10 research outputs found
Exploring efficient: numerical methods for differential equations
Numerical analysis is a way to do higher mathematical problems on a computer, a technique widely used by scientists and engineers to solve their problems. A major advantage of numerical analysis is that a numerical answer can be obtained even when a problem has no “analytical” solution. Results from numerical analysis are an approximation, which can be made as accurate as desired. The analysis of errors in numerical methods is a critically important part of the study of numerical analysis. Hence, we will see in this research that computation of the error is a must as it is a way to measure the efficiency of the numerical methods developed.
Numerical methods require highly tedious and repetitive computations that can only be done using the computer. Hence in this research, it is shown that computer programs must be written for the implementation of numerical methods. In the early part of related research the computer language used was Fortran. Subsequently more and more computer programs used the C programming language. Additionally, now computations can also be carried out using softwares like MATLAB, MATHEMATICA and MAPLE.
Many physical problems that arise from ordinary differential equations (ODEs) have magnitudes of eigenvalues which vary greatly, and such systems are commonly known as stiff systems. Stiff systems usually consist of a transient solution, that is, a solution which varies rapidly at the beginning of the integration. This phase is referred to as the transient phase and during this phase, accuracy rather than stability restricts the stepsize of the numerical methods used. Thus the generally the structure of the solutions suggests application of specific methods for non-stiff equations in the transient phase and specific methods for stiff equations during the steady-state phase in a manner whereby computational costs can be reduced.
Consequently, in this research we developed embedded Runge-Kutta methods for solving stiff differential equations so that variable stepsize codes can be used in its implementation. We have also included intervalwise partitioning, whereby the system is considered as non-stiff first, and solved using the method with simple iterations, and once stiffness is detected, the system is solved using the same method, but with Newton iterations. By using variable stepsize code and intervalwise partitioning, we have been able to reduce the computational costs.
With the aim of increasing the computational efficiency of the Runge-Kutta methods, we have also developed methods of higher order with less number of stages or function evaluations. The method used is an extension of the classical Runge-Kutta method and the approximation at the current point is based on the information at the current internal stage as well as the previous internal stage. This is the idea underlying the construction of Improved Runge-Kutta methods, so that the resulting method will give better accuracy.
Usually higher order ordinary differential equations are solved by converting them into a system of first order ODEs and using numerical methods suitable for first order ODEs. However it is more efficient, in terms of accuracy, number of function evaluations as well as computational time, if the higher order ODEs can be solved directly (without being converted to a system of first order ODEs), using numerical methods. In this research we developed numerical methods, particularly Runge-Kutta type methods, which can directly solve special third order and fourth order ODEs.
Special second order ODE is an ODE which does not depend on the first derivative. The solution from this type of ODE often exhibits a pronounced oscillatory character. It is well known that it is difficult to obtain accurate numerical results if the ODEs are oscillatory in nature. In order to address this problem a lot of research has been focused on developing methods which have high algebraic order, reduced phase-lag or dispersion and reduced dissipation. Phaselag is the angle between the true and approximate solution, while dissipation is the difference between the approximate solution and the standard cyclic solution. If a method has high algebraic order, high order of dispersion and dissipation, then the numerical solutions obtained will be very accurate. Hence in this research we have developed numerical methods, specifically hybrid methods which have all the above mentioned properties.
If the solutions are oscillatory in nature, it means that the solutions will have components which are trigonometric functions, that is, sine and cosine functions. In order to get accurate numerical solutions we thus phase-fitted the methods using trigonometric functions. In this research, it is proven that trigonometrically-fitting the hybrid methods and applying them to solve oscillatory delay differential equations result in better numerical results.
