Second-order Volterra Integro-differential Equations Research Articles

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.

ОБ АСИМПТОТИЧЕСКОЙ УСТОЙЧИВОСТИ РЕШЕНИЙ ЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА В СЛУЧАЕ НЕДИФФЕРЕНЦИРУЕМОСТИ СРЕЗАННЫХ ФУНКЦИЙ

Sufficient conditions are established for the asymptotic stability of any solution of a linear second-order Volterra integro-differential equation in the case when the truncated functions may be non-differentiable at some points of the semiaxis. To do this, first, using a non-standard method of reduction to a system, the equation under consideration is reduced to an equivalent system, and the method of transforming equations and the method of cutting functions are applied to this system. The truncated kernel and free term are associated with the coefficient of the new unknown function using the inequality between the geometric and arithmetic means of two functions. Using the method of integral inequalities, sufficient conditions are established for the boundedness on the semiaxis of all solutions of the system. Then the method of squaring equations is used to the first equation of the system and the Lyusternik-Sobolev lemma is applied. Finally, based on a non-standard substitution, we complete the proof of the asymptotic stability of solutions to a given second-order linear integro-differential equation. An illustrative example is given.

ОБ ОЦЕНКЕ СНИЗУ РЕШЕНИЙ СЛАБО НЕЛИНЕЙНОГО ВОЛЬТЕРРОВА ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ВТОРОГО ПОРЯДКА С ЗАПАЗДЫВАНИЯМИ

Sufficient conditions are established for lower bounds and tending to infinity as the argument of solutions to a weakly non-linear second-order Volterra integro-differential equation with delays on the half-axis grows unbounded. For this purpose, a non-standard method of reduction to a system, the method of transforming the equations of V. Volterra, the method of partial cutting, the method of integral inequalities with delays and the method of estimating solutions from below Yu.A. Ved are developed. It is shown that solutions with initial data from a certain manifold have lower bounds, and such solutions tend to infinity according to the exponential law with unlimited growth of the argument. This article proposes a technique for solving the problem of estimating from below on the half-line solutions of a weakly nonlinear Volterra-type integro-differential equation of the second order with delays, while the main role is played by the non-standard method of reduction to the system and the method of estimating solutions from below Yu.A.Ved (1965).

An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

An $hp$-Version of $C^0$ -Continuous Petrov-Galerkin Time-Stepping Method for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Analysis of Volterra integrodifferential equations with nonlocal and boundary conditions via Picard operator

This article investigates the existence and uniqueness of solutions to the second-order Volterra integrodifferential equations with nonlocal and boundary conditions through its equivalent integral equations and fixed point of Banach. Furthermore, utilizing the Picard operator theory, we obtain the dependence of solutions on the initial nonlocal data and on functions involved on the right-hand side of the equations.

Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Spectral Collocation Methods for Second-Order Volterra Integro-Differential Equations with Weakly Singular Kernels

Linear barycentric rational collocation method for solving second-order Volterra integro-differential equation

Second-order Volterra integro-differential equation is solved by the linear barycentric rational collocation method. Following the barycentric interpolation method of Lagrange polynomial and Chebyshev polynomial, the matrix form of the collocation method is obtained from the discrete Volterra integro-differential equation. With the help of the convergence rate of the linear barycentric rational interpolation, the convergence rate of linear barycentric rational collocation method for solving Volterra integro-differential equation is proved. At last, several numerical examples are provided to validate the theoretical analysis.

An hp-version of the C0-continuous Petrov-Galerkin method for second-order Volterra integro-differential equations

We present an hp-version of the C0-continuous Petrov-Galerkin method for linear second-order Volterra integro-differential equations. We combine the continuous and discontinuous Galerkin methodologies to obtain a natural discretization of the second-order derivative of arbitrary order. We derive a-priori error bounds in the L2- and H1-norm that are completely explicit in the local time steps, local approximation orders, and local regularity of the exact solution. Numerical experiments are provided to illustrate the theoretical results.

Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments

In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton–Jacobi–Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price.

HEREDITARY EFFECTS OF EXPONENTIALLY DAMPED OSCILLATORS WITH PAST HISTORIES

This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time.

Application of reproducing kernel Hilbert space method for solving second-order fuzzy Volterra integro-differential equations

In this article, we propose a new method that determines an efficient numerical procedure for solving second-order fuzzy Volterra integro-differential equations in a Hilbert space. This method illustrates the ability of the reproducing kernel concept of the Hilbert space to approximate the solutions of second-order fuzzy Volterra integro-differential equations. Additionally, we discuss and derive the exact and approximate solutions in the form of Fourier series with effortlessly computable terms in the reproducing kernel Hilbert space W_{2}^{3} [ a,b ] oplus W_{2}^{.3} [ a,b ]. The convergence of the method is proven and its exactness is illustrated by three numerical examples.

On Minimizing the Ultimate Ruin Probability of an Insurer by Reinsurance

We consider an insurance company whose reserves dynamics follow a diffusion-perturbed risk model. To reduce its risk, the company chooses to reinsure using proportional or excess-of-loss reinsurance. Using the Hamilton-Jacobi-Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation (VIDE) which we transform into a linear Volterra integral equation (VIE) of the second kind. We then proceed to solve this linear VIE numerically using the block-by-block method for the optimal reinsurance policy that minimizes the ultimate ruin probability for the chosen parameters. Numerical examples with both light- and heavy-tailed distributions are given. The results show that proportional reinsurance increases the survival of the company in both light- and heavy-tailed distributions for the Cramér-Lundberg and diffusion-perturbed models.

Existence results for some damped second-order Volterra integro-differential equations

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.

The stability of collocation methods for VIDEs of second order

Simplest results presented here are the stability criteria of collocation methods for the second‐order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with |λ| = 1 belong to a 1 × 1 Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure.

Collocation methods for second-order volterra integro-differential equations

We study the numerical solution of second-order Volterra integro-differential equations by means of collocation techniques in certain polynomial spline spaces. Suitable discretization of the resulting collocation equation yields implicit methods which may be viewed as extensions of m-stage implicit Runge-Kutta-Nyström methods for initial-value problems of second-order ordinary differential equations to second-order integro-differential equations. The attainable order of (local) convergence of these methods is analyzed in detail.

Implicit Runge-Kutta-Nyström methods for general second-order Volterra integro-differential equations

In this paper we derive m-stage Runge-Kutta-Nyström methods for the numerical solution of general second-order Volterra integro-differential equations. These implicit methods are based on collocation techniques in certain polynomial spline spaces. The question concerning the attainable order of convergence of such one-step methods is completely answered.

Asymptotic method in the theory of second-order volterra integrodifferential equations

Asymptotic method in the theory of second-order volterra integrodifferential equations