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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.

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Finding the large scale unconstrained minimizer using Newton method has required the calculation of large and complicated linear systems results from solving the Newton direction. Therefore, in this paper, we propose a method for solving large scale unconstrained optimization problems with tridiagonal Hessian matrices to reduce the complexity of calculating Newton direction. Our proposed method was a combination of Newton method and Accelerated Over Relaxation (AOR) iterative method. To evaluate the performance of the proposed method, combination of Newton method with Gauss-Seidel iteration and Newton method with Successive Over Relaxation (SOR) iteration were used as reference method. Finally, the numerical experiment illustrated that the proposed method produce results that are more efficient compared to the reference methods with less execution time and minimum number of iterations.

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