Volterra Integro-differential Equations Research Articles

A developed technique of homotopy analysis method for solving nonlinear systems of Volterra integro-differential equations of fractional order

In this paper, a developed technique of homotopy analysis method (HAM) is presented to solve nonlinear system of fractional order Volterra integro-differential equations. The fractional derivative is described in the Caputo form. The main thing we have done in this paper is that we have simplified the process of evaluating the fractional order integrations in the deformation equations than the standard HAM. The proposed technique is tested by some illustrative examples in order to demonstrate its accuracy, applicability and efficiency.

A priori and a posteriori error analysis for a system of singularly perturbed Volterra integro-differential equations

A priori and a posteriori error analysis for a system of singularly perturbed Volterra integro-differential equations

A fast time domain solver for the equilibrium Dyson equation

We consider the numerical solution of the real-time equilibrium Dyson equation, which is used in calculations of the dynamical properties of quantum many-body systems. We show that this equation can be written as a system of coupled, nonlinear, convolutional Volterra integro-differential equations, for which the kernel depends self-consistently on the solution. As is typical in the numerical solution of Volterra-type equations, the computational bottleneck is the quadratic-scaling cost of history integration. However, the structure of the nonlinear Volterra integral operator precludes the use of standard fast algorithms. We propose a quasilinear-scaling FFT-based algorithm which respects the structure of the nonlinear integral operator. The resulting method can reach large propagation times and is thus well-suited to explore quantum many-body phenomena at low energy scales. We demonstrate the solver with two standard model systems: the Bethe graph and the Sachdev-Ye-Kitaev model.

The Uniform Convergence of a DG Method for a Singularly Perturbed Volterra Integro-Differential Equation

The Uniform Convergence of a DG Method for a Singularly Perturbed Volterra Integro-Differential Equation

Nonlinear viscoelastic damping in a Poynting oscillator

This work considers a cylindrical rod, acting as a nonlinear viscoelastic spring, with an attached mass that can displace along and rotate about the rod’s axis, thereby inducing an extension and twist in the rod. The coupling between them produces the Poynting effect in the body. The material is modeled by the nonlinear single integral Pipkin–Rogers constitutive equation. The mass has been assumed to be at rest in a long-time equilibrium state and then given a small axial or rotational disturbance and released (plucked). The subsequent motion is governed by a linear Volterra integro-differential equation. For a particular choice of material parameters, analytical expressions for the time-dependent decay of the disturbance are obtained in terms of the extension/torsion coupling, material stiffness, the amount and rate of stress relaxation, and the inertia of the mass. For other choices of material parameters, the results provide insight into how these quantities affect the time-dependent decay. A number of plucking scenarios are treated such as extensional or rotational disturbances from the undeformed state, rotational plucking after a finite axial stretch, and extensional plucking after a finite rotation. Simultaneous rotational and extensional plucking leads to a system of Volterra integro-differential equations whose treatment is deferred to later work.

On the convergence analysis of efficient numerical schemes for singularly perturbed second order Volterra integro-differential equations

On the convergence analysis of efficient numerical schemes for singularly perturbed second order Volterra integro-differential equations

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations.

On Stability and Asymptotic Stability of Solutions for a Linear Fourth Order Volterra Integro-Differential Equation on the Half-Axis

On Stability and Asymptotic Stability of Solutions for a Linear Fourth Order Volterra Integro-Differential Equation on the Half-Axis

Extended Laplace Power Series Method for Solving Nonlinear Caputo Fractional Volterra Integro-Differential Equations

In this paper, we compile the fractional power series method and the Laplace transform to design a new algorithm for solving the fractional Volterra integro-differential equation. For that, we assume the Laplace power series (LPS) solution in terms of power q=1m,m∈Z+, where the fractional derivative of order α=qγ, for which γ∈Z+. This assumption will help us to write the integral, the kernel, and the nonhomogeneous terms as a LPS with the same power. The recurrence relations for finding the series coefficients can be constructed using this form. To demonstrate the algorithm’s accuracy, the residual error is defined and calculated for several values of the fractional derivative. Two strongly nonlinear examples are discussed to provide the efficiency of the algorithm. The algorithm gains powerful results for this kind of fractional problem. Under Caputo meaning of the symmetry order, the obtained results are illustrated numerically and graphically. Geometrically, the behavior of the obtained solutions declares that the changing of the fractional derivative parameter values in their domain alters the style of these solutions in a symmetric meaning, as well as indicates harmony and symmetry, which leads them to fully coincide at the value of the ordinary derivative. From these simulations, the results report that the recommended novel algorithm is a straightforward, accurate, and superb tool to generate analytic-approximate solutions for integral and integro-differential equations of fractional order.

