System Of Volterra Integral Equations Research Articles

Regularization of Solutions of One Class of Systems of Linear Volterra Integral Equations of the First Kind in the Space of Differentiable Vector Functions on the Axis

Regularizing operators according to Lavrentiev are constructed and sufficient conditions for the uniqueness of solutions to systems of linear Volterra integral equations of the first kind on the axis are established.

Stability of solutions of systems of Volterra integral equations

In this paper, we propose new sufficient conditions for stability of solutions of systems of Volterra linear integral equations and systems of linear integro-differential Volterra equations. Solution stability conditions for systems of Volterra linear integral equations are studied with perturbed right hand sides of the equations. These new sufficient conditions are expressed in terms of logarithmic norms of matrices consisting of coefficients of equations and derivatives of their kernels. For Volterra integro-differential equations we also provide sufficient conditions for stability of solutions when the initial conditions are perturbed.

Piecewise polynomial numerical method for Volterra integral equations of the fourth-kind with constant delay

This work studies the fourth-kind integral equation as a mixed system of first and second-kind Volterra integral equations (VIEs) with constant delay. Regularity, smoothing properties and uniqueness of the solution of this mixed system are obtained by using theorems which give the relevant conditions related to first and second-kind VIEs with delays. A numerical collocation algorithm making use of piecewise polynomials is designed and the global convergence of the proposed numerical method is established. Some typical numerical experiments are also performed which confirm our theoretical result.

Inverse problem for the system of viscoelasticity in anisotropic media with tetragonal form of elasticity modulus

For the reduced canonical system of integro-differential equations of viscoelasticity, direct and inverse problems of determining the velocity field of elastic waves and the relaxation matrix are considered. The problems are replaced by a closed system of Volterra integral equations of the second kind with respect to the Fourier transform in the variables $x_{1}$ and $x_{2}$ for the solution of the direct problem and unknowns of the inverse problem. Further, the method of contraction mappings in the space of continuous functions with a weighted norm is applied to this system. Thus, we prove global existence and uniqueness theorems for solutions of the problems.

The barycentric rational predictor-corrector schemes for Volterra integral equations

This paper introduces a family of barycentric rational predictor-corrector schemes based on the Floater–Hormann family of linear barycentric rational interpolants (LBRIs) for the numerical solution of classical systems of second-kind Volterra integral equations. Also, we introduce a family of LBRI-based predictor-corrector starting procedures that is essentially explicit and whose order of convergence can be as high as that of the main method. Numerical tests verify the theoretical results on the convergence order and stability and illustrate the efficiency and power of the developed family of methods in solving stiff equations.

A general collocation analysis for weakly singular Volterra integral equations with variable exponent

Abstract Piecewise polynomial collocation of weakly singular Volterra integral equations (VIEs) of the second kind has been extensively studied in the literature, where integral kernels of the form $(t-s)^{-\alpha }$ for some constant $\alpha \in (0,1)$ are considered. Variable-order fractional-derivative differential equations currently attract much research interest, and in Zheng & Wang (2020, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes. SIAM J. Numer. Anal., 58, 330–352), such a problem is transformed to a weakly singular VIE whose kernel has the above form with variable $\alpha = \alpha (t)$, then solved numerically by piecewise linear collocation, but it is unclear whether this analysis could be extended to more general problems or to polynomials of higher degree. In the present paper, the general theory (existence, uniqueness, regularity of solutions) of variable-exponent weakly singular VIEs is developed using novel techniques. These results then underpin an error analysis of collocation methods where piecewise polynomials of any degree can be used. This error analysis is also novel—it makes no use of the usual resolvent representation, which is a key technique in the error analysis of collocation methods for VIEs in the current research literature. Furthermore, all the above analysis for a scalar VIE can be extended to certain nonlinear VIEs and to systems of VIEs. The sharpness of the theoretical error bounds obtained for the collocation methods is demonstrated by numerical examples.

A Numerical Approach for the System of Nonlinear Variable-order Fractional Volterra Integral Equations

A Numerical Approach for the System of Nonlinear Variable-order Fractional Volterra Integral Equations

Using Homotopy Perturbation and Analysis Methods for Solving Different-dimensions Fractional Analytical Equations

The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method. The results indicate that this method is efficient for the linear and non - linear models with the dissipative terms.

