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.

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.

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Item Type: Project Material  |  Size: 135 pages  |  Chapters: 1-5
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