This research includes of the numerical modeling of fluids in two-dimensional cavity. The cavity flow is an important theoretical problem. In this research, modeling was carried out based on an ALTERNATING DIRECTION IMPLICIT via Vorticity-Stream function formulation. It evaluates different Reynolds numbers and grid sizes. Therefore, for the flow field analysis and prove of the ability of the SCHEME, the numerical solution was carried out for different values of the Reynolds numbers. Also, the behavior of the vortex flow in cavity was predicted. This research compares results of applied numerical model with the results of Chia et al. [1] and Chen & Pletcher [2]. Comparing the results with those of the benchmarks show that ALTERNATING DIRECTION IMPLICIT is an effective and suitable formulation for the solution of the Navier–Stokes equations.

In this paper, the fuzzy partial differential equation is investigated by using the strongly generalized differentiability concept. The ALTERNATING DIRECTION IMPLICIT (ADI) method is proposed for approximating the solution of the two-dimensional heat equation where the initial and boundary conditions are fuzzy numbers. The algorithm is illustrated by solving several examples.

In this paper, an ALTERNATING DIRECTION IMPLICIT (ADI) finite difference SCHEME is proposed for solving the two-dimensional time-dependent nonlinear Schrödinger equation. In the proposed SCHEME, the nonlinear term is linearized by using the values of the wave function from the previous time level at each iteration step. The resulting block tridiagonal system of algebraic equations is solved using the Gauss-Seidel method in conjunction with sparse matrix computation. The stability of the SCHEME is analyzed using matrix analysis and is found to be conditionally stable. Numerical examples are presented to demonstrate the efficiency, stability, and accuracy of the proposed SCHEME. The numerical results show good agreement with exact solutions.

In this article, at first standard linear Black-Scholes model and then some nonlinear Black-Scholes models will be considered and thereupon ALTERNATING DIRECTION explicit (ADE) method is applied firstly for solving the standard Black-Scholes model and then for Barles and Soner model which is one of the most complete and comprehensive nonlinear Black-Scholes models. Furthermore, the stability of this method has been considered and its accuracy will be compared with other numerical methods such as finite difference methods. Since in solving nonlinear Black-Scholes models by the ADE methods, we need to solve only some scalar nonlinear equations instead of a full nonlinear system of equations that we should solve in IMPLICIT methods, so this method can be a suitable choice for solving such models.

The extended trust region subproblem has been the focus of several research recently. Under various assumptions, strong duality and certain SOCP/SDP relaxations have been proposed for several classes of it. Due to its importance, in this paper, without any assumption on the problem, we apply the widely used ALTERNATING DIRECTION method of multipliers (ADMM) to solve it. The convergence of ADMM iterations to the first order stationary conditions is established. On several classes of test problems, the quality of the solution obtained by the ADMM for medium scale problems is compared with the SOCP/SDP relaxation. Moreover, the applicability of the method for solving large scale problems is shown by solving several large instances.

Knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering in which there is no need to be peculiar about how to select initial values. Due to properly converting the task of optimization to an equivalent convex optimization problem, the proposed data clustering algorithm can be indeed considered as a global minimizer. In this paper, a splitting method for solving the convex clustering problem is used called as Alterneting DIRECTION Method of Multipliers (ADMM), a simple but powerful algorithm that is well suited to convex optimization. We demonstrate the performance of the proposed algorithm on real data examples. The simulation result easily approve that the Modified Convex Data Clustering (MCDC) algorithm provides separation more than the Convex Data Clustering (CDC) algorithm. Furthermore, complexity of solving the second part of MCDC problem is reduced from O(n2) to O(n).