Matrix properties of Bernoulli and Euler polynomials and applications to delay integro differential equations

Delay differential, integral and integro differential equations play an important role in many applied mathematical sciences such as quantum mechanics, electrodynamics, population dynamic, chemical kinetics, control problems, electronic systems, probability, number theory, mechanics, astronomy, biology, economics, potential theory, electrostatics and industrial applications. In general, it is difficult to find the analitical solution of these type equations. So it is necessary to obtain their approximate solutions to numerical methods. In this thesis, a matrix method based on operational matrices of differentiation and integration together with standard collocation points is presented to find the approximate solution of high order linear delay integro-differential equations with variable coefficients under the initial and boundary conditions. For this aim, first the matrix forms of Bernoulli and Euler polynomials are computed. Then by using these matrix forms, delay integro-differential equation is reduced to two different systems of linear algebraic equations with unknown Bernoulli and Euler cofficients which can be solved by means of suitable numerical methods; thereby, the approximate solution of the problem is obtained in terms of Bernoulli and Euler polynomials by using " Matrix-Collocation method". In addition, an error analysis based on residual function is performed to estimate the error upper bounds of the obtained approximate solutions. Moreover, some numerival examples for each kind of the problem are provided to confirm the accuracy and efficiency of the presented method; the results are shown in tables and figures along with absolute errors and error estimations. Finally,discussions and recommentaries are given.