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Search Results: 1 - 10 of 20479 matches for " Mohamed El-Gamel "
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Sinc-Collocation Method for Solving Linear and Nonlinear System of Second-Order Boundary Value Problems  [PDF]
Mohamed El-Gamel
Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.311225
Abstract: Sinc methods are now recognized as an efficient numerical method for problems whose solutions may have singularities, or infinite domains, or boundary layers. This work deals with the sinc-collocation method for solving linear and nonlinear system of second order differential equation. The method is then tested on linear and nonlinear examples and a comparison with B-spline method is made. It is shown that the sinc-collocation method yields better results.
Numerical Solution of Troesch’s Problem by Sinc-Collocation Method  [PDF]
Mohamed El-Gamel
Applied Mathematics (AM) , 2013, DOI: 10.4236/am.2013.44098

A new algorithm is presented for solving Troeschs problem. The numerical scheme based on the sinc-collocation technique is deduced. The equation is reduced to systems of nonlinear algebraic equations. Some numerical experiments are made. Compared with the modified homotopy perturbation technique (MHP), the variational iteration method and the Adomian decomposition method. It is shown that the sinc-collocation method yields better results.

An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems  [PDF]
Mohamed El-Gamel, Mona Sameeh
Applied Mathematics (AM) , 2012, DOI: 10.4236/am.2012.38137
Abstract: In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.
Numerical Solutions for the Time-Dependent Emden-Fowler-Type Equations by B-Spline Method  [PDF]
Mohamed El-Gamel, W. El-bashbashy, Atallah El-Shenawy
Applied Mathematics (AM) , 2014, DOI: 10.4236/am.2014.54056

A numerical method based on B-spline is developed to solve the time-dependent Emden-Fow- ler-type equations. We also present a reliable new algorithm based on B-spline to overcome the difficulty of the singular point at x = 0. The error analysis of the method is described. Numerical results are given to illustrate the efficiency of the proposed method.

Two Very Accurate and Efficient Methods for Solving Time-Dependent Problems  [PDF]
Mohamed El-Gamel, Waleed Adel, M. S. El-Azab
Applied Mathematics (AM) , 2018, DOI: 10.4236/am.2018.911083
In this paper, collocation method based on Bernoulli and Galerkin method based on wavelet are proposed for solving nonhomogeneous heat and wave equations. The two methods have the linear systems solved by suitable solvers. Several examples are given to examine the performance of these methods and a comparison is made.
Chelyshkov-Tau Approach for Solving Bagley-Torvik Equation  [PDF]
Mohamed El- Gamel, Mahmoud Abd-El-Hady, Magdy El-Azab
Applied Mathematics (AM) , 2017, DOI: 10.4236/am.2017.812128
There are few numerical techniques available to solve the Bagley-Torvik equation which occurs considerably frequently in various offshoots of applied mathematics and mechanics. In this paper, we show that Chelyshkov-tau method is a very effective tool in numerically solving this equation. To show the accuracy and the efficiency of the method, several problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that Chelyshkov-tau method is superior to other existing ones and is highly accurate.
Quantum Entanglement as a Consequence of a Cantorian Micro Spacetime Geometry  [PDF]
Mohamed S. El Naschie
Journal of Quantum Information Science (JQIS) , 2011, DOI: 10.4236/jqis.2011.12007
Abstract: Building upon the pioneering work of J. Bell [1] and an incredible result due to L. Hardy [2] it was shown that the probability of quantum entanglement of two particles is a maximum of 9.0169945 percent [2]. This happens to be exactly the golden mean to the power of five (?5) [3-7]. Although it has gone largely unnoticed for a long time, this result was essentially established independently in a much wider context by the present author almost two decades ago [3-6]. The present work gives two fundamentally different derivations of Hardy’s beautiful result leading to precisely the same general conclusion, namely that by virtue of the zero measure of the underlying Cantorian-fractal spacetime geometry the notion of spatial separability in quantum physics is devoid of any meaning [7]. The first derivation is purely logical and uses a probability theory which combines the discrete with the continuum. The second derivation is purely geometrical and topological using the fundamental equations of a theory developed by the author and his collaborators frequently referred to as E-infinity or Cantorian spacetime theory [3-7].
The Missing Dark Energy of the Cosmos from Light Cone Topological Velocity and Scaling of the Planck Scale  [PDF]
Mohamed S. El Naschie
Open Journal of Microphysics (OJM) , 2013, DOI: 10.4236/ojm.2013.33012
Abstract: The paper presents an exact analysis leading to an accurate theoretical prediction of the amount of the mysteriously missing hypothetical dark energy density in the cosmos. The value found, namely 95.4915028% is in full agreement with earlier analysis, the WMAP and the supernova cosmic measurements. The work follows first the strategy of finding a critical point which separates a semi-classical regime from a fully relativistic domain given by topological unit interval velocity parameter then proceeds to wider aspects of a topological quantum field of fractal unit interval. This idea of a critical velocity parameter was first advanced by Sigalotti and Mejias in 2006 who proposed a critical value equal\"\" . A second interesting proposal made in 2012 by Hendi and Sharifzadeh set the critical point at 0.8256645. The present analysis is based upon a light cone velocity quantized coordinate. This leads to the same quantum relativity energy mass relation found in earlier publications by rescaling that of Einstein’s special relativity. Two effective quantum gravity formulae are obtained. The first is for the ordinary measurable energy of the quantum particle\"\" while the second is for dark energy density of the quantum wave which we cannot measure directly and we can only infer its existence from the measured accelerated expansion of the universe E(D) = \"\"where\"\" . The critical velocity parameter in this case arises naturally to be \"\". The results so obtained are validated using a heuristic Lorentzian transformation. Finally the entire methodology is put into the wider perspective of a fundamental scaling theory for the Planck scale proposed by G. Gross.
What Is the Missing Dark Energy in a Nutshell and the Hawking-Hartle Quantum Wave Collapse  [PDF]
Mohamed S. El Naschie
International Journal of Astronomy and Astrophysics (IJAA) , 2013, DOI: 10.4236/ijaa.2013.33024

We reason that in quantum cosmology there are two kinds of energy. The first is the ordinary energy of the quantum particle which we can measure. The second is the dark energy of the quantum wave by quantum duality. Because measurement collapses the Hawking-Hartle quantum wave of the cosmos, dark energy cannot be detected or measured in any conventional manner. The quantitative results are confirmed using some exact solutions for the hydrogen atom. In particular the ordinary energy of the quantum particle is given by E(0) = (\"\"/2)(mc2) where \"\" is Hardy’s probability of quantum entanglement, \"\" =(\"\" - 1)/2 is the Hausdorff dimension of the zero measure thin Cantor set modeling the quantum particle, while the dark energy of the quantum wave is given by E(D) = (5\"\"/2)(mc2) where \"\" is the Hausdorff dimension of the positive measure thick empty Cantor set modeling the quantum wave and the factor five (5) is the Kaluza-Klein spacetime dimension to which the measure zero thin Cantor set D(0) = (0,\"\") and the thick empty set D(-1) = (1,\"\") must be lifted to give the five dimensional analogue sets namely \"\"

Topological-Geometrical and Physical Interpretation of the Dark Energy of the Cosmos as a “Halo” Energy of the Schrödinger Quantum Wave  [PDF]
Mohamed S. El Naschie
Journal of Modern Physics (JMP) , 2013, DOI: 10.4236/jmp.2013.45084

The paper concludes that the energy given by Einstein’s famous formula E = mc2 consists of two parts. The first part is the positive energy of the quantum particle modeled by the topology of the zero set. The second part is the absolute value of the negative energy of the quantum Schr?dinger wave modeled by the topology of the empty set. We reason that the latter is nothing else but the so called missing dark energy of the universe which accounts for 94.45% of the total energy, in full agreement with the WMAP and Supernova cosmic measurement which was awarded the 2011 Nobel Prize in Physics. The dark energy of the quantum wave cannot be detected in the normal way because measurement collapses the quantum wave.

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