The authors consider systems of linear integro-differential equations of Fredholm-Volterra type in the form
under the mixed conditions
where , , and are real-valued column matrices with dimension and indicates the th-order derivative and . The aim of this study is to get a solution as truncated Chebyshev series defined by
where denotes the Chebyshev polynomials of the first kind, are unknown Chebyshev coefficients, and is chosen any positive integer such that .
The authors transform the system (1) and the given conditions (2) into matrix equations via Chebyshev collocation points. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Chebyshev coefficients of the solution function. An interesting feature of this method is that when system of integro-differential equations (1) has linearly independent polynomial solution of degree or less than , the method can be used for finding the analytical solution. Besides, when the truncation limit is increased, there exists a solution, which is closer to the exact solution. Some numerical results are also given to illustrate the efficiency of the method.