Uniform convergence analysis of the BDF2 scheme on Bakhvalov-type meshes for a singularly perturbed Volterra integro-differential equation

A singularly perturbed Volterra integro-differential problem is considered. At first, this problem is discretized by using the variable two-step backward differentiation formula (BDF2) and the trapezoidal formula on a Bakhvalov-type mesh to approximate the first-order derivative term and the integral term, respectively. Then, the stability and convergence analysis of the proposed numerical method are carried out. It is shown that the proposed numerical method is second-order uniformly convergent with respect to perturbation parameter ɛ in the discrete maximum norm. Finally, the theoretical find is illustrated by numerical experiments.

On stability analysis of a fractional volterra integro delay differential equation in the context of Mittag–Leffler kernel

This paper deals with a nonlinear multi-derivative fractional Volterra integro-differential equation with finite delay involving a non-singular Mittag–Leffler kernel. Krasnoselskii's fixed-point theorem is used to obtain the current result. Also, we establish the generalization of Pachapate's inequality and provide its applications for the existence, uniqueness, stability, and data dependence of the suggested problem. All the results are justified by an example.

Multistep Runge–Kutta methods for Volterra integro-differential equations

In this paper, we investigate multistep Runge–Kutta methods for Volterra integro-differential equations. First, we derive order conditions for methods of order p and stage order q=p−1 for p⩽4. Then, stability properties of these methods with respect to the basic and convolution test equations are analyzed. V0-stable methods of order 2 and 3 are obtained. Finally, some numerical experiments are given to verify the obtained theoretical results.

Stability inequalities and numerical solution for neutral Volterra delay integro-differential equation

The aim of the present paper is to introduce a new numerical method for solving Volterra integro-differential equation involving delay and neutral types. Using the numerical quadrature formula we form a finite difference scheme on a uniform mesh. The presented numerical method obtains a second-order convergence in discrete maximum norm. Furthermore, we illustrate the efficacy of the proposed method by constructing examples.

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

Exponential Stability of Integro-Differential Volterra Equation on Time Scales

Abstract We study the Volterra integro-differential equation on time scales and provide sufficient conditions for boundness of all solutions of considered equation. Using that result, we present the conditions for exponential stability of considered equation. All the results proved on the general time scale include results for both integral and discrete Volterra equations.

On the Method of Lyapunov Functionals for a Linear First-Order Volterra Integrodifferential Equation with Delay on the Semiaxis

On the Method of Lyapunov Functionals for a Linear First-Order Volterra Integrodifferential Equation with Delay on the Semiaxis

A Legendre spectral method for multidimensional partial Volterra integro-differential equations

We present a Legendre spectral method for solving multidimensional partial Volterra integro-differential equations. The main idea of our approach is first to employ some function transformations and variable transformations to transform the equations into new partial Volterra integro-differential equations, and then the Legendre spectral method is used to solve the equivalent equations. We derive error bounds in L∞- and L2-norms, which indicate that the errors of solution and the first order partial derivative decay exponentially. Three numerical examples are displayed to confirm the reliability of the Legendre spectral analysis.

Numerical Approximation of Volterra Integro-Differential Equations of the Second Kind Using Boole's Quadrature Rule Method

Numerical Approximation of Volterra Integro-Differential Equations of the Second Kind Using Boole's Quadrature Rule Method

Application of Effective Semi-Analytical Algorithms for Neutral and Retarded Volterra Integrodifferential Equations

Application of Effective Semi-Analytical Algorithms for Neutral and Retarded Volterra Integrodifferential Equations