In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm–Volterra integro-differential equations under the initial boundary conditions. In addition, the results are compared with the results of other methods. | Turk J Math (2018) 42: 2546 – 2562 © TÜBİTAK doi: Turkish Journal of Mathematics Research Article An exponential method to solve linear Fredholm–Volterra integro-differential equations and residual improvement Şuayip YÜZBAŞI∗, Department of Mathematics, Faculty of Science, Akdeniz University, Antalya, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: In this paper, a collocation approach based on exponential polynomials is introduced to solve linear Fredholm– Volterra integro-differential equations under the initial boundary conditions. First, by constructing the matrix forms of the exponential polynomials and their derivatives, the desired exponential solution and its derivatives are written in matrix forms. Second, the differential and integral parts of the problem are converted into matrix forms based on exponential polynomials. Later, the main problem is reduced to a system of linear algebraic equations by aid of the collocation points, the matrix operations, and the matrix forms of the conditions. The solutions of this system give the coefficients of the desired exponential solution. An error estimation method is also presented by using the residual function and the exponential solutions are improved by the estimated error function. Numerical examples are solved to show the applicability and the effectiveness of the method. In addition, the results are compared with the results of other methods. Key words: Collocation method, exponential polynomials, exponential solutions, Fredholm–Volterra integro-differential equations, initial boundary conditions, residual improvement 1. Introduction Differential, integral, and integro-differential equations contribute to the modeling of many problems in science and engineering. In this study, we introduce an exponential method together with residual error estimation and residual correction method for .
