A collocation method based on the euler and genocchi operational matrix for solving high-order linear complex differential equations in a rectangular domain

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the given initial conditions. On the basis of the presented approach, the matrix forms of the Euler and Genocchi polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is constructed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Euler and Genocchi coefficients are determined and hence the approximate solutions are obtained. Also, an error analysis based on the use of the Euler polynomials is provided under several conditions. To illustrate the efficiency of this method, some numerical examples are presented. Keywords:Complex Differential Equations, Collocation Method, Euler Polynmials, Genocchi Polynomials.