collocation method

In this study, the Burgers equation is analyzed in both numerically and mathematically by considering various finite element based techniques including Galerkin, Taylor-Galerkin and collocation methods for spatial variation of the equation. The obtained time dependent ordinary differential equation system is approximately solved by α-family of time approximation. All these methods are theoretically explained using cubic B-spline basis and weight functions for a strong form of the model equation. Von Neumann matrix stability analysis is performed for each of these methods and stability criteria are determined in terms of the problem parameters. Some challenging examples of the Burgers equation are numerically solved and compared with the literature and exact solutions. Also, the proposed techniques have been compared with each other in terms of their advantageous and disadvantageous depending on the problem types. The more advantageous method of the three, comparison to other two, has been found out for the special cases of the present problem in detail.

Due to the easy adaption of radial basis functions (RBFs), a directRBF collocation method is considered to develop an approximate scheme to solvefractional delay differential equations (FDDEs). The method of RBFs is a method of scattered data interpolation that has many application in different fields. In spite of easy implementation of other high-order methods and finite difference schemes for solving a problem of fractional order derivatives, the challenge of these methods is their limited accuracy, locality, complexity and high cost of computing in discretization of the fractional terms, which suggest that global scheme such as RBFs that are more accurate way for discretizing fractional calculus and would allow us to remove the ill-conditioning of the system of discrete equations. Applications to a variety ofproblems confirm that the proposed method is slightly more efficient than thoseintroduced in other literature and the convergence rate of our approach is high.

In this study, a collocation method based on the Bernoulli polynomials is presented to find approximate solutions of a system of high-order linear Volterra integral equations (VIEs) with variable coefficients. In fact, the approximate solution of the problem in the truncated Bernoulli series form is obtained by this method. In addition, the method is presented with error and stability analysis. To show the accuracy and the efficiency of the method, numerical examples are implemented and the comparisons are given by the other methods.