A new hermite collocation method based on the integral for solving neutral and pantograph type differential equations

In this thesis, a new collocation method based on integration and using the Hermite polynomials is presented for solving neutral and pantograph type differential equations. As the first step of the presented method, the highest order derivative in the equation is approximated in the form of a finite Hermite series and this approximation is written in matrix form. The matrix forms of the approximation of the other derivatives of the unknown function and the function itself are obtained by taking the necessary number of integrals. These matrix forms are substituted in the considered problem. The differential equation problem is transformed into a system of linear algebraic equations by using collocation points and matrix operations. This algebraic system is solved and the unknown coefficients are calculated. The solution of the problem is found by writing these coefficients in the matrix form of the approximation of the unknown function. On the other hand, the residual error estimation technique is used to estimate the error of approximate solutions. Using the residual function, an error problem is created and the error problem is solved with the presented new method and the error function is estimated. After the method is created, numerical examples are solved to show the computational efficiency of the method and the error estimation technique. Obtained data in the numerical examples are compared with the results of known techniques from the literature. The obtained results show the accuracy and usability of the method.