Solution Of Volterra Research Articles

Numerical Solution of Volterra's Integral Equations by Collocation and Taylor Method

In this article, two numerical methods (collocation method and Taylor method) are introduced to solve Volterra integral equation system. In the collocation method, using Bessel polynomials and collocation points, we convert the device into a matrix form and solve the device using the matrix form and obtain an approximate solution for the device. This answer is such that the larger the N becomes, the approximate answer is closer to the exact answer of the device. In Taylor method, With the use of the Taylor series, the system of integral equations is transformed into a matrix equation, and by integrating the output of this new system, a system of algebraic equations is created. By working through this device, we can obtain a rough solution for the Integral Equations device. Using these two methods when the answers are polynomial, we can get real answers.

Collocation method using auto-correlation functions of compact supported wavelets for solving Volterra’s population model

In this paper, we present two numerical collocation methods for approximating the solution of Volterra’s population model by utilizing auto-correlation functions of scaling functions of Daubechies wavelets. By using the properties of these functions, we compute the Volterra integral exactly at dyadic points, and then reduce the integro-differential population model to a system of algebraic equations. Our numerical results demonstrate the effectiveness and accuracy of these methods, and we compare our numerical results with other approaches described in the literature. Additionally, we investigate an error bound for our schemes.

Collocation methods based on barycentric rational interpolation for Volterra integro‐differential equations with weakly singular kernels

This paper is concerned with the numerical solution of Volterra integro‐differential equations with weakly singular kernels. A smoothing transformation is first used to convert the original equation to a new equation with better regularity. A collocation method based on barycentric rational interpolation is introduced. The convergence and supercovergence of the numerical solution are studied in detail. Some numerical results are illustrated to confirm the theoretical prediction of the rate of convergence and superconvergence.

Control Problem Concerned With the Process of Heating a Thin Plate

Previously, a mathematical model for the following problem was considered. On a part of the border of the right rectangle there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some region reaches some given value. In this paper, we consider a boundary control problem associated with a parabolic equation on a right rectangle. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation. In addition, the definition of the generalized solution of the given initialboundary problem is given in the article and the existence of such a solution is proved. The solution of Volterra’s integral equation was found by the Laplace transform method and the existence theorem for admissible control functions was proved. It is also shown that the initial value of the admissible control function is equal to zero using the change of variable in the integral equation. The proof of this comes from the fact that the kernels of the integral equations are positive and finite, and the system has a single-valued solution.Ранее была рассмотрена математическая модель следующей задачи. На части границы правого прямоугольника расположен нагреватель с регулируемой температурой. Требуется найти такой режим его работы, чтобы средняя температура в каком-либо районе достигала некоторого заданного значения. В данной работе рассматривается задача граничного управления, связанная с параболическим уравнением на правом прямоугольнике. На части границы рассматриваемой области указано значение решения с управляющим параметром. Ограничения на управление задаются таким образом, чтобы среднее значение решения в некоторой части рассматриваемой области принимало заданное значение. Вспомогательная задача решается методом разделения переменных, а рассматриваемая задача сводится к интегральному уравнению Вольтерра. Кроме того, в статье дается определение обобщенного решения данной начально-краевой задачи и доказывается существование такого решения. Методом преобразования Лапласа найдено решение интегрального уравнения Вольтерра и доказана теорема существования допустимых управляющих функций. Также показано, что начальное значение допустимой функции управления равно нулю с помощью замены переменной в интегральном уравнении. Доказательство этого исходит из того, что ядра интегральных уравнений положительны и конечны, а система имеет однозначное решение.

Existence of solution for Volterra–Fredholm type stochastic fractional integro‐differential system of order μ ∈ (1, 2) with sectorial operators

The mainspring of the study is to investigate the out‐turn of stochastic Volterra–Fredholm integro‐differential inclusion of order with sectorial operator of the type . The existence results of our proposed problem is derived by employing Martelli's fixed point approach. We do not limit the theoretical results of fractional stochastic equation to local condition but extend to nonlocal condition, and physical interpretation of our obtained results is proved with an appropriate illustration.

On the numerical solution of Volterra and Fredholm integral equations using the fractional spline function method

In this article, the researchers develop a new type of spline function with fractional order which constructs two distinct formulas for the proposed method by using fractional boundary conditions and fractional continuity conditions. These methods are used to solve linear Volterra and Fredholm-integral equations of the second kind. The convergence analysis is studied. Moreover, some numerical examples are provided and compared to illustrate the efficiency and applicability of the proposed methods.

Numerical solution of Volterra–Fredholm integral equation systems by operational matrices of integration based on Bernstein multi-scaling polynomials

Numerical solution of Volterra–Fredholm integral equation systems by operational matrices of integration based on Bernstein multi-scaling polynomials

Daubechies Wavelet Scaling Function Approach to Solve Volterra’s Population Model

In this paper, we focus on a collocation approach based on Daubechies wavelet scaling functions for approximating the solution of Volterra’s model of population growth of a species with a closed system. We present that the integral and derivative terms, which appear in Volterra’s model of the population, will be computed exactly in dyadic points. Utilizing this collocation technique, Volterra’s population model reduces into a system of nonlinear algebraic equations. In addition, an error bound for our method will be explored. The numerical results demonstrate the applicability and accuracy of our method.

