Matrix properties of Lerch and Pell polynomials and applications to linear partial differential equations

In this thesis, a matrix collocation method based on Lerch polynomials and a matrix collocation method based on Pell polynomials are developed to obtain the approximate solutions of linear partial differential equations under the Cauchy, Dirichlet, Neumann or Robin conditions, which correspond to Cauchy, Dirichlet, Neumann or Robin problems. In used methods, the coefficients are reduced to the matrix forms, which correspond to the system of algebric equations and the approximate solution of the problem is reached. From the obtained results, it is observed that implementation of the proposed methods is efficient and easy. In study, firstly, utilization in the fields of the science and engineering, the historical development process and the solution methods of the partial differential equations are examined. Thereafter, general information of partial differential equations, definition of Lerch polynomials and their graphics along with definition of Pell polynomials and their graphics are given. Behind, by using matrix relations of the mentioned polynomials and their derivatives, Lerch matrix collocation method and Pell matrix collocation method are explained for linear partial differential equations, respectively. Also, numerical examples are performed for linear partial differential equations as Laplace, Poisson, Helmholtz, telegraph, convection diffusion, 1-D heat, damped wave, vibration, under the Cauchy, Dirichlet, Neumann and Robin conditions. Some numerical examples together with residual error analysis are performed to illustrate the efficiency of the method and the obtained results are scrutinized and interpreted.