System Of Volterra Integral Equations Research Articles

Series Solution for Single and System of Non-linear Volterra Integral Equations

In this paper, Taylor expansion has been used for solving non-linear Volterra integral equations (VIEs) of the second kind. This method allows us to overcome the difficulty caused by integrals and non-linearity; also, it has more precise and rapidly convergent to the exact solution. Two examples are presented for illustrate the performance of this method.

Using Bernstein multi-scaling polynomials to obtain numerical solution of Volterra integral equations system

In this paper, first, Bernstein multi-scaling polynomials (BMSPs), which are generalization of Bernstein polynomials (BPs), are introduced and some of their properties are explained. Then, a new method based on BMSPs to achieved numerical solution for system of nonlinear integral equations is proposed. The proposed method converted the system of integral equations to a nonlinear system. To evaluate the efficiency of the proposed method, some systems of nonlinear integral equations are solved, and their numerical solutions are compared with other similar methods.

Optimization Problem for the Integral Model of Developing Systems

This paper addresses an integral model of development a large electric power system using the example of the Unified Energy System of Russia. The model takes account of the age structure of the plants main equipment. Besides, generating equipment is divided into components depending on the types of energy resources used. The mathematical model is presented by a system of nonclassical Volterra integral equations of the first kind. One of the equations describes the balance between the total available capacity of the electric power system, commissioning of new equipment, and dismantling of obsolete one. The other equations define the shares of different types of power plants in the total composition of the electric power system equipment. Based on the considered model, we set a problem that searches the optimal lifetimes of electric power system capacities for a given demand for electricity. The optimality criterion is a functional reflecting cost of commissioning and operating the capacities. The specifics of the optimization problem are that the optimization parameter is in the lower limit of integration. An algorithm for solving of this optimal control problem numerically is developed. The influence of economic indices on the solution to the optimal control problem is studied. Calculations of the optimal development of the Unified Energy System of Russia until 2050 are carried out using real-life data.

Numerical solutions of system of two-dimensional Volterra integral equations via Legendre wavelets and convergence

The main target of the paper is solving a system of two-dimensional Volterra integral equations (2-DVIE) and receiving convincing solutions. We change the problem to a system of simpler equations by Legendre wavelets. By merging properties of wavelets to system and reducing the number of variables we build more powerful tools than some other methods. 3 examples and proof of convergence verify the applicability of the new numerical technique.

Convergence analysis of the product integration method for solving the fourth kind integral equations with weakly singular kernels

In this paper, we consider product integration method based on orthogonal polynomials to solve mixed system of Volterra integral equations of the first and second kind with weakly singular kernels. For investigation of the theoretical and numerical analysis of the mixed systems, the notions of the tractability index and ν-smoothing property are extended for a weakly singular Volterra integral operator. Convergence analysis of the product integration method is derived. Finally, the proposed method is illustrated by two examples, which confirm the theoretical prediction of the error estimation.

Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel

In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel; then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.

Задача Коши для системы интегральных уравнений типа Вольтерра, описывающей движение конечной массы самогравитирующего газа

Задача Коши для системы интегральных уравнений типа Вольтерра, описывающей движение конечной массы самогравитирующего газа

The Numerical Solutions of Systems of Nonlinear Integral Equations with the Spline Functions

The main goal of this work is to develop an effective technique for solving nonlinear systems of Volterra integral equations. The main tools are the cardinal spline functions on small compact supports. We solve a system of algebra equations to approximate the solution of the system of integral equations. Since the matrix for the algebraic system is nearly triangular, It is relatively painless to solve for the unknowns and an approximation of the original solution with high precision is accomplished. In order to enhance the accuracy, several cardinal splines are employed in the paper. Our schemes were compared with other techniques proposed in recent papers and the advantage of our method was exhibited with several numerical examples.

Semi-analytical solution of a McKean–Vlasov equation with feedback through hitting a boundary

In this paper, we study the nonlinear diffusion equation associated with a particle system where the common drift depends on the rate of absorption of particles at a boundary. We provide an interpretation of this equation, which is also related to the supercooled Stefan problem, as a structural credit risk model with default contagion in a large interconnected banking system. Using the method of heat potentials, we derive a coupled system of Volterra integral equations for the transition density and for the loss through absorption. An approximation by expansion is given for a small interaction parameter. We also present a numerical solution algorithm and conduct computational tests.

