We discuss a new single sweep alternating group explicit iteration method, along with a third-order numerical method based on off-step discretization on a variable mesh to solve the nonlinear ordinary differential equation subject to given natural boundary conditions. Using the proposed method, we have solved Burgers’ equation both in singular and nonsingular cases, which is the main attraction of our work. The convergence of the proposed method is discussed in detail. We compared the results of the proposed iteration method with the results of the corresponding double sweep alternating group explicit iteration methods to demonstrate computationally the efficiency of the proposed method. 1. Introduction Consider the general nonlinear ordinary differential equation subject to essential boundary conditions where are finite constants. We assume that for (i) is continuous,(ii) and exist and are continuous,(iii) and for some positive constant . These conditions ensure that the boundary value problem (1) and (2) possesses a unique solution (see Keller [1]). With the advent of parallel computers, scientists are focusing on developing finite difference methods with the property of parallelism. Working on this, in the early 1980s, Evans [2, 3] introduced the Group Explicit methods for large linear system of equations. Further he discussed the Alternating Group Explicit (AGE) method to solve periodic parabolic equations in a coupled manner. Mohanty and Evans applied AGE method along with various high order methods [4, 5] for the solution of two-point boundary value problems. Later, Sukon and Evans [6] introduced a Two-parameter Alternating Group Explicit (TAGE) method for the two-point boundary value problem with a lower order accuracy scheme. In 2003 Mohanty et al. [7] discussed the application of TAGE method for nonlinear singular two point boundary value problems using a fourth-order difference scheme. In 1990, Evans introduced the Coupled Alternating Group Explicit method [8] and applied it to periodic parabolic equations. Many scientists are applying these parallel algorithms to solve ordinary and partial differential equations [9–11]. Recently, Mohanty [12] has proposed a high order variable mesh method for nonlinear two-point boundary value problem. Mohanty and Khosla [13, 14] also devised a new third-order accurate arithmetic average variable mesh method for the solution of the boundary value problem (1) and (2), using three grid points, which is applicable to both singular and nonsingular problems. No special technique is required to handle singular
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