Numerical Analysis Methods and Applications. .::. Invited Lecture

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An efficient method for solving 2-D fractional Schrödinger equation with the Riesz-Feller derivative

Nasser Hassan Sweilam , Muneer M. Abou Hasan

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

ABSTRACT

In this paper, we present an accurate numerical method for solving a space fractional Schrödinger equation in two dimensions. The quantum Riesz-Feller fractional derivative is used to define the fractional derivatives. The weighted average non-standard finite difference method is implemented to study the behavior of the model problem. The stability analysis of the proposed method is given by a recently proposed procedure similar to the standard John von Neumann stability analysis, moreover the truncation error is analyzed. Some numerical test examples are presented with variety values of derivative order α,1<α≤2 and skewness θ. It is found that the proposed method is easy to implement, effective and convenient for solving the proposed model.

FREE CONVECTION BOUNDARY LAYER FLOW DUE TO OXYTACTIC MICROORGANISMS ABOVE AN INCLINED PLATE EMBEDDED IN A NON-DARCY POROUS MEDIUM SATURATED BY A NANOFLUID

F.M. Hady1 , R. A. Mohamed , Omima A. Abo zaid , A. Mahdy

1Department of Mathematics, Faculty of Sciences, Assiut University, Assiut 71515, Egypt

2Department of Mathematics, Faculty of Sciences, South Valley University, Qena 83523, Egypt

2Department of Mathematics, Faculty of Sciences, South Valley University, Qena 83523, Egypt

2Department of Mathematics, Faculty of Sciences, South Valley University, Qena 83523, Egypt

ABSTRACT

Free convection flow with heat and mass transfer of a water-based nanofluid containing nanoparticles and motile microorganisms over a semi-infinite inclined flat plate embedded in a non Darcy porous medium is investigated numerically. The partial differential equations (PDEs) are transformed to nonlinear ordinary differential equations (NODEs) using similarity transformations, which are then solved numerically using fourth-order Runge–Kutta method. Both Brownian motion and thermophoresis effects are incorporated into the nanfluid model. The effect of inclination angle on the flow is tested. Also the effects of the governing parameters on the dimensionless quantities like velocity, temperature, nanoparticle concentration, density of motile microorganisms, local Nusselt, local Sherwood and local density numbers for both nanoparticles and motile microorganism density are explored. From this extensive study, it is found that the dimensionless velocity decreases near the wall with an increase in the inclination angle and bioconvection Rayleigh number, but the opposite behavior become clear to stay away from the wall, in addition to the dimensionless temperature increases with increasing bioconvection Rayleigh number and the inclination angle. It is also found that nanofluid and bioconvection parameters have strong effects on local Nusselt, Sherwood and density numbers.

PROJECT THE FOR COMPUTER SCIENCE COURSE PROJECT MAN-
AGEMENT
Department of Computer Science & Technology
Federal University of Technology Nigeria.
Minna - Zungeru Rd, Minna,Nigeria,1983
+2340903 849 1766
federaluniversitytechnologymin@gmail.com
ABSTRACT
Project-based assignments are widely used in Computer Science courses to give students hands-on experience
in using the learned knowledge to solve problems. However, despite its importance, management on student
projects is usually done in an ad-hoc fashion, with varying degrees of interaction and guidance to students.
This paper presents a general, Wiki-based project management framework: project activities are centralized
and captured by a dedicated Wiki site, consisting of Wiki pages created by both the instructor and students.
The Project Wiki framework supports all types of project activities, and enables flexible and multi-dimension
interaction patterns for instructor, individual student and the whole class.
INTRODUCTION
Project-based assignments have been widely used in Computer Science courses [2, 11, 10]; in fact, most
recently published CS textbooks include companion web sites offering course project materials, for example,
see [7, 4]. By doing course projects, students deepen their understanding on course subject matter and
gain hands-on experience in using their learned knowledge to solve problems — the main goal in active
learning. However, despite its importance, management on student projects is usually done in an ad-hoc
fashion: instructors use different tools and techniques, specific to a particular course or instructor, with
varying degrees of interaction and guidance to students.
To address these issues, this paper presents a general, Wiki-based project management framework which
can be easily modified and used in typical computer science courses. Using Project Wiki, project activities are
centralized and captured by a dedicated Wiki site, consisting of Wiki pages created by both the instructor and
students. Instructors create template pages to guide student through projects and highlight key requirements;
students create and maintain their own Wiki pages for project planning and logging; instructors review
student pages to check project status and give feedback; students can also review and add comments to each
other’s Wiki; finally, the whole class can access and edit common group Wiki pages to engage in broader
community-style collective learning.
The contributions of this paper are the following:
•
we developed a general and flexible Wiki-based project management framework;
•
we present the structure and setup of the Project Wiki framework, and explore enabled activities;
•
we describe our experience using it in a project-oriented compiler class.
The rest of the paper is organized as follows: Section 2 describes the background and motivation of
Wiki-based project management; Section 3 describes the Project Wiki framework, and the users, activities,
and typical settings. Section 4 presents our experience using Project Wiki in a project-oriented compiler
class. We conclude in Section 5.

On the Efficiency of Modified Variational Iteration Method for the Solution of a Class of Nonlinear Partial Differential Equation

Dr. Morufu Oyedunsi OLAYIWOLA

Department of Mathematical & Physical Sciences, Osun State University, Osogbo, Nigeria.

ABSTRACT

In this paper, a Modified Variational Iteration Method is discussed. Typical application to some nonlinear partial differential equations is presented.
The results show a rapid convergent of the method with less iteration and without linearization.

Multilevel Hamiltonian Monte Carlo for Quantifying Uncertainty in Reservoir Simulation

Doaa Elsakout , Mike Christie , Gabriel Lord

Heriot-Watt University

Heriot-Watt University

Heriot-Watt University

ABSTRACT

Uncertainty quantification is an important task in reservoir simulation. The main idea of uncertainty quantification is to compute the distribution of a quantity of interest, for example field oil production rate (FOPR). This uncertainty then feeds into the decision making process.
A statistically valid way of quantifying the uncertainty is by a Markov Chain Monte Carlo (MCMC) method, such as Metropolis-Hastings (Standard MCMC). Standard MCMC can be prohibitively expensive when the function evaluations take a long time, as in the case of reservoir simulation. Hamiltonian Monte Carlo accelerates the convergence of Standard MCMC, but may lead to a large increase in computational cost because it requires the gradient.
In this paper, we present a new way to accelerate convergence for MCMC called Multilevel Hamiltonian Monte Carlo (MLHMC). The idea behind the MLHMC technique is a simple telescoping sum which represents the quantity of interest (e.g. the mean), on the finest grid in terms of the same quantity on a coarse grid plus a series of corrections. First, numerous cheap Hamiltonian Monte Carlo realizations are run on the coarse grid. A small fraction of realizations are then run on the finer grids to compute correction terms. This reduces the computational cost and simulation errors significantly. MLHMC is a combination of the Multilevel Monte Carlo method with a Hamiltonian Monte Carlo algorithm.
MLHMC has been implemented on a two real fields called Teal South and Scapa to assess the uncertainty. We show that (a) MLHMC get the same p-quantiles as Hamiltonian Monte Carlo within the sampling error, (b) MLHMC is significantly faster than Hamiltonian Monte Carlo and decreases the computational cost for HMC. Also, we show how the choice of the experimental design affects the solution.

Solving Time-Fractional Order Telegraph Equation Via Sinc-Legendre Collocation Method

N. H. Sweilam , A. M. Nagy , Adel A. El-Sayed

Departement of mathematics, Faculty of science, Cairouniversity

Departement of mathematics, Faculty of science, Banha university

Departement of mathematics, Faculty of science, Fayoum university

ABSTRACT

In this paper, We introduce a numerical method for solving fractional order telegraph equation.
The method depends basically on the fact that an expansion of approximated solution in a series of Sinc function and shifted Legendre polynomials. The fractional derivative is expressed in the Caputo definition of fractional derivatives. The expansion coefficients are then determined by
reducing the time fractional order telegraph equation with its boundary and initial conditions to a system of a algebraic equations for these coefficients. This system may be solved numerically by using the Newton's iteration method. Several numerical examples are introduced to demonstrate the reliability and effectiveness of the introduced method.

Single and Dual Solutions of Fractional Order Differential Equations Based on Modified VIM and Simpson Rule

Mourad S. Semary , Hany N. Hassan , Ahmed G. Radwan

aDepartment of Basic Engineering Sciences, Benha Faculty of Engineering, Benha University, Egypt

aDepartment of Basic Engineering Sciences, Benha Faculty of Engineering, Benha University, Egypt

Engineering Mathematics and Physics Department, Cairo University, 12613, Egypt. Nanoelectronics Integrated Systems Center (NISC), Nile University, Egypt.

ABSTRACT

An efficient method is proposed for solving fractional differential equations with strongly terms like (exp, sin, cos,…). The proposed method uses a combination of variational iteration method (VIM) with an auxiliary parameter and Simpson rule. This method effectively applies to Bratu problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. The proposed approach is also tested on other fractional differential equations. The results demonstrate reliability, simplicity and efficiency of the approach developed.

Mono-Implicit Runge Kutta Method for Singularly Perturbed Delay Differential Equations

Fathalla A. Rihan

Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, 15551, UAE

ABSTRACT

In this paper, we provide unconditionally stable method based on mono-implicit Runge-Kutta schemes for numerical approximations of singularly perturbed stiff delay differential equations. Numerical Stability analysis is also investigated.
The schemes are suitable for both stiff and non-stiff initial value problems.
Numerical simulations are provided to show the effectiveness of the methods for stiff problems.

Delay differential model of tumor-immune system with Immuno-Chemotherapy and optimal Control

Fathalla A. Rihan

Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, 15551, UAE

ABSTRACT

A delay differential model with optimal control is presented to describe the dynamics of tumour-immune interactions in the presence of immuno-chemotherapy treatments. The role of interleukin-2 (IL-2) in stimulation of the effector cells and tumour dynamics is considered in the model with
a discrete time-delay to justify
the time required to stimulate the effector cells.
An expression for the length of the time-delay to preserve stability is deduced.
Two optimal control variables are
incorporated to identify the best treatment
strategy with
minimum side effects by blocking the production of new tumour cells and keeping the level of normal cells above the average of its carrying capacity.
%Existence of the optimal controls and optimality system are established.
Pontryagin's maximum principle is applicable to characterize the optimal controls. An algorithm, to approximate the solution of the optimal control problem, is suggested by solving the state system (forward) and adjoint system (backward) in time. The numerical simulations show that combination therapy protocol of immuno-chemotherapy reduces the tumour cells load in few months of therapy.

Simulations of Fluid-Structure Interactions with Complex Free Surface Flows using an Incompressible Smoothed Particle Hydrodynamics

Abdelraheem Mahmoud Aly Abdallah

Assistant Professor in Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt

ABSTRACT

In this study, we simulated fluid-structure interaction (FSI) with complex free surface flows using Incompressible Smoothed Particle Hydrodynamics (ISPH) method in two and three dimensions. The governing equations are discretized and solved using ISPH method. In the current algorithm of ISPH method, the pressure was evaluated by solving pressure Poisson equation using a semi-implicit algorithm based on the projection scheme to ensure divergence free velocity field and density invariance condition. In addition, we improved the boundary treatment between the moving rigid body and surrounding fluid. The motions of a rigid body have been computed by direct integration of fluid pressure at the position of each particle on the body surface. The equations of translational and rotational motions were integrated in time to update the position of the rigid body at each time step. In this study, an. The force exerted on the moving rigid boundary particles by the particles surrounding it is calculated by SPH approximation. The applicability and efficiency of current ISPH method are tested by comparison with reference experimental results.

Numerical solution of fractional integro-differential equations by least squares
method and shifted Chebyshev polynomials of the third kind method

K. R. Mohamed1 , A. M. S. Mahdy , E. M. H. Mohamed3

;3 Department of Mathematics, Faculty of Science, Alazhar University, Cairo, Egypt

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

;3 Department of Mathematics, Faculty of Science, Alazhar University, Cairo, Egypt

ABSTRACT

In this paper, an implementation of an efficient numerical method of linear fractional
integro-differential equations by least squares method with aid of shifted Chebyshev
polynomials of the third kind method. The fractional derivative is described in the Caputo
sense. The method is based upon shifted Chebyshev polynomials of the third kind approximations is introduced. Some numerical examples are presented to illustrate the theoretical results and compared with the results obtained by other numerical methods. We have computed the numerical results using Mathematica 9 programming.

A Matrix Iterative Techniqe for The Solution of Fredholm Integral Equations of The Second Kind

Nermein.A.Saber , Shoukralla.E.S , El-Serafi.S.A

teaching assistant at BUC university

head of mathematic department at Future university

proffessor of mathematics, Menofia university

ABSTRACT

A matrix iterative algorithm is given for the approximate solution of fredholm Integral equations of the second kind. Tve algorithm modifies the ideas of the iterated kernel via Hilbert matrix. Thus reducing the required solution so that only one coefficient matrix is computed.

