An algorithm for solving Fredholm integro-differential equations

Abstract

The aim of this paper is to present an efficient numerical procedure for solving linear second order Fredholm integro-differential equations. The scheme is based on B-spline collocation and cubature formulas. The analysis is accompanied by numerical examples. The results demonstrate reliability and efficiency of the proposed algorithm.

Hermite wavelets collocation method for solving a fredholm integro-differential equation with fractional Caputo-Fabrizio derivative

“…The authors, in [3,15], examined the nonlinear Fredholm integro-differential equation when the derivative of the unknown function is inside of the integral operator. On the other hands, many numerical methods are applied to get an approximate solution for such equations, Laplace decomposition method [1], Rational approximation [11], B-spline method [18,14], Homotopy perturbation method [20], Modified variational iteration method [10], CAS wavelet operational matrix [6,19]. Also, in [8], the author studied the existence and uniqueness of the following nonlinear Volterra integro-differential equation with Caputo derivative…”

Numerical study for a second order Fredholm integro-differential equation by applying Galerkin-Chebyshev-wavelets method

“…Recently, there has been a growing interest in the Integro-Differential Equations (IDEs), for these kinds of equations can be found in modeling real phenomena in many fields of sciences, physics, chemistry, biology and engineering problems, such as epidemic models [1,2], the Boltzmann kinetic equation [3], and the Vlasov and Landau equations [4]. However, the exact solution to such equations is usually difficult to obtain, so the researchers use different numerical methods to approach the exact solution such as the Pade approximation, the Legendre-Galerkin method [5], the Hermite wavelet [6], the Haar wavelet [7], the Chebyshev wavelet collocation method [8,9], the wavelet-Galerkin method [10], the Laguerre wavelets collocation method [11], the Laplace decomposition method [12], Bernoulli polynomials [13], and the B-spline method [14,15].…”

Exponential spline for the numerical solutions of linear Fredholm integro-differential equations

“…Recently, many authors have investigated the numerical methods for integral equations. These methods include a cubic spline approximation in C 2 to the solution of the Volterra integral equation of the second kind [33], quintic B-spline method [30], Bernstein operational matrix of derivative [4], hybrid of block pulse functions and normalized Bernstein polynomials [5], iterative method [49], sinc-collocation method [48], bivariate splines on nonuniform partitions [36], Jacobi operational matrices for solving delay or advanced integro-differential equations [40], the tau approximation for the Volterra-Hammerstein integral equations [21], b-spline collocation and cubature formulas [12] and [37], wavelet method [6], Walsh function method [35], Chebyshev finite difference method [13], differential transform method [7], Legendre polynomial method [39], an approximating solution, based on Lagrange interpolation and spline functions, to treat functional integral equations of Fredholm type and Volterra type [20], CAS wavelets method [22], an efficient matrix method based on Bell polynomials for solving nonlinear Fredholm-Volterra integral equations [32], collocation methods [10], Taylor polynomial methods [46], and Bernoulli matrix method [9]. Xuhao Li and Patricia J.Y.…”