On the analytical solution of the two-phase Couette flow with wall transpiration

The two-phase Couette flow with transpiration through both walls is considered, where there is a constant blowing v0 at the lower wall and a corresponding suction at the upper wall. The interface between both fluids is initially flat and, hence, stays flat as it moves upward at the constant speed of the transpiration velocity v0. The corresponding initial value problem is subject to three dimensionless numbers consisting of the Reynolds number Re and the viscosity and density ratios, ϵ and γ. The solution is obtained by the unified transform method (Fokas method) in the form of an integral representation depending on initial and all boundary values including the Dirichlet and Neumann values at the interface. The unknown values at the moving interface are determined by a system of linear Volterra integral equations (VIEs). The VIEs are of the second kind with continuous and bounded kernels. Hence, the entire two-phase spatiotemporal 1 + 1 system has dimensionally reduced. The system of VIEs is solved via a standard marching method. For the numerical computation of the complex integral contours, a parameterized hyperbola is used. The influence of the dimensionless numbers Re, γ, and ϵ is studied exemplarily. The most notable effect results from ϵ that gives rise to a kink in the velocity at the moving interface. Both ratios, ϵ and γ, allow for very different flow regimes in each fluid phase such as nearly pure Couette flows and transpiration dominated flows with strongly curved velocity profiles. Those regimes are mainly determined by the effective Reynolds number in the respective phases.

Dimensional changes during shear without normal tractions (the Poynting effect) in nonlinear viscoelastic fiber-reinforced solids

When a rectangular block of a nonlinear material is subjected to a simple shearing deformation, specific normal tractions are required to ensure that the distances between the faces of the block, i.e. its dimensions, do not change. This work investigates the time-dependent dimensional changes during shear in the absence of these normal tractions (the Poynting effect) that occur in a block composed of an incompressible nonlinearly viscoelastic fiber-reinforced solid. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for viscoelasticity. This constitutive equation is used because (1) it exhibits the essential features of nonlinear viscoelasticity; (2) it is straightforward to include the material symmetry restrictions due to the reinforcing fibers. A system of nonlinear Volterra integral equations is formulated for the dimensional changes in the block. Numerical solutions are presented for the case when the standard reinforcing model for nonlinearly elastic fiber-reinforced materials is incorporated in the Pipkin–Rogers constitutive framework. The results illustrate how the time-dependent dimensional changes depend on the fiber orientation and the viscoelastic properties of the fibers relative to those of the matrix.

The asymptotic approximations to linear weakly singular Volterra integral equations via Laplace transform

In this paper, the asymptotic expansions for the solution about zero and infinity are formulated via Laplace transform for linear Volterra integral equation with weakly singular convolution kernel. The expansions about zero and infinity, as well as their Pade approximations, are used to approximate the solution when the argument is small and large, respectively, and the Pade approximations are more accurate. The methods are also valid to solve some other Volterra type integral equations including linear Volterra integro-differential equations, fractional integro-differential equations, and system of singular Volterra integral equations of the second kind with convolution kernels. Some examples confirm the correctness of the methods and the effectiveness of the asymptotic expansions. They show that numerical methods are only necessary in a small interval in practical computation when uniform high precision evaluations are needed for solving these kinds of Volterra integral equations.

A new code for Volterra integral equations based on natural Runge-Kutta methods

We describe some issues related to the development of a new code for numerical solution of systems of Volterra integral equations of the second kind. This code is based on A- and V0-stable natural Volterra Runge-Kutta method of order p=4 implemented in a variable stepsize environment. This method was derived by Conte et al. (2014) [13]. The numerical experiments on many equations and systems of equations illustrate that the code achieves the expected order of accuracy, and confirm its efficiency and robustness.

Chebyshev spectral collocation method for system of nonlinear Volterra integral equations

We investigate Chebyshev spectral collocation method for system of nonlinear Volterra integral equations. We choose Chebyshev Gauss points as collocation points, and approximate integral terms by Legendre Gauss quadrature formula. The provided convergence analysis shows that numerical errors decay exponentially, which is one of the most prominent features of spectral methods. Numerical experiments are carried out to confirm theoretical results. We have never seen any paper investigating the spectral method for the system of nonlinear Volterra integral equations (VIEs). The present method and corresponding convergence analysis would be useful in studying numerical methods for the system of integral equations and partial differential equations.

