Nonlinear Volterra Integro-differential Equations Research Articles

A new algorithm for the solution of nonlinear two-dimensional Volterra integro-differential equations of high-order

In this manuscript, a new numerical approach is developed for approximating the solution of the one-dimensional and two-dimensional Volterra integro-differential equations (2D-VIDEs) of high-order. New operational matrices of integration, based on shifted Jacobi polynomials, are presented and used in combination with the collocation method to convert the 2D-VIDE into a system of algebraic equations. In addition, the convergence analysis of the suggested approach is investigated. The efficiency of the algorithm is demonstrated by means of several examples and the results are compared with those given using other numerical schemes.

ON FRACTIONAL VOLTERRA INTEGRODIFFERENTIAL EQUATIONS WITH FRACTIONAL INTEGRABLE IMPULSES

We consider a class of nonlinear fractional Volterra integrodifferential equation with fractional integrable impulses and investigate the existence and uniqueness results in the Bielecki’s normed Banach spaces. Further, Bielecki-Ulam type stabilities have been demonstrated on a compact interval. A concrete example is provided to illustrate the outcomes we acquired.

Uniformly Boundedness in Nonlinear Volterra Integro-Differential Equations with Delay

Uniformly Boundedness in Nonlinear Volterra Integro-Differential Equations with Delay

An hp-Version Chebyshev Spectral Collocation Method for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels

An hp-Version Chebyshev Spectral Collocation Method for Nonlinear Volterra Integro-Differential Equations with Weakly Singular Kernels

Theoretical and numerical analysis for Volterra integro-differential equations with Itô integral under polynomially growth conditions

In this paper, we theoretically and numerically deal with nonlinear Volterra integro-differential equations with Itô integral under a one-sided Lipschitz condition and polynomially growth conditions. It is proved that both the exact solutions and vector fields are bounded and satisfy a Hölder condition in the pth moment sense. Analogously, the boundedness and Hölder condition in the pth moment sense are preserved by the semi-implicit Euler method for sufficiently small step-size. Moreover, by the local truncated errors, we prove the strong convergence order 1. Finally, numerical simulations on stochastic control models and stochastic Ginzburg–Landau equation illustrate our results.

Solutions of Volterra integral and integro-differential equations using modified Laplace Adomian decomposition method

Abstract In this paper, an effectual and new modification in Laplace Adomian decomposition method based on Bernstein polynomials is proposed to find the solution of nonlinear Volterra integral and integro-differential equations. The performance and capability of the proposed idea is endorsed by comparing the exact and approximate solutions for three different examples on Volterra integral, integro-differential equations of the first and second kinds. The results shown through tables and figures demonstrate the accuracy of our method. It is concluded here that the non orthogonal polynomials can also be used for Laplace Adomian decomposition method. In addition, convergence analysis of the modified technique is also presented.

An iterative multistep kernel based method for nonlinear Volterra integral and integro-differential equations of fractional order

An iterative multistep kernel based method is proposed for the nonlinear Volterra integral equations and nonlinear Volterra integro-differential equations of fractional order, which can produce reliable globally smooth numerical solutions. An error estimate of the positive definite kernel interpolation and, the convergence and error analysis of the proposed iterative scheme are investigated. Here, we focused on positive definite radial basis kernels and further, a new and applicable shape parameter selection strategy is proposed. The proposed multi-step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long-time simulations. In order to show the efficiency and versatility of the proposed method, some numerical experiments are reported. The comparison of the numerical results with the analytical solutions and the best-reported results in the literature confirm the good accuracy and applicability of the proposed method.

Fifth Order Multistep Block Method for Solving Volterra Integro-Differential Equations of Second Kind

In the present paper, the multistep block method is proposed to solve the linear and non-linear Volterra integro-differential equations (VIDEs) of the second kind using constant step size. The proposed block method of order five consists of two point block method presented as in the simple form of Adams Moulton type. The numerical solutions are obtained at two new values simultaneously at each of the integration step. In VIDEs, the unknown function appears in the form of derivative and under the integral sign. The approximation of the integral part is estimated using the Boole’s quadrature rule. The stability region is shown, and the numerical results are presented to illustrate the performance of the proposed method in terms of accuracy, total function calls and execution times compared to the existing method.

A note on the qualitative analysis of Volterra integro-differential equations

The aim of paper is to analyse some qualitative properties of solutions of nonlinear Volterra integro-differential equations (VIDDEs) with constant delay and Volterra integro-differential equations (VIDEs) without delay by means of the Razumikhin method. A suitable Lyapunov function is defined and then applied to that IDEs such that some former results can be obtained under weaker conditions and additionally some new results are given on the qualitative properties of that IDEs such as uniformly stability and integrability of solutions. For the VIDDEs with constant delay, the improved less conservative conditions of the stability, asymptotically stability and the boundedness of solutions and the new conditions of uniformly stability, integrability of solutions are all derived subject to the functions appeared in the considered IDEs. Moreover, the established stability, asymptotically stability and boundedness conditions of VIDDEs with constant delay simplify, extend and improve some previous works can be found in the literature and remove some unnecessary conditions. Finally, the validity of the presented results is indicated by some numerical examples using MATLAB-Simulink.

