Solving linear Fredholm integro-differential equation by Nyström method

Abstract

The study of the solution's existence and uniqueness for the linear integrodifferential Fredholm equation and the application of the Nyström method to approximate the solution is what we will present in this paper. We use the Neumann theorem to construct a sufficient condition that ensures the solution's existence and uniqueness of our problem in the Banach space C 1 [a, b]. We have applied the Nyström method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method's convergence. The Nyström method discretizes the integro-differential equation into solving a linear system. Only with the existence and uniqueness condition, we show the solution's existence and uniqueness of the linear system and the convergence of the numerical solution to the exact solution in infinite norm sense. We present two theorems to give a good estimate of the error. Also, to show the efficiency and accuracy of the Nyström method, some numerical examples will be provided at the end of this work.

Spline Quasi-Interpolation Numerical Methods for Integro-Differential Equations With Weakly Singular Kernels

“…Numerous numerical techniques have emerged to address Fredholm integrodifferential equations (FIDE for short) with smooth kernels, [7,9,18,19,20]. Notable among these are the Adomian decomposition [8], the homotopy analysis method [11], and Chebyshev and Taylor collocation methods [22].…”

First-order numerical method for the singularly perturbed nonlinear Fredholm integro-differential equation with integral boundary condition

“…Significant efforts have been made in the last 50 years to treat the Volterra/Fredholm integral equations numerically [18,19,20,21,22,23,24,25,26,27,28,29]. More specically, in [30,31], a difference scheme of the exponential type on a uniform mesh is considered.…”

An efficient numerical method for a singularly perturbed Fredholm integro-differential equation with integral boundary condition

“…Furthermore, researchers employed fitted analytical approaches because of the difficulty of obtaining accurate solutions to these types of problems. Some of these methods are reproducing kernel Hilbert space method [ 7 ], Nyström method [ 38 ], Touchard polynomials method [ 2 ], Tau method [ 20 , 32 ], Collocation and Kantorovich methods [ 37 ], Galerkin method [ 12 , 41 , 43 ], Boole collocation method [ 14 ], parameterization method [ 17 ], Legendre collocation matrix method[ 44 ], variational iteration technique [ 19 ]. The increasing interest in recent years is not limited to only FIDEs, but also the numerical solutions of linear and nonlinear Volterra or Volterra-Fredholm integro-differential equations are increasing in popularity.…”