Algorithm For Boundary Value Problems Research Articles

Extended Residual Power Series Algorithm for Boundary Value Problems

In this article, modification of the residual power series method (RPSM) is proposed for higher order boundary value problems (BVPs). The proposed algorithm is tested against various linear and nonlinear BVPs of orders nine up to thirteen. For the efficiency check of RPSM, obtained series solutions are compared with other available results in the literature. Analysis indicates that RPSM is better in terms of accuracy as compared to other mentioned schemes. As RPSM is applicable to BVPs without linearization, discretization, and perturbation, hence practically it is the best suitable solution tool for more complex BVPs in science and engineering.

A new finite difference algorithm for boundary value problems involving transmission conditions

Abstract The finite difference method (FDM) is used to find an approximate solution to ordinary and partial differential equations of various type using finite difference equations to approximate derivatives. The idea is to replace ordinary or partial derivatives appearing in the boundary-value problem by finite differences that approximate them. There is an extensive literature on this topic. But, as a rule, ordinary differential equations or partial differential equations were studied without an internal singular point and without corresponding transmission conditions. It is our main goal here to develop finite difference method to deal with an boundary value problem involving additional transmission conditions at the interior singular point. In this study, we have proposed a new modification of classical FDM for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transmission conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To demonstrate the efficiency and reliability of the proposed modification of FDM an illustrative example is solved b y this method. The obtained results are compared with those obtained by the standard FDM and by the analytical method. Corresponding graphical illustrations are also presented.

A GPU-based multi-level algorithm for boundary value problems

A novel geometric single-grid multi-level (GSGML) algorithm is presented for the numerical solution of boundary value problems (BVPs). The algorithm is specifically designed for parallel scalability with high occupancy of streaming processors inside many-core architectures like the Graphics Processing Unit (GPU). The algorithm consists of iterative, superposed operations on a single grid, and is composed of two full-grid routines: a restriction and a coarsened interpolation–relaxation. The restriction is a local average of scalar or vector fields, and the interpolation–relaxation is a simultaneous application of coarsened finite-difference operators and interpolations in complementary subsets of the grid. The routines are scheduled in a saw-like refining cycle. Convergence to machine precision is achieved repeating the full cycle using residuals and superposition. The algorithm is attractive for its computational simplicity and provides a competitive GPU-based fast solver for elliptic partial differential equations like the Poisson–Helmholtz equation. Applications shown in this work include the deformation of two-dimensional grids, the computation of three-dimensional streamlines for a singular trifoil-knot vortex and the calculation of three-dimensional electric potentials in heterogeneous dielectric media.

The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations

This paper introduces the improved LS-SVM algorithms for solving two-point and multi-point boundary value problems of high-order linear and nonlinear ordinary differential equations. To demonstrate the reliability and powerfulness of the improved LS-SVM algorithms, some numerical experiments for third-order, fourth-order linear and nonlinear ordinary differential equations with two-point and multi-point boundary conditions are performed. The idea can be extended to other complicated ordinary differential equations.

Solution of initial and boundary value problems by the variational iteration method

The Variational Iteration Method (VIM) is an iterative method that obtains the approximate solution of differential equations. In this paper, it is proven that whenever the initial approximation satisfies the initial conditions, VIM obtains the solution of Initial Value Problems (IVPs) with a single iteration. By using this fact, we propose a new algorithm for Boundary Value Problems (BVPs): linear and nonlinear ones. Main advantage of the present method is that it does not use Green’s function, however, it has the same effect that it produces the exact solution to linear problems within a single, but simpler, integral. In order to show the effectiveness of the method we give some examples including linear and nonlinear BVPs.

COMPUTATIONAL OPTIMIZATION FOR INTEGRAL TRANSFORM ALGORITHMS APPLIED TO THE LID-DRIVEN CAVITY FLOW PROBLEM

An analysis is presented of the computational optimization for integral transform algorithms using the streamfunction only formulation of the Navier-Stokes equations, as applied to the steady incompressible laminar flow of a Newtonian fluid in two-dimensional formulation. The classical lid-driven rectangular cavity flow problem is considered in order to revise the conventional development of the Generalized Integral Transform Technique (GITT). The GITT is applied transforming the partial differential equation into a system of coupled ordinary differential equations, which is numerically solved by a general algorithm for boundary value problems, using a subroutine from the IMSL library with automatic error control. The conventional algorithm written in FORTRAN language is modified and tested seeking its optimization. A few different strategies of applying the technique are considered to achieve improved computational performance and allowing the inspection of convergence rates in the eigenfunction expansion of the original potentials for high Reynolds numbers. These different algorithm alternatives are analyzed and the relative merits are discussed. Results for different values of the Reynolds number and cavity aspect ratios are presented in tabular and graphical forms and fully converged results are critically compared against previously published findings.

Renorm-group symmetry for functionals of boundary value problem solutions

Recent advances in generalizing the renorm-group algorithm for boundary value problems of mathematical physics and the related concept of the renorm-group symmetry, previously formulated with reference to models based on differential equations, are revisited. The algorithm and symmetry are now formulated for models with nonlocal (integral) equations. Examples illustrate how the updated algorithm applies to models with nonlocal terms appearing as linear functionals of the solution.

Optimal Korobov Coefficients for Good Lattice Points in Quasi Monte Carlo Algorithms

The optimal coefficients in the sense of Korobov serve for obtaining good lattice points sets. By using these sets for multidimensional numerical integration, numerical solution of integral equation and other related applications, the relative error of corresponding Quasi Monte Carlo algorithms can be kept relatively small. This study deals with finding optimal coefficients for good lattice points for high dimensional problems in weighted Sobolev and Korobov spaces. Two Quasi Monte Carlo algorithms for boundary value problems are proposed and analyzed. For the first of them the coefficients that characterize the good lattice points are found “component-by-component”: the (k + l)th coefficient is obtained by one-dimensional search with all previous k coefficients kept unchanged. For the second algorithm, the coefficients depending on single parameter are found in Korobov's form. Some numerical experiments are made to illustrate the obtained results.

Parallel algorithms for boundary value problems

In the present paper we discuss a general approach to solve boundary value problems numerically in a parallel environment. The basic algorithm consists of two steps: the local step, where all the P available processors work in parallel, and the global step, where one processor solves a tridiagonal linear system of the order P. The main advantages of this approach are twofold: First, this suggested approach is very flexible, especially in the local step, and thus the algorithm can be used with any number of processors and with any of the SIMD or MIMD machines. Second, the communication complexity is very small and thus can be used as easily with shared memory machines. Several examples uing this strategy are discussed.

Decoupling and Stability of Algorithms for Boundary Value Problems

The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated.