الفهرس 
1 Chebyshev Method for Neutral Equations with State- Dependent DelaysOnly 14 pages are availabe for public view
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Abstract
This thesis, which consists of three basic chapters, is concerned with the El-Gendi method [25] for the numerical solution of several types of neutral functional differential equations.
Chapter 1: Contains a class of neutral functional differential equaations with state-dependent d’elays. We give a brief survey about two types of such equations described by
a)The scalar initial-value problem for state-dependent delay differential equations of neutral type
x’(t) = F(t, x(t), x(t - a(t, x(t))), x’(t - (3(t, x(t)))), t E [a, b],
w’th the initial condition
x(t) = g(t),
t E [r, a]’
r ~ a < b, where r < a(t, x(t)) < t, r < (3(t, x(t)) < t, and 9 is a given initial function.
b) Neutral functional differential equations (NFDEs) with state-dependent delays (Hereditary systems)
~(x(t) + q(t)x(t - T(t,X(t)))) = J(t,x(t),x(t - O’(t,x(t)))) t > 0,
with the initial condition
x(t) = g(t),
t E [-r,O].
This is the single delay version (m = 1, l = 1) of the more general equaation,
d m
dt(x(t) + Eqi(t)X(t - Ti(t,X(t)))
= J(t, x(t), x(t - O’l(t, x(t)))), ... , x(t - O’[(t, x(t))).
Also in this chapter we apply the EI-Gendi [25] method for the numerrical solution of (NFDEs) to the above two types, The local error and the
convergence of the El-Gendi method are discussed and numerical exammples and a comparison with other methods are given.
Chapter 2: Contains a class of nonlinear Volterra integro-differential equations (VIDEs) with unbounded and bounded delay. In this chapter we give a brief survey about two types of such equations described by
a) The Volterra Integro- Differential Equations with an Infinite delay (VIDEIs) as the form
y(m)(t) = f(t, y(t), ... , y(m-l)(t)) + J~oo k(t, s, y(s), ... , y(m-l)(s))ds, t > 0,
y(t) = ¢(t), on (-00,0] for some given function ¢ E Cm-1( -00,0], with In = 1,2. For m = 1 it takes the form
y’(t) = f(t, y(t)) + J~oo k(t, s, y(t), y(s))ds, t > 0,
y(t) = ¢(t), -00 < t < O.
b)The Volterra Integro-Differential Equations (VIDEs) with any other type of delay takes the form
y’(t) = f(t, y(t), y(t - T(t))) + tt(t) k(t, s, y(s))ds, t > 0,
y(t) = ¢(t), t < 0
with continuous functions T(t) and (3(t) satisfying 0 < T(t) < TO, (30 < ~(t) ~ t. We use application of the El-Gendi [25] method to treat nuumerically the above two types of problems, also the convergence of the method is discussed and numerical examples and a comparison with other methods are given.
Chapter 3: We present the application of anyone-step method vvstage El-Gendi method [25] to obtain approximate solutions for singular neutral equations (SNFDEs) of the type.