Solving linear Fredholm integro-differential equation by Nyström method

. The study of the solution’s existence and uniqueness for the linear integro-differential Fredholm equation and the application of the Nystr ¨ om 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 Nystr ¨ om method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method’s convergence. The Nystr ¨ om 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 Nystr¨om method, some numerical examples will be provided at the end of this work.