Analytical Approach to Approximate the Solution of Volterra and Fredholm Integral Equations

Analytical Approach to Approximate the Solution of Volterra and Fredholm Integral Equations

Piecewise barycentric interpolating functions for the numerical solution of Volterra integro‐differential equations

This investigation presents an effective numerical scheme using a new set of basis functions, namely, the piecewise barycentric interpolating functions, to find the approximate solution of Volterra integro‐differential equations (VIDEs). The operational matrices of integration and product for the PBIFs are provided. Then these operational matrices are utilized to reduce the VIDEs to a system of algebraic equations. Applying the Floater–Hormann weights, the convergence analysis of the PBIFs method is studied. Finally, several numerical examples are provided to illustrate the efficiency and validity of the proposed method in acceptable computational times, and the results are compared with some existing numerical methods.

The numerical approximation for the solution of linear and nonlinear integral equations of the second kind by interpolating moving least squares

In this paper, the interpolating moving least-squares (IMLS) method is discussed. The interpolating moving least square methodology is an effective technique for the approximation of an unknown function by using a set of disordered data. Then we apply the IMLS method for the numerical solution of Volterra–Fredholm integral equations, and finally some examples are given to show the accuracy and applicability of the method.

Numerical solution of Volterra–Fredholm integral equations with Hosoya polynomials

In this study, Volterra–Fredholm integral equation is solved by Hosoya polynomials. The solutions obtained with these methods were compared on the figure and table. And error analysis was done. MATLAB programming language has been used to obtain conclusions, tables, and error analysis within a certain algorithm.

Volterra–Stieltjes integral equations and impulsive Volterra–Stieltjes integral equations

In this paper, we prove existence and uniqueness of solutions of Volterra– Stieltjes integral equations using the Henstock–Kurzweil integral. Also, we prove that these equations encompass impulsive Volterra–Stieltjes integral equations and prove the existence and uniqueness for these equations. Finally, we present some examples to illustrate our results.

An Iterative Approach for Solving Nonlinear Volterra–Fredholm Integral Equation Using Tension Spline

According to mathematical literature, the parametric spline functions have not been used directly for solving the integral equations so far. In this work, we develop the tension spline approximation to obtain the numerical solution of Volterra–Fredholm integral equation. First, the tension spline method was obtained and then the combination of this method with the quasi-linearization method has been used for solving the nonlinear Volterra–Fredholm integral equation. The error analysis of the method is given. Illustrative examples are included to verify the effectiveness and applicability of the presented approach.

On the structure of solutions of Volterra interval‐valued integro‐differential equations

In this paper, we discuss the structure of the solutions for linear interval‐valued Volterra integro‐differential equations (VIDEs) based on gH‐difference. The considered VIDEs consist of six forms. Among those six forms, three forms are obtained by using the definition of generalized Hukuhara difference (or gH‐difference). We obtain solutions of that linear interval‐valued VIDEs and discuss their properties. Some examples illustrate the theory.

An efficient algorithm for numerical solution of fractional integro-differential equations via Haar wavelet

In this paper, Haar wavelet collocation technique is developed for the solution of Volterra and Volterra–Fredholm fractional integro-differential equations. The Haar technique reduces the given equations to a system of linear algebraic equations. The derived system is then solved by Gauss elimination method. Some numerical examples are taken from literature for checking the validation and convergence of the proposed method. The maximum absolute errors are compared with the exact solution. The maximum absolute and mean square root errors for different number of collocation points are calculated. The results show that Haar method is efficient for solving these equations. The experimental rates of convergence for different number of collocation point is calculated which is approximately equal to 2. Fractional derivative is described in the Caputo sense. All algorithms for the developed method are implemented in MATLAB (R2009b) software.

Numerical Solution of Volterra–Hammerstein Delay Integral Equations

In this article, a new numerical scheme based on the Chelyshkov wavelets is presented for finding the numerical solutions of Volterra–Hammerstein delay integral equations arising in infectious diseases. First, properties of Chelyshkov polynomials and Chelyshkov wavelets are discussed. Then, integral and derivative operators of these wavelets are constructed, for first time. The Chelyshkov wavelets and their operators are used to reduce these integral equations to a system of algebraic equations, and again these algebraic systems have been solved numerically by Newton’s iterative scheme. Finally, the error analysis of the proposed method is investigated and its accuracy is illustrated on three numerical examples. Also the results achieved by Chelyshkov wavelets method have been compared with that of by the Bernoulli wavelets scheme and the B-spline wavelets scheme.

Existence Solution of Volterra Hammerstein Equation in L^p Space

In this paper we use the fixed-point theorem of Latrach, Taoudi and Zeghal under some conditions to find a solution for Volterra_Hammerstein integral equation in the Banach space L^p ([0,m],R). We use this fixed point theorem with new assumptions.

Analytical Solution of Volterra–Fredholm Integral Equations Using Hybrid of the Method of Contractive Mapping and Parameter Continuation Method

In this paper, an analytical method is proposed to solve Volterra–Fredholm integral equations. The method is based on a hybrid of the method of contractive mapping and parameter continuation method. The existence and uniqueness of the solution will be discussed. Error analysis of the proposed method is also investigated. Finally, illustrative examples are given to demonstrate the validity and applicability of the proposed method.

New approximation method for Volterra nonlinear integro-differential equation

In this work, we build a new numerical method to approximate the solution of Volterra’s nonlinear integro-differential equation. This method needs fewer conditions to converge, compared to the direct Nytröm method. Numerical tests show its efficiency. This new method is more practical and compatible with the Volterra nonlinear integro-differential equation.