Volterra Integro-differential Equations Research Articles

Numerical solution based on the Haar wavelet collocation method for partial integro-differential equations of Volterra type

In this paper, a numerical investigation of a class of parabolic Volterra integro-differential equations has been carried out. Basically, the finite difference method associated with the Haar wavelet collocation technique is pursued. The main idea relies on semi-discretizing the parabolic-VIDEs by considering the finite differences in time and approximating the integral term within the trapezoidal rule. Afterwards, using a uniform mesh, the spatial derivative is approximated by the Haar wavelet method. Stability and convergence of the proposed approach are established in the frame of Sobolev space. Numerical examples are used to illustrate the effectiveness of this approach under different test scenarios. The error analysis, using both L 2 and L ∞ norms, shows that the method has high efficiency computationally with a great deal of accuracy. It is observed that these numerical results are in excellent agreement with the analytical solution. In addition, the methods and results obtained in this study are compared with those reported in recent literature.

Numerical solutions for second-order neutral volterra integro-differential equations: Stability analysis and finite difference method

This work deals with the initial-value problem for a second-order neutral Volterra integro-differential equation. First, we give the stability inequality indicating stability of the problem with respect to the right-side and initial conditions. Further, we develop a finite difference method that uses for differential part second difference derivative, for the integral part appropriate composite trapezoidal and midpoint rectangle rules followed by second-order accurate difference quantities at intermediate points. Error estimate for the approximate solution is established. In support of theoretical results, numerical results are performed by employing the proposed numerical technique.

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Radial Kernels Collocation Method for the Solution of Volterra Integro-Differential Equations

Radial kernel interpolation is an advanced method in approximation theory for the construction of higher order accurate interpolants for scattered data up to higher dimensional spaces. In this manuscript, we formulate a radial kernel collocation approach for solving problems involving the Volterra integro-differential equations using two radial kernels: The Generalized Multi-quadrics and the linear Laguerre-Gaussians. This was achieved by simplifying the Volterra integral problem's solution to an algebraic system of equations. The impact of the shape parameter present in every kernel on the method's accuracy is examined and found to be significant. Two examples were used to illustrate the process; the numerical results are shown as tables and graphs. MATLAB 2018a was employed in the process.

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Representations of the solutions for volterra integro-differential equations in hilbert spaces

Numerical computational technique for solving Volterra integro-differential equations of the third kind using meshless collocation method

The primary goal of this study is to give an approximate algorithm for solving Volterra integro-differential equations (VIDEs) of the third kind using meshless collocation techniques. The basic framework of the novel approach is based on a collocation scheme and radial basis functions (RBFs) created on scattered points. This technique requires no background approximation cells, and the algorithm is powerful, has greater stability, and does not require much computer memory. This approach represents the solution of VIDEs of the third kind by interpolating the RBFs based on the Gauss–Legendre quadrature formula. The problem is reduced to a system of algebraic equations that can be easily solved. A description of the technique for the proposed equations is provided. Furthermore, the error analysis of this scheme is examined. A few numerical experiments are presented to prove the reliability and precision of the suggested approach for solving VIDEs of the third kind. Certain problems were compared with analytical solutions, the moving least squares method, and other methods to prove the effectiveness and applicability of the approach described.

Fifth-Kind Chebyshev Spectral Collocation Treatment for the Volterra Integro-Differential Equation of the Third Kind

In this research work, we offer an efficient spectral numerical scheme for handling the Volterra integro-differential of the third kind via Chebyshev polynomials of the fifth kind. The celebrated fundamentals and relations were stated with some details. With the aid of the spectral collocation method and the properties of Chebyshev polynomials, the singular Volterra integro-differential equation with its initial condition was transformed into a system of algebraic equations. The convergence and error analyses were discussed in depth. The validity and applicability of the method were tested and verified through three numerical examples, with the absolute errors reported in tables an graphs.

A Numerical Scheme of a Fractional Coupled System of Volterra Integro-Differential Equations with the Caputo Fabrizio Fractional Derivative

In this paper, we have developed a new computational scheme for solving coupled systems of fractional order Volterra-type integro-differential equation (FVIDE). We construct new operational matrices, which serve as building blocks for converting the FVIDE into a Sylvester-type algebraic structure that is more easily solvable. The fractional derivatives and their inverses (integrals) are considered in the Caputo-Fabrizio sense. This computational scheme is based on Legendre polynomials. The assessment of the proposed method’s convergence involves utilizing appropriate error norms to measure the level of convergence. The algorithm is designed in such a way that it can be simulated using any computational package; we have used Matlab for simulating the proposed scheme. Graphical visualization and tabulation of data are performed to confirm convergence and to analyze errors.

Study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments

In this paper, we give new results on the study of a class of fractional order non linear neutral abstract Volterra integro-differential equations with deviated arguments, considered in a Banach space. Using the method suggested by Hernández et al. (2010), which is based on the theory of resolvent operators for integral equations and fixed point theorems, we prove the main results on the existence and uniqueness of the mild solution of this type of problems which have not been studied before. We also give an example to which our main results are applicable.

