A Taylor Collocation Method for Solving Systems of Two-Dimensional Volterra Integral Equations with Proportional Delays

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 Volterra integral equations of the first kind: the system of Volterra linear integral equations of the first kind in the interval stability of solutions of Volterra integral equations of the second kind in the semiaxis the system of Volterra integral equations of the first kind in the semiaxis the system of Volterra linear integral equations of the first kind in the space of square-integrable functions. The system of Volterra linear integral equations of the first kind of the convolution type and commutative ring: the system of Volterra integral equations of the first kind of the convolution type commutative ring and a system of equations in it application to inverse problems. The system of Volterra integral equations of the first kind with two independent variables: the system of two-dimensional Volterra linear integral equations of the first kind the system of Volterra nonlinear two-dimensional integral equations of the first kind. Volterra operator equations: Volterra operator equations of the first kind in the scales of Hilbert spaces Volterra operator equations of the first kind in the scales of Banach spaces Volterra operator equations of the first kind with commuting kernels on uniqueness of solution of Volterra operator equations nonlinear Volterra operator equations of the first kind. Problems of integral geometry: the problem of integral geometry in the three dimensional band the problem of integral geometry in a two-dimensional band the problem of integral geometry with spatial curves the equations connected with problems of integral geometry.

Mathematical modeling for many problems in different disciplines, such as engineering, chemistry, physics and biology leads to integral equation, or system of integral equations. It’s the reason of great interest for solving these equations. There are some analytical and numerical methods for solving Volterra integral equations, but extension of these methods to systems of such integral equations is not easy to employ. Adomian decomposition method, well address in [1,2] has been used to solved some of these systems such as systems of differential equations, systems of integral equations and even systems of integro-differential equation [3,4,5]. Applying this method needs some computations which is sometimes boring, having a program to do all computations would be interesting and helpful. In this article a maple program is prepared to solve a system of Volterra integral equations of the second kind, linear or non-linear.

PurposeThe purpose of this paper is to solve systems of linear Volterra integral equations of the first kind by the Adomian decomposition method (ADM). An elegant and reliable technique is outlined to find canonical form of ADM.Design/methodology/approachThe approximate solution of systems of linear Volterra integral equations is calculated in the form of series with easily computable components. In this work, some methods based on substitution techniques are presented to permit the application of ADM to systems of integral equations of the first kind.FindingsThe approach developed in this work is valuable as a tool for scientists and applied mathematicians. It provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to systems of integral equations of the first kind, without linearization or discretization. The presented technique has many advantages over the traditional methods because it takes into account some systems of integral equations of first kind where the kernels present some singularities.Research limitations/implicationsA reliable method for obtaining approximate solutions of linear systems of integral equations of the first kind using the ADM which avoids the tedious work needed by traditional techniques has been developed.Practical implicationsThe research provides a new efficient method for solving systems of integral equations of first kind using ADM. The convergence result is investigated also and some numerical examples are given to illustrate the importance of the analysis presented.Originality/valueThe technique is both innovative and efficient, and an original approach for solving any kind of systems of linear Volterra integral equations of the first kind.

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.

In this chapter we consider three systems of singular integral equations. Specifically we are interested in the following systems of Fredholm integral equations $$\displaystyle{ u_{i}(t) =\int _{ 0}^{1}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,1],\ 1 \leq i \leq n }$$ (7.1.1) $$\displaystyle{ u_{i}(t) =\int _{ 0}^{\infty }g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,\infty ),\ 1 \leq i \leq n }$$ (7.1.2) and the system of Volterra integral equations $$\displaystyle{ u_{i}(t) =\int _{ 0}^{t}g_{ i}(t,s)f_{i}(s,u_{1}(s),u_{2}(s),\cdots \,,u_{n}(s))ds,\ \ t \in [0,T],\ 1 \leq i \leq n }$$ (7.1.3) where T > 0 is fixed. The nonlinearities \(f_{i},\ 1 \leq i \leq n\) in the above systems may be singular in the independent variable and may also be singular at \(u_{j} = 0,\ j \in \{ 1,2,\cdots \,,n\}.\)

This paper describes a procedure for solving the system of linear Volterra integral equations by means of the Sinc collocation method. A convergence and an error analysis are given; it is shown that the Sinc solution produces an error of order O(exp(−c N 1/2)), where c>0 is a constant. This approximation reduces the system of integral equations to an explicit system of algebraic equations. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.

The aim of this paper is for finding the numerical solution (sometimes exact) for non-linear system of Volterra integral equations of the second kind (NSVIEK2) by using block-by-block method. Which avoid the need for special starting procedures, but uses numerical quadrature rule. Also some illustrative examples are presented, to elucidate the accuracy of this method.

Superconvergence of system of Volterra integral equations by spectral approximation method

The main aim of this paper is to use the operational matrices of fractional integration of Haar wavelets to find the numerical solution for a nonlinear system of two-dimensional fractional partial Volterra integral equations. To do this, first we present the operational matrices of fractional integration of Haar wavelets. Then we apply these matrices to solve systems of two-dimensional fractional partial Volterra integral equations (2DFPVIE). Also, we present the error analysis and convergence as well. At the end, some numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.

Product integration methods for solving a system of nonlinear Volterra integral equations

In this paper, a new modified homotopy perturbation method (NHPM) is introduced for solving systems of Volterra integral equations of the second kind. Theorems of existence and uniqueness of the solutions to these equations are presented. Comparison of the results of applying the NHPM with those of the homotopy perturbation method and Adomian's decomposition method leads to significant consequences. Several examples, including the system of linear and nonlinear Volterra integral equations, are given to demonstrate the efficiency of the new method.

In this article, we apply the Single Term Walsh Series (STWS) method to solve the systems of Volterra integral equations of first kind (SVIEF). Because of the ill-posedness of the first kind problem, the SVIEF is converted into system of Volterra integral equations of second kind (SVIES). The STWS method for the SVIES with variable coefficient is presented. Numerical examples for linear and non-linear problems are experimented to show the efficiency and simplicity of the STWS method. The obtained numerical solutions (results) have been compared with the existing method.

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

In this paper, a new numerical technique is introduced to find the solution of the system of Volterra integral equations based on symmetric Bernstein polynomials. The use of Bernstein polynomials to find the numerical solutions of differential and integral equations increased due to its fast convergence. Here, the numerical solution of the system of Volterra integral equations on any finite interval [m,n] is obtained by replacing the unknown functions with the generalized Bernstein basis functions. The proposed technique converts the given system of equations into the system of algebraic equations which can be solved by using any standard rule. Further, Hyers–Ulam stability criteria are used to check the stability of the given technique. The comparison between exact and numerical solution for the distinct nodes is demonstrated to show its fast convergence.