الفهرس 
GENERAL INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
The main goal of this thesis, which consists of five chapters, is to study the numerical
solution of functional, neutral and optimal control differential equations using
ultraspherical polynomials.
The thesis consists of five chapters:
In chapter one, the ultraspherical polynomials, the basic concepts of optimization
and optimal control parabolic problems governed by distributed parameter systems
(OCPDPS) are reviewed briefly. The results of the second part of this chapter
have been published in
”International Journal of Mathematical Archive” [35].
In chapter two, we are interested in the numerical solution of an initial-value
problem for neutral functional differential equations (NFDEs). The -stage continuousimplicit Runge-Kutta methods for NFDEs and a finite ultraspherical expansionto approximate and f(.) on the interval Ii = (ti, ti+1], i = 0(1)N − 1 arestudied. Also, an easily implemented numerical method for NFDEs is derived. Finally, we present some numerical examples; which show that the presented method for ( )provides a noticeable improvement in the efficiency over some previously suggested methods and the present method for ( (0.1)1). The results of this chapter have been published in ”International Journal of Computational Engineering Research” [50].
In chapter three, an optimal control parabolic problems governed by distributed
parameter systems (OCPDPS) is studied. The correctness of the considered optimal
control problem is proved, a theorem concerning the sufficient differentiability conditions of the cost functional and its gradient formula based on solving the adjointsystem are also proved.
In chapter four, the numerical solutions are given to solve both the optimal boundary control where one wishes to find the heating regime v(t) and the optimal distributed control of the heat propagation are governed by the parabolic equation with control in the form v = v(x, t). For the first optimal control problem, the functional is minimized by modified partial quadratic interpolation method (MPQIM)without using the analytical gradient formula, but the second problem is minimized by Conjugate Gradient Method (CGM) with analytical gradient formula. The results of this chapter and chapter three have been also published in ”ARPN Journal of Science and Technology” [65].
In chapter five, an optimal control parabolic distributed parameter systems(OCPDPS) is studied. The parabolic state with v = v(t) is solved by ultraspherical polynomials method ( ) which depends on the approximate function y(x, t) and its differentiation yx(xi, t), yxx(xi, t) and the fully implicit method for time, the integral used to find the first term of the cost function. The functional is minimized by the combined modified partial quadratic interpolation and ultraspherical polynomials method. Illustrative examples are presented to show the comparison between the results in chapter four and this chapter.
All computations are made with the compiler of FORTRAN 90.