Explicit analytical solutions of an incommensurate system of fractional differential equations in a fuzzy environment

Fuzzy fractional models have attracted considerable attention because of their comprehensive and broader understanding of real-world problems. Analytical studies of these models are often complex and difficult. Therefore, it is beneficial to develop a suitable and comprehensive technique to solve these models analytically. In this paper, an explicit analytical technique for solving two-dimensional incommensurate linear fuzzy systems of fractional Caputo differential equations (FLSoCFDEs) considering generalized Hukuhara differentiability (gH-differentiability) is presented and demonstrated. This extracted explicit solution is presented for different classes of such systems, including homogeneous and non-homogeneous cases with commensurate and incommensurate fractional orders. Moreover, the potential solution of FLSoCFDEs in terms of the Mittag-Leffler function involving double series is presented. The originality of the proposed technique is that the fuzzy Cauchy problem is transformed into a system of fuzzy linear Volterra integral equations of second kind and then the solution is extracted using the iterative Picard scheme based on the Banach fixed point theorem. Moreover, several interesting results are derived from FLSoCFDEs in terms of the Mittage-Leffler function for both homogeneous and inhomogeneous cases. To understand the proposed technique, we solve a diffusion process problem (a biological model) and several mass-spring systems as applications. Their graphs are analyzed to illustrate and support the theoretical results.

Boundary control problem for the heat transfer equation associated with heating process of a rod

In this paper, we consider a boundary control problem for a parabolic equation in a segment. In the part of the domain’s bound it is a given value of the solution and it is required to find controls to get the average value of the solution. The given control problem is reduced to a system of Volterra integral equations of the first kind. By the mathematical-physics methods it is proved that like this control functions exist over some domain, the necessary estimates were found and obtained.

The Homotopy Analysis Method to Solve the Nonlinear System of Volterra Integral Equations and Applying the Genetic Algorithm to Enhance the Solutions

This paper presents the application of the Homotopy Analysis Method (HAM) for solving nonlinear system of Volterra integral equations used to obtain a reasonably approximate solution. The HAM contains the auxiliary parameter h which provides a simple way to adjust and control the convergence region of the solution series. The results show that the HAM is a very effective method as well. The results were compared with the solutions obtained by developing a homotopy analysis method using the genetic algorithm (HAM-GA), considering the residual error function as a fitness function of the genetic algorithm.

On a System of Linear Volterra Integral Equations with Variable Integration Limits

On a System of Linear Volterra Integral Equations with Variable Integration Limits

Fredholm boundary-value problem for the system of fractional differential equations.

This paper deals with the study of Fredholm boundary-value problem for the system of fractional differential equations with Caputo derivative. The boundary-value problem is specified by linear vector functional such that the number of it components does not coincide with the dimension of the system of fractional differential equations. We first considered the general solution of the system of fractional differential equations that consist with the general solution of the associated homogeneous system and the arbitrary particular solution of the inhomogeneous system. The particular solution we found is a solution of the system of linear Volterra integral equations of the second kind with weakly singular kernels. Further, by using the theory of pseudo-inverse matrices, we established conditions that should be imposed on the coefficients of the original problem to guarantee that the indicated boundary conditions are satisfied. Moreover, a family of linearly independent solutions of this boundary-value problem is constructed. The specific examples are provided to verify the effectiveness of the proposed approach.

A higher-order numerical scheme for system of two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy

<abstract><p>We give a modified block-by-block method for the nonlinear fractional order Volterra integral equation system by using quadratic Lagrangian interpolation based on the classical block-by-block method. The core of the method is that we divide its domain into a series of subdomains, that is, block it, and use piecewise quadratic Lagrangian interpolation on each subdomain to approximate $ \mathit{\boldsymbol{\kappa}}(x, y, s, r, u(s, r)) $. Our proposed method has uniform accuracy and its convergence order is $ O(h_x^{4-\alpha}+h_y^{4-\beta}) $. We give a strict proof for the error analysis of the method, and give several numerical examples to verify the correctness of the theoretical analysis.</p></abstract>