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Numerical analysis of nonlinear Volterra integrodifferential equations for viscoelastic rods and plates

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

Representations of Solutions for Volterra Integro-Differential Equations in Hilbert Spaces

Euler wavelets method for optimal control problems of fractional integro-differential equations

This research introduces an efficient direct method for finding the numerical solution to optimal control problems involving linear and nonlinear fractional integro-differential equations using orthonormal Euler wavelets. Using orthonormal Euler wavelets, fractional integral, and product operational matrices have been obtained. By using these fractional operational matrices and collocation points, optimal control problems of fractional Volterra integro-differential equations have been reduced to a system of linear or non-linear equations. We establish the convergence analysis and error bound of the proposed method. Orthonormal Euler wavelets have been utilized to solve numerical examples and determine the approximate value of the cost function by approximating state and control functions. This has been done in order to verify the effectiveness of the proposed numerical technique.

On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations

On the convergence of Galerkin methods for auto-convolution Volterra integro-differential equations

Second-order numerical method for a neutral Volterra integro-differential equation

This paper is dedicated to obtaining an approximate solution for a neutral second-order Volterra integro-differential equation. Our method is the second-order accurate finite difference scheme on a uniform mesh. The error analysis is carried out and numerical results are given to support the proposed approach.

Finite Difference with Quintic B-Splines for Solving a system of nonlinear Volterra Integro-Differential Equations of integer order

This study presents a novel numerical approach for solving systems of nonlinear Volterra Integro-Differential Equations (VIDEs) of integer order by integrating the Finite Difference Method (FDM) with Quintic B-Splines for. The proposed method aims to address the challenges posed by the nonlinearity and integral terms inherent in VIDEs while providing enhanced accuracy and stability in the numerical solution. An error estimate of the approximate solution is proved. Finally, Some example are presented to illustrate the simplicity and the effectiveness of the propose method

Analytical Approaches for Computing Exact Solutions to System of Volterra Integro-Differential Equations

Analytical Approaches for Computing Exact Solutions to System of Volterra Integro-Differential Equations

An hp-version error estimate of spectral collocation methods for weakly singular Volterra integro-differential equations with vanishing delays

An hp-version error estimate of spectral collocation methods for weakly singular Volterra integro-differential equations with vanishing delays

A generic numerical method for treating a system of Volterra integro-differential equations with multiple delays and variable bounds

PurposeThis study aims to treat a novel system of Volterra integro-differential equations with multiple delays and variable bounds, constituting a generic numerical method based on the matrix equation and a combinatoric-parametric Charlier polynomials. The proposed method utilizes these polynomials for the matrix relations at the collocation points.Design/methodology/approachThanks to the combinatorial eligibility of the method, the functional terms can be transformed into the generic matrix relations with low dimensions, and their resulting matrix equation. The obtained solutions are tested with regard to the parametric behaviour of the polynomials with $\\alpha$, taking into account the condition number of an outcome matrix of the method. Residual error estimation improves those solutions without using any external method. A calculation of the residual error bound is also fulfilled.FindingsAll computations are carried out by a special programming module. The accuracy and productivity of the method are scrutinized via numerical and graphical results. Based on the discussions, one can point out that the method is very proper to solve a system in question.Originality/valueThis paper introduces a generic computational numerical method containing the matrix expansions of the combinatoric Charlier polynomials, in order to treat the system of Volterra integro-differential equations with multiple delays and variable bounds. Thus, the method enables to evaluate stiff differential and integral parts of the system in question. That is, these parts generates two novel components in terms of unknown terms with both differentiated and delay arguments. A rigorous error analysis is deployed via the residual function. Four benchmark problems are solved and interpreted. Their graphical and numerical results validate accuracy and efficiency of the proposed method. In fact, a generic method is, thereby, provided into the literature.

A new application of the reproducing kernel method for solving linear systems of fractional order Volterra integro-differential equations

In this article, a new implementation of the reproducing kernel method is presented for solving systems of fractional-order Volterra integro-differential equations. Unlike previous implementations, this method does not rely on the Gram-Schmidt process. The reproducing kernel method utilizes various components, including space, inner product, bases, and points. Furthermore, the system of fractional-order Volterra integro-differential equations involves Caputo’s fractional derivative and Volterra integral. However, when using the reproducing kernel method to solve these systems, challenges such as longer execution time and lower accuracy may arise compared to other methods. The present method has overcome these challenges with features such as easy implementation, high accuracy, and lower execution time.

AN EFFICIENT SPECTRAL METHOD FOR NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For Volterra integro-differential equations (VIDEs) with weakly singular kernels, their solutions are singular at the initial time. This property brings a great challenge to traditional numerical methods. Here, we investigate the numerical approximation for the solution of the nonlinear model with weakly singular kernels. Due to its characteristic, we split the interval and focus on the first one to save operation. We employ the corresponding singular functions as basis functions in the first interval to simulate its singular behavior, and take the Legendre polynomials as basis functions in the other one. Then the corresponding hp-version spectral method is proposed, the existence and uniqueness of solution to the numerical scheme are proved, the hp-version optimal convergence is derived. Numerical experiments verify the effectiveness of the proposed method.