A new collocation method based on charlier polynomials and integral to solve lane-emden, pantograph and riccati differential equations

The aim of this thesis is to create a new collocation method based on Charlier polynomials that will help us solve multi-pantograph, Lane-Emden, Riccati and generalized pantograph type differential equations numerically. The method starts by approximating the highest order derivative of unknown term in the differential equation as a finite series of Charlier polynomials. This approximation is expressed in matrix form and the necessary number of integrals are taken to find the matrix form of the unknown function and the other derivative terms in the equation. When the integrals are taken, the conditions given by the differential equation are used in the relevant places. After these steps, the differential equation problem is converted to a linear/nonlinear system of algebraic equations using collocation points. By solving this algebraic system, the unknown coefficients in the approximation are obtained. The solution of the problem is obtained by replacing the unknown function into matrix form. Then, an error estimation method based on the residual function is presented. With this error estimation technique, an error problem is created and the error problem is solved with the same solution method and the error is estimated. The method and error estimation technique are explained by solving numerical examples. The effectiveness of the method is shown by making comparisons with known methods from the literature. It is observed that the results of the presented method are better than other methods. All calculations are made with the code written in Matlab.