BOUNDARY-VALUE PROBLEM FOR THE MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATION WITH CAPUTO DERIVATIVE

The extensive application of fractional differential equations and boundary-value problems for these equations promotes the development of the theory and the appearance of numerous publications in this field. One of the types of such equations are equations containing more than one differential operator of fractional order. This paper deals with the study of linear boundary-value problem for the multi-term fractional differential equation with the Caputo derivative. We considered the left fractional Caputo derivative, which is convenient for the description of systems with memory. The boundaryvalue problem is specified by linear vector functional such that the number of it components does not coincide with the number of the orders of the derivative. Assume that the coefficients of the equation are continuous functions and the orders of the derivative are commensurate. A multi-term fractional differential equation is reduced to an equivalent system of differential equations containing only one fractional operator. The general solution of the system of fractional differential equations consisting of a general solution of the associated homogeneous system and the arbitrary particular solution of the inhomogeneous system is considered. The particular solution we found, which is also a solution of the system of linear Volterra integral equations of the second kind with square summable kernels. The question of the solvability of the boundary-value problem for the multi-term fractional differential equations was studied. We considered the critical case, i.e. case when the homogeneous problem has nontrivial solutions. By using the theory of pseudo-inverse matrices, the necessary and sufficient conditions for solvability of the given problem are established. Moreover, a family of linearly independent solutions of this boundary-value problem is constructed.

Numerical Simulations of the Fractional Systems of Volterra Integral Equations within the Chebyshev Pseudo-Spectral Method

In this article, we find the solutions to fractional Volterra-type integral equation nonlinear systems through a Chebyshev pseudo-spectral method (CPM). The fractional derivative is described in the Caputo manner. The suggested method’s accuracy and reliability are confirmed by the results. The proposed method is implemented for solving various nonlinear systems; the results we obtained were compared with the exact solution and other method solutions. The graphical representation and tables show that our method’s error quickly converges as compared to other methods. By comparing the proposed method’s solution with the actual solution and other methods, we can confirm that CPM is more accurate and closer to the exact solution. We display the pointwise solution in the tables, which verifies the proposed method’s accuracy at each point and aids in a better comprehension of the suggested approach. Moreover, the results of using the suggested method at different fractional orders are examined, showing that when a value moves from a fractional order to an integer order, the result is closer to the precise solution. Furthermore, the proposed technique for handling fractional-order linear and non-linear physical problems in science and engineering is straightforward to implement.

New approach to solve system of volterra integral equations

A new approach is used to solve Volterra integral equations approximately. Discretization is a new approximating function that replaces complicated functions with simpler functions so that integral operations can bemore easily performed. This approach is a Subdomain Collocation approach (SC). The researcher showed the efficiency of the prescribed approach by solving some numerical tests.

Superconvergence of system of Volterra integral equations by spectral approximation method

In this article, we apply Jacobi spectral Galerkin, multi-Galerkin methods and their iterated versions to approximate the system of Volterra integral equations for smooth as well as weakly singular kernels and obtain the superconvergence results. Before doing this, first, we develop the regularity properties of the system of linear Volterra integral equations. We show that the Jacobi spectral iterated Galerkin approximation yields better converge rates over Jacobi spectral Galerkin method. We make the improvement of the superconvergence rates further for both smooth and weakly singular kernels in Jacobi spectral iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.

A novel method with convergence analysis based on the Jacobi wavelets for solving a system of two-dimensional Volterra integral equations

ABSTRACT The main objective of this paper is to introduce a collocation-based method for solving a system of two-dimensional Volterra integral equations. The proposed approach is based on the Jacobi wavelets collocation procedure. In addition, the Gegenbauer wavelets method has been implemented to show the comparison of error norms obtained by the proposed method. This proposed approach is applied to reduce the system of Volterra integral equations into a system of algebraic equations. Furthermore, some theorems are elucidated to establish the convergence analysis of the proposed method. Some numerical problems are presented in order to show the accuracy and effectiveness of the proposed scheme. Furthermore, comparison tables and some figures are depicted in support of numerical analysis of the proposed scheme.

System of Non-Linear Volterra Integral Equations in a Direct-Sum of Hilbert Spaces

We use the contraction mapping theorem to present the existence and uniqueness of solutions in a short time to a system of non-linear Volterra integral equations in a certain type of direct-sum H[a; b] of a Hilbert space V[a; b]. We extend the local existence and uniqueness of solutions to the global existence and uniqueness of solutions to the proposed problem. Because the kernel function is a transcendental function in H[a; b] on the interval [a; b], the results are novel and very important in numerical approximation.