolfi gauge functions, we define a new notion of BG-contraction mapping and proximal BG-contraction mapping. After that, we prove some best proximity point results. Hence, we generalize and improve results existing in the literature such as [12]. Also, we provide nontrivial example to demonstrate the effectiveness of our result. Finally, we give an existence and uniqueness result for the solution of nonlinear Freldhom and Volterra integral equations via our result.

A High-Order Approximate Solution for the Nonlinear 3D Volterra Integral Equations with Uniform Accuracy

In this paper, we present a high-order approximate solution with uniform accuracy for nonlinear 3D Volterra integral equations. This numerical scheme is constructed based on the three-dimensional block cubic Lagrangian interpolation method. At the same time, we give the local truncation error analysis of the numerical scheme based on Taylor’s theorem. Through theoretical analysis, we reach the conclusion that the optimal convergence order of this high-order numerical scheme is 4. Finally, we verify the effectiveness and applicability of the method through four numerical examples.

A numerical algorithm for a class of nonlinear fractional Volterra integral equations via modified hat functions

A numerical algorithm via a modified hat functions (MHFs) has been proposed to solve a class of nonlinear fractional Volterra integral equations of the second kind. A fractional-order operational matrix of integration is introduced. In a new methodology, the operational matrices of MHFs and the powers of weakly singular kernels of integral equations are used as a structure for transforming the main problem into a number of systems consisting of two equations for two unknowns. Relative errors for the approximated solutions are investigated. Convergence analysis of the proposed method is evaluated and convergence rate is addressed. Finally, the extraordinary accuracy of the utilized approach is illustrated by a few examples. The results, absolute and relative errors are illustrated in some tables and diagrams. In addition, a comparison is made between the absolute errors obtained by the proposed method and two other methods; one uses a hybrid approach and the other applies second Chebyshev wavelet.

Iterative Continuous Collocation Method for Solving Nonlinear Volterra Integral Equations

This paper is concerned with the numerical solution of nonlinear Volterra integral equations. The main purpose of this work is to provide a new numerical approach based on the use of continuous collocation Lagrange polynomials for the numerical solution of nonlinear Volterra integral equations. It is shown that this method is convergent. The results are compared with the results obtained by other well-known numerical methods to prove the effectiveness of the presented algorithm.

Moment bounds for a class of stochastic nonlinear fractional Volterra integral equations of the second kind

This paper studies and compares the second moment (Energy growth) bounds for solutions to a class of stochastic fractional Volterra integral equations of the second kind, under some Lipschitz continuity conditions on the parameters. The result shows that both solutions exhibit exponential growth but at different rates. The existence and uniqueness of the mild solutions are established via the Banach fixed point theorem.

Chelyshkov polynomials strategy for solving 2-dimensional nonlinear Volterra integral equations of the first kind

Chelyshkov polynomials strategy for solving 2-dimensional nonlinear Volterra integral equations of the first kind

Global stability of an influenza A model with vaccination

In this paper, we establish an influenza A epidemic model with vaccination and asymptomatic patient. The most important is that the method of the stability proof at the endemic equilibrium. We convert the model to a nonlinear Volterra integral equation, and prove the stability at the origin of this integral equation to obtain the local stability of the endemic equilibrium. Furthermore, combining with the uniqueness and the global attractiveness, we give the global stability of the endemic equilibrium. The method can be seen as a supplement for proving the global stability of the epidemic model, especially for the high-dimensional system.

Numerical solution of two-dimensional linear and nonlinear Volterra integral equations using Taylor collocation method

The main purpose of this work is to provide a numerical approach for two-dimensional Volterra integral equations (2D-VIEs). An algorithm based on the use of Taylor polynomials is developed to construct a collocation solution v∈Sp−1,p−1(−1)(ΠN,M) for approximating the solution of 2D-VIEs. It is shown that this algorithm is convergent. Some numerical examples are included to demonstrate the accuracy of the presented method.

The Numerical Solution of Volterra Integral Equation of Second Kind Using Quartic Non-Polynomial Spline Function

In this study, the non-polynomial spline function of fourth order is given to solve the second kind of the nonlinear Volterra integral equation. Numerical examples are show to clarify the applications of the proposed method and to compare the calculated results with other methods. In addition, there is a comparison of the absolute value of the error that estimated by of the suggested method is compared by those that estimated through other methods and the exact solution. The obtained results are very encouraging compared by other numerical methods.

A Higher-Order Numerical Scheme for Two-Dimensional Nonlinear Fractional Volterra Integral Equations with Uniform Accuracy

In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order O(hx4−α+hy4−β) for 0<α,β<1. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method.

Galerkin Approximation for Stochastic Volterra Integral Equations with Doubly Singular Kernels

This paper is concerned with the more general nonlinear stochastic Volterra integral equations with doubly singular kernels, whose singular points include both s=t and s=0. We propose a Galerkin approximate scheme to solve the equation numerically, and we obtain the strong convergence rate for the Galerkin method in the mean square sense. The rate is min{2−2(α1+β1),1−2(α2+β2)} (where α1,α2,β1,β2 are positive numbers satisfying 0<α1+β1<1, 0<α2+β2<12), which improves the results of some numerical schemes for the stochastic Volterra integral equations with regular or weakly singular kernels. Moreover, numerical examples are given to support the theoretical result and explain the priority of the Galerkin method.

On the solution of Volterra integral equations with decomposable kernel functions

We consider the solution of a class of nonlinear Volterra integral equations, x(t)=g(t)+∫⁡t0tk(t,s;x(s))ds, where t≥t0 and the kernel function k is finitely decomposable, and derive variation of parameters formulae that provide the solution of corresponding perturbed nonlinear equations. We attain this by relating the integral equations to a certain class of initial value problems for ODEs, for which variation of parameters are also formulated.

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays

Iterated collocation methods for nonlinear third-kind Volterra integral equations with proportional delays

Symmetric Spectral Collocation Method for a Kind of Nonlinear Volterra Integral Equation

In this paper, we develop an efficient spectral method for numerically solving the nonlinear Volterra integral equation with weak singularity and delays. Based on the symmetric collocation points, the spectral method is illustrated, and the convergence results are obtained. In the end, two numerical experiments are carried out to confirm the theoretical results.

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