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The primary goal of this paper is to investigate the effectiveness of the 4-point Explicit Group (4-point EG) iterative method for solving one-dimensional unsteady advection-diffusion problems via similarity transform. By using this transformation approach, the proposed problem can be reduced into the corresponding two-point boundary volume problem. By imposing the second-order central finite difference discretization scheme, then the corresponding approximation equation can be derived to construct a system of linear equations. Having a large linear system, the 4-point EG iterative method has been used to solve the generated system of linear equations. The formulation of the 4-point EG method is also derived. Some numerical experiments are conducted that to verify the 4-point EG method is more effective than the Gauss-Seidel (GS) method.

The point of this study is to explore and elucidating the performance of the four-point Half-Sweep EGSOR (4HSEGSOR) iterative method to solve fractional two-point boundary value problems by using Caputo's fractional operator and family of finite differences (FD) schemes. To apply the iterative methods, linear system needs to be constructed via the discretization process with fractional order to get the approximation equation of the linear fractional two point boundary value problem by using the Caputo's derivative operator. Then the generated linear system has been solved using the proposed 4HSEGSOR iterative method. In the addition, the formulation and application of the 4HSEGSOR method to solve the problems are also presented. Three numerical examples and comparison are used to illustrate with tested FSSOR and HSSOR methods. The numerical results reveals the effectiveness of 4HSEGSOR method compared with tested iterative methods.