ABSTRACT
In this thesis, we constructed a four-step
fourth derivative exponentially fitted integrator of order six and a six-step
third derivative exponentially fitted integrator of order eight for the
numerical integration of initial value problems in first order ordinary
differential equations. The integrators
which possess free parameters are based on predictor-corrector mode. The
constructed formula of order six and eight are casted into an exponentially
fitted formula. The stability analysis of the new
methods was examined and the methods were implemented using Fortran program to
solve some initial value problems in ordinary differential equations. Finally,
the numerical results show that the new methods compete favourably with the
existing methods in the literature.
CHAPTER ONE
INTRODUCTION
1.1 Background to the study
A differential equation is an
equation involving a relation between an unknown function and one or more of
its derivatives. Differential equations are among the most important
mathematical tools used in producing models in the physical sciences, biological
sciences, social and management sciences and engineering. They occur in
connection with the mathematical description of problems that are encountered
in various branches of science. Consequently, it constitutes a large and very
important aspect of today’s mathematics.
Differential
equation is a process by which solutions can be sort to some real life
problems. These problems can either be solved by the use of analytical
techniques or by numerical methods. Since most ordinary differential equations
are not analytically solvable, numerical methods are often better option. Many
methods have been proposed and used by different authors with the aim of
providing accurate solutions to the various types of differential equations.
Differential equation is divided into two parts, ordinary differential equation
and partial differential equations; here our work is centred on proposing a
technique that can solve problems in ordinary differential equations, although
many of such methods already exist. Our focus here is on numerical solutions to
ordinary differential equations with particular emphasis on the use of linear
multistep methods.
Stiff
differential systems including the building energy simulation problems, are
difficult and costly to compute. Standard explicit solvers are compact, and
time stepping with them is cheap, but many active increments are required.
Implicit solvers offer stability for any time increment at the cost of a lot of
computation per step. What is needed is a method that can take a long time cheaply.
Exponential fitting methods offer this option. Abhulimen (2006).
The
rational behind the development of this kind of numerical integrator is that
exponentially fitted formulae possess a large region of absolute stability when
compared to conventional ones, Hochbruck, Lubich, Selhfer (1998).
In
the last decades, several authors such as Enright (1974), Enright and Pryce
(1983), Brown (1977), Cash (1981), Jackson and Kenue (1974) Voss (1988),
Okunuga (1994), Abhulimen and Okunuga (2008), and Abhulimen and Omeike (2011)
developed second derivative integrators for the numerical solutions of stiff
differential equations. These integrators however were found to be A-stable,
particularly for stiff problems whose solutions have exponential functions.
================================================================
Item Type: Postgraduate Material | Attribute: 135 pages | Chapters: 1-5
Format: MS Word | Price: N3,000 | Delivery: Within 30Mins.
================================================================
No comments:
Post a Comment