Convergence analysis of spectral methods for high-order nonlinear Volterra integro-differential equations

In this paper, we propose and analyze Chebyshev spectral collocation approximation for high-order nonlinear Volterra integro-differential equations. Under reasonable assumptions on the nonlinearity, it is shown that this numerical method converges exponentially in both $$L^\infty $$ -norm and $$L^2$$ -norm. Numerical results of several test examples are presented and comparisons are made with some existing numerical methods to prove the superiority and the effectiveness of the proposed method.

The combined reproducing kernel method and Taylor series for handling nonlinear Volterra integro-differential equations with derivative type kernel

The reproducing kernel method is applied to Volterra nonlinear integro-differential equations. In this technique, the nonlinear term is replaced by its Taylor series. The exact solution is represented in the form of series in the reproducing Hilbert kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions. Some numerical examples are solved in two different spaces and parameters of n. Measurements of the experimental data is an indications of stability and convergence on the reproducing kernel.

Multiple Perturbed Collocation Tau Method for Solving Nonlinear Integro-Differential Equations

The purpose of the study was to investigate the numerical solution of non-linear Fredholm and Volterra integro-differential equations by the proposed method called Multiple Perturbed Collocation Tau Method (MPCTM). We assumed a perturbed approximate solution in terms of Chebyshev polynomial basis function and then determined the derivatives of the perturbed approximate solution which are then substituted into the special classes of the problems considered. Thus, resulting into n-folds integration, the resulting equation is then collocated at equally spaced interior points and the unknown constants in the approximate solution are then obtained by Newton’s method which are then substituted back into the approximate solution.Illustrative examples are given to demonstrate the efficiency, computational cost and accuracy of the method. The results obtained with some numerical examples are compared favorable with some existing numerical methods in literature and with the exact solutions where they are known in closed form.Keywords: Nonlinear Problems, Tau Method, Integro-Differential, Newton’s method.

Convergence analysis of fitted numerical method for a singularly perturbed nonlinear Volterra integro-differential equation with delay

In this paper, the initial value problem for a quasilinear singularly perturbed delay Volterra integro-differential equation was considered. By the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form, a fitted difference scheme is constructed and analysed. It is shown that the method displays first order uniform convergence in perturbation parameter. Some numerical results are given to confirm the theoretical analysis.

Numerical solution of the general Volterra nth-order integro-differential equations via variational iteration method

In this paper, we use the variational iteration method (VIM) to construct approximate solutions for the general [Formula: see text]th-order integro-differential equations. We show that his method can be effectively and easily used to solve some classes of linear and nonlinear Volterra integro-differential equations. Finally, some numerical examples with exact solutions are given.

Nonlocal Boundary Value Problem for a Nonlinear Fredholm Integro-Differential Equation with Degenerate Kernel

We obtain a criterion for the unique solvability of a nonlocal boundary value problem for a second-order nonlinear Volterra integro-differential equation with degenerate kernel. Several informative examples are given.

Boundedness and asymptotic stability of nonlinear Volterra integro-differential equations using Lyapunov functional

In this paper, I consider Lyapunov functionals combined with the Laplace transform to obtain boundedness results regarding the solutions of the nonlinear Volterra integro-differential equationsx′(t)=A(t)x(t)+B(t)+∫0tC(t,s)f(x(s))ds+g(x(t)).Asymptotic stability results regarding the zero solution are carried out for the case where B(t) is identically zero. Numerical examples are proposed to perform the given results.

Systems of nonlinear Volterra integro-differential equations of arbitrary order

In this paper, a new approximate method for solving of systems of nonlinear Volterra integro-differential equations of arbitrary (integer and fractional) order is introduced. For this purpose, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) based on the classical Chebyshev polynomials of the first kind has been introduced that can be used to obtain the solution of the integro-differential equations (IDEs). Also, we construct the fractional derivative operational matrix of order $\alpha$ in the Caputo's definition for GFCFs. This method reduced a system of IDEs by collocation method into a system of algebraic equations. Some examples to illustrate the simplicity and the effectiveness of the propose method have been presented.

Stability of nonlinear Volterra equations and applications

General nonlinear Volterra integro-differential equations are considered. Explicit criteria for uniform asymptotic stability and exponential stability of such equations are given. Applications to models of growth of biological populations and of grazing systems are presented.

An efficient algorithm for solving Volterra integro-differential equations based on Alpert’s multi-wavelets Galerkin method

This paper is concerned with the application of the multi-wavelets Galerkin method to linear and nonlinear second order Volterra integro-differential equations. Using the operational matrices of integration and wavelet transform matrix, we reduce the problem to a set of algebraic equations. For linear type, we use thresholding to reduce the computational effort. We demonstrate that our solutions may be efficiently represented in a multi-wavelets basis, due to flexible vanishing moments property of this type of multi-wavelets. The L 2 convergence of the scheme for the proposed equation has been investigated. A series of numerical tests is provided to demonstrate the validity and applicability of the technique.

New qualitative criteria for solutions of Volterra integro-differential equations

In this paper, we consider certain non-linear scalar Volterra integro-differential equations and Volterra integro-differential systems of first order. We investigate the boundedness, stability, uniformly asymptotic stability, integrability and square integrability of solutions to the scalar equations and the system considered. The technique used to prove the results of the paper is based on the second method of Lyapunov. From the obtained results, we extend and improve some related results that can be found in the literature.