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Two-point Boundary Value Problem (two-point + boundary_value_problem)
Selected AbstractsStress analyses of laminates under cylindrical bendingINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2008Tarun Kant Abstract A semi-analytical approach for evaluation of stresses and displacements in composite and sandwich laminates under cylindrical bending subjected to transverse load has been developed in this paper. Two dimensional (2D) partial differential equations (PDEs) of such a laminate are obtained by imposing plane-strain conditions of elasticity. The fundamental dependent variables are so selected in this formulation that they satisfy the continuity of displacements and transverse interlaminar stresses at the laminate interface through the thickness. The set of governing PDEs are transformed into a set of coupled first-order ordinary differential equations (ODEs) in thickness direction by assuming suitable global orthogonal trigonometric functions for the fundamental variables satisfying the boundary conditions. These ODEs are numerically integrated by a specially formulated ODE integrator algorithm involving transformation of a two-point boundary value problem (BVP) into a set of initial value problems (IVPs). Numerical studies on both composite and sandwich laminates for various aspect ratios are performed and presented. Accuracy of the present approach is demonstrated by comparing the results with the available elasticity solution. It is seen that the present results are in excellent agreement with the elasticity solutions. Some new results for sandwich laminates and for uniform loading condition are presented for future reference. Copyright © 2006 John Wiley & Sons, Ltd. [source] A high-order finite difference method for 1D nonhomogeneous heat equationsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2009Yuan Lin Abstract In this article a sixth-order approximation method (in both temporal and spatial variables) for solving nonhomogeneous heat equations is proposed. We first develop a sixth-order finite difference approximation scheme for a two-point boundary value problem, and then heat equation is approximated by a system of ODEs defined on spatial grid points. The ODE system is discretized to a Sylvester matrix equation via boundary value method. The obtained algebraic system is solved by a modified Bartels-Stewart method. The proposed approach is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of our approximation method along with comparisons with those generated by the standard second-order Crank-Nicolson scheme as well as Sun-Zhang's recent fourth-order method. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009 [source] A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) spatial grids in one-dimensional space geometryJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 12 2004aw K. Bieniasz Abstract The validity for finite-difference electrochemical kinetic simulations, of the extension of the Numerov discretization designed by Chawla and Katti [J Comput Appl Math 1980, 6, 189,196] for the solution of two-point boundary value problems in ordinary differential equations, is examined. The discretization is adapted to systems of time-dependent reaction-diffusion partial differential equations in one-dimensional space geometry, on nonuniform space grids resulting from coordinate transformations. The equations must not involve first spatial derivatives of the unknowns. Relevant discrete formulae are outlined and tested in calculations on two example kinetic models. The models describe potential step chronoamperometry under limiting current conditions, for the catalytic EC, and Reinert-Berg CE reaction mechanisms. Exponentially expanding space grid is used. The discretization considered proves the most accurate and efficient, out of all the three-point finite-difference discretizations on such grids, that have been used thus far in electrochemical kinetics. Therefore, it can be recommended as a method of choice. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1515,1521, 2004 [source] Computational optimal control of the terminal bunt manoeuvre,Part 2: minimum-time caseOPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 5 2007S. Subchan Abstract This is the second part of a paper studies trajectory shaping of a generic cruise missile attacking a fixed target from above. The problem is reinterpreted using optimal control theory resulting in a minimum flight time problem; in the first part the performance index was time-integrated altitude. The formulation entails non-linear, two-dimensional (vertical plane) missile flight dynamics, boundary conditions and path constraints, including pure state constraints. The focus here is on informed use of the tools of computational optimal control, rather than their development. The formulation is solved using a three-stage approach. In stage 1, the problem is discretized, effectively transforming it into a non-linear programming problem, and hence suitable for approximate solution with DIRCOL and NUDOCCCS. The results are used to discern the structure of the optimal solution, i.e. type of constraints active, time of their activation, switching and jump points. This qualitative analysis, employing the results of stage 1 and optimal control theory, constitutes stage 2. Finally, in stage 3, the insights of stage 2 are made precise by rigorous mathematical formulation of the relevant two-point boundary value problems (TPBVPs), using the appropriate theorems of optimal control theory. The TPBVPs obtained from this indirect approach are then solved using BNDSCO and the results compared with the appropriate solutions of stage 1. The influence of boundary conditions on the structure of the optimal solution and the performance index is investigated. The results are then interpreted from the operational and computational perspectives. Copyright © 2007 John Wiley & Sons, Ltd. [source] | ||||||||||