Numerical Solution of Fractional Volterra-Fredholm Integro-Differential Equation Using Lagrange Polynomials

في هذا البحث، ستراتيجيات جديدة لإيجاد الحل العددي للمعادلات الخطية الكسورية التفاضلية - التكاملية فولتيرا- فريدهولم (LFVFIDE) تم دراستها. الطرق المتبعه على ثلاث انواع من متعددات الحدود لاكرانج وهي: متعددة حدود لاكرانج الأصلية (OLP) ، متعددة حدود لاكرانج ذات الدعامة المركزية (BLP) و متعددة حدود لاكرانج المعدلة  (MLP).كما تم اقتراح خوارزمية عامة واعطاء  أمثلة لبرهنة فعالية الطرق وتنفيذها. وأخيرًا ، تم استخدام مقارنة بين الطرق المقترحة والطرق الأخرى لحل هذا النوع من المعادلات.In this study, a new technique is considered for solving linear fractional Volterra-Fredholm integro-differential equations (LFVFIDE's) with fractional derivative qualified in the Caputo sense. The method is established in three types of Lagrange polynomials (LP’s), Original Lagrange polynomial (OLP), Barycentric Lagrange polynomial (BLP), and Modified Lagrange polynomial (MLP). General Algorithm is suggested and examples are included to get the best effectiveness, and implementation of these types. Also, as special case fractional differential equation is taken to evaluate the validity of the proposed method. Finally, a comparison between the proposed method and other methods are taken to present the effectiveness of the proposal method in solving these problems

Numerical Solution of Linear Volterra-Fredholm Integro-Differential Equations Using Lagrange Polynomials

In this paper, we introduce a numerical method for solving linear Volterra-Fredholm integro-differential Equations (LVFIDE’s) of the first order. To solve these equations, we consider the polynomial approximation from original Lagrange polynomial approximation, barycentric Lagrange polynomial approximation, and modified Lagrange polynomial approximation. Finally, some examples are included to improve the validity and applicability of the techniques. Keywords: Linear Volterra-Fredholm integro-differential equation, Original Lagrange polynomial, Barycentric Lagrange Polynomial, Modified Lagrange polynomial

Numerical Solution For Mixed Volterra-Fredholm Integral Equations Of The Second Kind By Using Bernstein Polynomials Method

In this paper, we have used Bernstein polynomials method to solve mixed Volterra-Fredholm integral equations(VFIE’s) of the second kind, numerically. First we introduce the proposed method, then we used it to transform the integral equations to the system of algebraic equations. Finally, the numerical examples illustrate the efficiency and accuracy of this method. Keywords: Bernestein polynomials method, linear Volterra-Fredholm integral equations

Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions

This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods

An exponentially convergent Volterra-Fredholm method for integro-differential equations

Extending the authors’ recent work [15] on the explicit computation of error bounds for Nystrom solvers applied to one-dimensional Fredholm integro-differential equations (FIDEs), presented herein is a study of the errors incurred by first transforming (as in, e.g., [21]) the FIDE into a hybrid Volterra-Fredholm integral equation (VFIE). The VFIE is solved via a novel approach that utilises N-node Gauss-Legendre interpolation and quadrature for its Volterra and Fredholm components respectively: this results in numerical solutions whose error converges to zero exponentially with N, the rate of convergence being confirmed via large- N asymptotics. Not only is the exponential rate inherently far superior to the algebraic rate achieved in [21], but also it is demonstrated, via diverse test problems, to improve dramatically on even the exponential rate achieved in [15] via direct Nystrom discretisation of the original FIDE; this improvement is confirmed theoretically

Approximate Optimal Control of Volterra-Fredholm Integral Equations Based on Parametrization and Variational Iteration Method

This article presents appropriate hybrid methods to solve optimal control problems ruled by Volterra-Fredholm integral equations. The techniques are grounded on variational iteration together with a shooting method like procedure and parametrization methods to resolve optimal control problems ruled by Volterra - Fredholm integral equations. The resulting value shows that the proposed method is trustworthy and is able to provide analytic treatment that clarifies such equations and is usable for a large class of nonlinear optimal control problems governed by integral equations

Solution of the Volterra-Fredholm integral equations via the Bernstein polynomials and least squares approach

We develop a numerical scheme to solve a general category of VolterraFredholm integral equations. For this purpose, the Bernstein polynomials and their features have been used. We convert the main equation into a set of algebraic equations in which the coefficient matrix is obtained by the least squares approximation approach. The error analysis is given to corroborate the precision of the proposed method. Numerical results are presented to demonstrate the success of the scheme for solving integral equations.Publisher's Versio

Spectral collocation method for compact integral operators

We propose and analyze a spectral collocation method for integral equations with compact kernels, e.g. piecewise smooth kernels and weakly singular kernels of the form $\frac{1}{|t-s|^\mu}, \; 0\u3c\mu\u3c1.$ We prove that 1) for integral equations, the convergence rate depends on the smoothness of true solutions $y(t)$. If $y(t)$ satisfies condition (R): $\|y^{(k)}\|_{L^\infty[0,T]}\leq ck!R^{-k}$}, we obtain a geometric rate of convergence; if $y(t)$ satisfies condition (M): $\|y^{(k)}\|_{L^{\infty}[0,T]}\leq cM^k$, we obtain supergeometric rate of convergence for both Volterra equations and Fredholm equations and related integro differential equations; 2) for eigenvalue problems, the convergence rate depends on the smoothness of eigenfunctions. The same convergence rate for the largest modulus eigenvalue approximation can be obtained. Moreover, the convergence rate doubles for positive compact operators. Our numerical experiments confirm our theoretical results