General Approach of the Research Work for The Research Work is in the field of Numerical Analysis
and Computational Science. It starts with presenting more than 100-papers in international, local conferences in addition
to periodicals. Thirty-three of them published in international periodicals were chosen: 10 are published in England, 8 in
U.S.A, 8 in India, 3 in Romania, 3 in Czechoslovakia and 1 in Holland.
Subjects Classification of the 33-International
Publications
A. 16-Papers on Several Subjects of Numerical Analysis:
I) 1-Papers: for Theory of Ordinary Differential Equations [1].
II) 2-Papers: for Numerical Multistep Methods for Initial Value Problems for Ordinary Differential
Equations [2], [3].
III) 2-Papers: for Linear Algebra as Fuzzy Rank [4], and computational Linear Algebra in Control
Theory [5].
IV) 5-Papers: for Spectral Analysis and Chebyshev Expansion to solve Boundary Value Problems
[10], Initial Value Problems [9], Two Point Boundary Value Problems [6], Computational Fluid Dynamics as Blusius Equation
[8] and Laminar Boundary Layer Flow [7].
V) 2-Papers: for Multigrid Method for Anisotropic Elliptic Boundary Value Problems [11]; also
that with Mixed Derivative Coefficients in 2-D and 3-D [12].
VI) 2-Papers: for Flux Limiter High Order High Resolution Methods for Initial Value Problem
for Hyperbolic Conservation Equation, as by Shock Waves Problems [13], [14].
VII) 2- Papers: for Adomian Decomposition Method for Nonlinear Initial Boundary Value Problems for
Partial Differnetial Equations as Burger-Huxley, Burger-Fisher [16], or the Solitary Wave Solution of KdV Equation [15].
B. 17-Papers on Restrictive Approximations:
It is a new approach called Restrictive Approximations. It is the restriction which acts as a combination
between the Interpolation and Fitting of functions. These functions are of one variable, several variables, scalars, matrices
or difference operators. The main idea-applied for finite difference methods is to adapt it such that its solution must agree
for the exact solution not only for the initial and boundary points but also for the other given group of points of certain
level of time. The papers vary due to the types of the Initial Boundary Value Problems (IBVP), if they are Parabolic, Hyperbolic,
Mixed, with Constant or Variable Coefficients, Singularly Perturbed, One Dimension or More. Also for the problems representing
Physical Phenomena like Convection Diffusion, Shrodinger Equations or of Nonlinear Type as KdV Equation or that appears in
Fluid Dynamics as Burger, Fisher, Huxley Equations or there generalized types.
A Survey of the Papers can take the Directions:
1. Restrictive Padé Approximation as an Implicit Method:
I) Parabolic IBVP: 1-D case [17], the Singularly Perturbed case [24], Convection-Diffusion Equation
[18] and 2-D Shrodinger Equation [21].
II) Hyperbolic IBVP: 1-D case [22], 2-D case [23] and the Variable Coefficients Equation [25].
III) Nonlinear problem: generalized Burger Equation [26] and Burger-Fisher and Generalized Fisher
Equations [30].
2. Restrictive Taylor Approximation as an Explicit Methods:
I) Parabolic IBVP: 1-D case [20], 2-D case [29], Singularly Perturbed Equation [27] and Convection-Diffusion
Equation [28].
II) Hyperbolic IBVP: 1-D case [19].
3. General Papers:
I) Comparison between Restrictive Pade, Restrictive Taylor and Adomain Decomposition Methods for
the KdV Equation [31].
II) Solvability, Existence and Uniquenes of Pade Approximation [32], Convergence of Padé Algorithm
to the Exact Solution of IBVP of Certain Parabolic and Hyperbolic Types [33].
The Advantages of Restrictive Approximation Methods
1. The solvability of this Approach as the Existence and Uniqueness of the
Solution.
2. The use of more simple method by Restrictive Taylor Approach with Explicit
Finite Difference Algorithms.
3. The Stability Condition depends on some parameters in addition to the
mish sizes of distance and time as usual.
4. In spite of the Estimation of the local Truncation Error Upper Bound by
the Method for solving all of the considered problems is given. It is proved that the Local Truncation Error is zero, i.e.
the Considered Solution is the Exact one which satisfies the given Differential Equation of the problem - only it happens
for types of Hyperbolic and Parabolic linear problems; it is true for Continuous and Discontinuous Initial and Boundary Conditions
as well. Without loss of generality the solution by the Restrictive Approach Method is of Solution which converges to the
Exact Solution not as usual as the manner for the Finite Difference Methods, in which the Solution converges to an Approximate
Solution, where the series of the Truncation Error is Convergent.
5. For the non-test problems, the exact solution at the used-chosen points
are not known as usual. In this case, we can use an expensive highly accurate finite difference method, as Doglass Method
for example, also with a suitable tiny mesh sizes for time and distances as far as the given facilities of computations are
accepted. It’s not too expensive method because it is applied to small number of the used-chosen points.