These are the highlights of my research journey, though a lot of work has also been done in developing numerical methods which are multistep in nature, for solving higher order ODEs, as well as implementation of methods developed for solving fuzzy differential equations and partial differential equations, which are not covered here
Numerical treatment of special second order ordinary differential equations: general and exponentially fitted methods
2010 - 2011The aim of this research is the construction and the analysis of new families of numerical
methods for the integration of special second order Ordinary Differential Equations
(ODEs). The modeling of continuous time dynamical systems using second order ODEs
is widely used in many elds of applications, as celestial mechanics, seismology, molecular
dynamics, or in the semidiscretisation of partial differential equations (which leads to
high dimensional systems and stiffness). Although the numerical treatment of this problem
has been widely discussed in the literature, the interest in this area is still vivid,
because such equations generally exhibit typical problems (e.g. stiffness, metastability,
periodicity, high oscillations), which must efficiently be overcome by using suitable
numerical integrators. The purpose of this research is twofold: on the one hand to construct
a general family of numerical methods for special second order ODEs of the type
y00 = f(y(t)), in order to provide an unifying approach for the analysis of the properties
of consistency, zero-stability and convergence; on the other hand to derive special
purpose methods, that follow the oscillatory or periodic behaviour of the solution of the
problem...[edited by author]X n. s
Multi-Value Numerical Modeling for Special Di erential Problems
2013 - 2014The subject of this thesis is the analysis and development of new numerical methods
for Ordinary Di erential Equations (ODEs). This studies are motivated by the
fundamental role that ODEs play in applied mathematics and applied sciences in
general. In particular, as is well known, ODEs are successfully used to describe
phenomena evolving in time, but it is often very di cult or even impossible to nd
a solution in closed form, since a general formula for the exact solution has never
been found, apart from special cases. The most important cases in the applications
are systems of ODEs, whose exact solution is even harder to nd; then the role played
by numerical integrators for ODEs is fundamental to many applied scientists. It is
probably impossible to count all the scienti c papers that made use of numerical
integrators during the last century and this is enough to recognize the importance
of them in the progress of modern science. Moreover, in modern research, models
keep getting more complicated, in order to catch more and more peculiarities of
the physical systems they describe, thus it is crucial to keep improving numerical
integrator's e ciency and accuracy.
The rst, simpler and most famous numerical integrator was introduced by Euler
in 1768 and it is nowadays still used very often in many situations, especially in educational
settings because of its immediacy, but also in the practical integration of
simple and well-behaved systems of ODEs. Since that time, many mathematicians
and applied scientists devoted their time to the research of new and more e cient
methods (in terms of accuracy and computational cost). The development of numerical
integrators followed both the scienti c interests and the technological progress
of the ages during whom they were developed. In XIX century, when most of the calculations
were executed by hand or at most with mechanical calculators, Adams and
Bashfort introduced the rst linear multistep methods (1855) and the rst Runge-
Kutta methods appeared (1895-1905) due to the early works of Carl Runge and
Martin Kutta. Both multistep and Runge-Kutta methods generated an incredible
amount of research and of great results, providing a great understanding of them
and making them very reliable in the numerical integration of a large number of
practical problems.
It was only with the advent of the rst electronic computers that the computational
cost started to be a less crucial problem and the research e orts started to
move towards the development of problem-oriented methods. It is probably possible
to say that the rst class of problems that needed an ad-hoc numerical treatment was
that of sti problems. These problems require highly stable numerical integrators
(see Section ??) or, in the worst cases, a reformulation of the problem itself.
Crucial contributions to the theory of numerical integrators for ODEs were given
in the XX century by J.C. Butcher, who developed a theory of order for Runge-Kutta
methods based on rooted trees and introduced the family of General Linear Methods
together with K. Burrage, that uni ed all the known families of methods for rst
order ODEs under a single formulation. General Linear Methods are multistagemultivalue
methods that combine the characteristics of Runge-Kutta and Linear
Multistep integrators... [edited by Author]XIII n.s
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described
Stability Analysis of Plates and Shells
This special publication contains the papers presented at the special sessions honoring Dr. Manuel Stein during the 38th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference held in Kissimmee, Florida, Apdl 7-10, 1997. This volume, and the SDM special sessions, are dedicated to the memory of Dr. Manuel Stein, a major pioneer in structural mechanics, plate and shell buckling, and composite structures. Many of the papers presented are the work of Manny's colleagues and co-workers and are a result, directly or indirectly, of his influence. Dr. Stein earned his Ph.D. in Engineering Mechanics from Virginia Polytechnic Institute and State University in 1958. He worked in the Structural Mechanics Branch at the NASA Langley Research Center from 1943 until 1989. Following his retirement, Dr. Stein continued his involvement with NASA as a Distinguished Research Associate
Proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress
Published proceedings of the 2018 Canadian Society for Mechanical Engineering (CSME) International Congress, hosted by York University, 27-30 May 2018
Annual Report of the Board of Regents of the Smithsonian Institution, showing the operations, expenditures, and condition of the Institution to July, 1889
Annual Report of the Smithsonian Institution. 1 July. HMO 224 (pts. 1 and 2), 51-1, v20-21. 18llp. [2779-2780] Research related to the American India