Collocation methods for integro-differential algebraic equations with index 1

Abstract The notion of the tractability index based on the $\nu $-smoothing property of a Volterra integral operator is introduced for general systems of linear integro-differential algebraic equations (IDAEs). It is used to decouple the given IDAE system of index $1$ into the inherent system of regular Volterra integro-differential equations (VIDEs) and a system of second-kind Volterra integral equations (VIEs). This decoupling of the given general IDAE forms the basis for the convergence analysis of the two classes of piecewise polynomial collocation methods for solving the given index-$1$ IDAE system. The first one employs the same continuous piecewise polynomial space $S_m^{(0)}$ for both the VIDE part and the second-kind VIE part of the decoupled system. In the second one the VIDE part is discretized in $S_m^{(0)}$, but the second-kind VIE part employs the space of discontinuous piecewise polynomials $S_{m - 1}^{(- 1)}$. The optimal orders of convergence of these collocation methods are derived. For the first method, the collocation solution converges uniformly to the exact solution if and only if the collocation parameters satisfy a certain condition. This condition is no longer necessary for the second method; the collocation solution now converges to the exact solution for any choice of the collocation parameters. Numerical examples illustrate the theoretical results.

Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind

In this paper, optimal homotopy asymptotic method (OHAM) and its implementation on subinterval, called multistage optimal homotopy asymptotic method (MOHAM), are presented for solving linear and nonlinear systems of Volterra integral equations of the second kind. To illustrate these approaches two examples are presented. The results confirm the efficiency and ability of these methods for such equations. The results will be compared to find out which method is more accurate. Advantages of applying MOHAM are also illustrated.

An Analytical Approximate Solution of Linear, System of Linear and Non Linear Volterra Integral Equations Using Variational Iteration Method

In this paper we have solved Linear Volterra Integral Equations (LVIE), System of Linear Volterra Integral Equation (SLVIE) and non Linear Volterra Integral Equation (NLVIE) using variational Iteration Method (VIM). VIM is a powerful tool to solve the differential equations as well as integral equations. VIM gives consecutive approximations which converge rapidly to the exact solution in the given domain for the given initial condition, if it exists, without any transformation or any restrictive assumptions to non linear terms, which may change the physical behaviour of the problem, otherwise few approximations are enough only for numerical purposes. To exhibit the ability, accuracy and reliability of the method we have provided two problems of Linear Volterra Integral Equations, one problem of system of Linear Volterra Integral Equation and one problem of Non Linear Volterra Integral Equation. The study highlights the heaviness of the method.

Global dynamics for a class of infection-age model with nonlinear incidence

In this paper, we propose an HBV viral infection model with continuous age structure and nonlinear incidence rate. Asymptotic smoothness of the semi-flow generated by the model is studied. Then we caculate the basic reproduction number and prove that it is a sharp threshold determining whether the infection dies out or not. We give a rigorous mathematical analysis on uniform persistence by reformulating the system as a system of Volterra integral equations. The global dynamics of the model is established by using suitable Lyapunov functionals and LaSalle's invariance principle. We further investigate the global behaviors of the HBV viral infection model with saturation incidence through numerical simulations.

The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

The exact solutions of a system of linear weakly singular Volterra integral equations (VIE) have been a difficult to find. The aim of this paper is to apply reproducing kernel Hilbert space (RKHS) method to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential equations (LODEs). The approximate solutions are represent in the form of series in the reproducing kernel space . By comparing with the exact solutions of two examples, we saw that RKHS is a powerful, easy to apply and full efficiency in scientific applications to build a solution without linearization and turbulence or discretization.

On Sinc discretization for systems of Volterra integral-algebraic equations

Integral-algebraic equations (IAEs) are coupled systems of Volterra integral equations of the first and second kind which naturally arise in many applications in mathematical physics. In this paper, we solve the IAEs of index-1 by Sinc-collocation discretization and prove that the discrete solutions converge to the true solutions of the IAEs exponentially. The discrete solutions are determined by linear systems which can be effectively solved by suitable iteration methods. Numerical examples are given to illustrate the effective performance of our method.