287 research outputs found
Parameter identification for an abstract Cauchy problem by quasilinearization
A parameter identification problem is considered in the context of a linear abstract Cauchy problem with a parameter-dependent evolution operator. Conditions are investigated under which the gradient of the state with respect to a parameter possesses smoothness properties which lead to local convergence of an estimation algorithm based on quasi-linearization. Numerical results are presented concerning estimation of unknown parameters in delay-differential equations
Method of the quasilinearization for nonlinear impulsive differential equations with linear boundary conditions
The method of quasilinearization for nonlinear impulsive differential equations with linear boundary conditions is studied. The boundary conditions include periodic boundary conditions. It is proved the convergence is quadratic
Maneuver simulations of flexible spacecraft by solving TPBVP
The optimal control of large angle rapid maneuvers and vibrations of a Shuttle mast reflector system is considered. The nonlinear equations of motion are formulated by using Lagrange's formula, with the mast modeled as a continuous beam. The nonlinear terms in the equations come from the coupling between the angular velocities, the modal coordinates, and the modal rates. Pontryagin's Maximum Principle is applied to the slewing problem, to derive the necessary conditions for the optimal controls, which are bounded by given saturation levels. The resulting two point boundary value problem (TPBVP) is then solved by using the quasilinearization algorithm and the method of particular solutions. In the numerical simulations, the structural parameters and the control limits from the Spacecraft Control Lab Experiment (SCOLE) are used. In the 2-D case, only the motion in the plane of an Earth orbit or the single axis slewing motion is discussed. In the 3-D slewing, the mast is modeled as a continuous beam subjected to 3-D deformations. The numerical results for both the linearized system and the nonlinear system are presented to compare the differences in their time response
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
An extended method of quasilinearization for nonlinear impulsive differential equations with a nonlinear three-point boundary condition
In this paper, we discuss an extended form of generalized quasilinearization technique for first order nonlinear impulsive differential equations with a nonlinear three-point boundary condition. In fact, we obtain monotone sequences of upper and lower solutions converging uniformly and quadratically to the unique solution of the problem
Generalized quasilinearization method for nonlinear boundary value problems with integral boundary conditions
The quasilinearization method coupled with the method of upper and lower solutions is used for a class of nonlinear boundary value problems with integral boundary conditions. We obtain some less restrictive sufficient conditions under which corresponding monotone sequences converge uniformly and quadratically to the unique solution of the problem. An example is also included to illustrate the main result
On paired decoupled quasi-linearization methods for solving nonlinear systems of differential equations that model boundary layer fluid flow problems.
Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.Two numerical methods, namely the spectral quasilinearization method (SQLM) and the spectral
local linearization method (SLLM), have been found to be highly efficient methods for solving
boundary layer flow problems that are modeled using systems of differential equations. Conclusions
have been drawn that the SLLM gives highly accurate results but requires more iterations
than the SQLM to converge to a consistent solution. This leads to the problem of figuring out how
to improve on the rate of convergence of the SLLM while maintaining its high accuracy.
The objective of this thesis is to introduce a method that makes use of quasilinearization in pairs
of equations to decouple large systems of differential equations. This numerical method, hereinafter
called the paired quasilinearization method (PQLM) seeks to break down a large coupled
nonlinear system of differential equations into smaller linearized pairs of equations. We describe
the numerical algorithm for general systems of both ordinary and partial differential equations. We
also describe the implementation of spectral methods to our respective numerical algorithms. We
use MATHEMATICA to carry out the numerical analysis of the PQLM throughout the thesis and
MATLAB for investigating the influence of various parameters on the flow profiles in Chapters 4, 5
and 6.
We begin the thesis by defining the various terminologies, processes and methods that are applied
throughout the course of the study. We apply the proposed paired methods to systems of ordinary
and partial differential equations that model boundary layer flow problems. A comparative study is
carried out on the different possible combinations made for each example in order to determine the
most suitable pairing needed to generate the most accurate solutions. We test convergence speed
using the infinity norm of solution error. We also test their accuracies by using the infinity norm of
the residual errors. We also compare our method to the SLLM to investigate if we have successfully
improved the convergence of the SLLM while maintaining its accuracy level. Influence of
various parameters on fluid flow is also investigated and the results obtained show that the paired
quasilinearization method (PQLM) is an efficient and accurate method for solving boundary layer
flow problems. It is also observed that a small number of grid-points are needed to produce convergent
numerical solutions using the PQLM when compared to methods like the finite difference
method, finite element method and finite volume method, among others. The key finding is that
the PQLM improves on the rate of convergence of the SLLM in general. It is also discovered that
the pairings with the most nonlinearities give the best rate of convergence and accuracy
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