An improved upper bounds for the sum of the eigenvalues of some trees

Abdallah W. Aboutahoun , Fatma A. El-Safty

Associative Professor, Department of Mathematics, Faculty of Science, Alexandria University

Assistant Lecturer, Department of Mathematics, Faculty of Science, Damanhour University

ABSTRACT

Let G =(V,E) be a simple connected graph with n and |E| edges. The Laplacian matrix of the graph G is defined by L(G)=D(G)-A(G), where D(G) and A(G) denote the diagonal matrix of vertex-degrees and the adjacency matrix of G, respectively. Let S_k (G) be the sum of the largest k eigenvalues of L(G). In this paper, we study the Laplacian spectrum of trees. We give an improved upper bounds on S_k (T) for some trees T, where it is proved in literature that S_k (T)≤|E|+2k-1 for any tree T.

Exact Computations of Trigonometric Sums by Hermite Interpolation

Mahmoud Annaby , Hassan A. Hassan

Cairo University

Cairo University

ABSTRACT

We use Hermite interpolation formula to derive closed forms for some trigonometric sums.
The used nodes are the zeros of Chebyshev polynomials of first and
second kinds. We show that some of the resulting formulas are
integral valued and we give asymptotic formulas.

Comparitve Studies for the Fractional Optimal Control in
Transmission Dynamics of West Nile Virus

D. G. Mohamed , N. H. Sweilam , O. M. Saad

Department of Mathematics, Faculty of Science, Helwan University

Department of Mathematics, Faculty of Science, Cairo University

Department of Mathematics, Faculty of Science, Helwan University

ABSTRACT

In this paper, numerical studies for the nonlinear fractional-order optimal control
problem (FOCP) for the transmission dynamics of West Nile Virus (WNV) are presented.
Two numerical methods are used to study the FOCP. The methods are, the
iterative optimal control method (IOCM) and the generalized Euler method (GEM).
The fractional derivatives are described in the Caputo sense. Pontryagin's maximum
principle is applied to solve this FOCP, where the optimality system is composed
of nine nonlinear ordinary dierential equations (NODES). Comparative studies are
presented, it can be concluded that IOCM is better than GEM.

Numerical Study for the Fractional Optimal Control Problem Using Different Method

A. M. S. Mahdy , A. A. H. Mtawa

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

ABSTRACT

In this paper, we presents an numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in Caputo sence. We obtain the approximate solutions for fractional optimal control of linear systems, which have a quadratic performance index. Using the homotopy Perturbation Sumudu transform method (HPSTM) and Picard method (PM) are applied for solving the extreme conditions obtained from the Pontryagins maximum principle. We construct a fractional optimal feedback contol law. The ressults reveal that the
proposed methods are very effiective and simple.

Shifted Chebyshev polynomials of the third kind solution for the multi-order
nonlinear fractional dierential equations

A. M. S. Mahdy , R. T. Shwayyea2

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt

2Department of Mathematics, Faculty of Science, Al-Qadisiyah University, Iraq

ABSTRACT

In this paper, an efficient numerical method for solving the multi-order nonlinear fractional
differential equations (NFDEs) is considered. The fractional derivative is described in the Caputo sense. The method is based upon shifted Chebyshev polynomials of the third kind approxima-
tions. The properties of shifted Chebyshev polynomials of the third kind approximations are
utilized to reduce multi-order NFDEs to a system of nonlinear of algebraic equations, which
solved by the well known method, Newton method. Numerical simulation of multi-order NFDEs
is presented and the results are compared with the traditional methods, variational iteration
method and the homotopy perturbation method.

Professor, Department of Mathematics and Computer Science, Faculty of Science, Alexandria University.

Associate Professor, Department of Mathematics and Computer Science, Faculty of Science, Alexandria University.Abdallah Aboutahoun

Ph.D. Student, Department of Mathematics and Computer Science, Faculty of Science, Alexandria University.

ABSTRACT

In this paper some nonlinear semi-definite programming problem is considered, which represents the ${\cal H}_\infty$--synthesis problem for discrete-time systems. Two numerical methods are presented for tackling this problem. The first one is a modified Lagrangian method while the second is an augmented Lagrangian method. The difficulty of finding a starting feasible point of the numerical methods with respect to the positive definite constraints of the problem is handled by solving an eigenvalue assignment problem. The two methods are tested numerically on wide
range of test problems from the literature.

On finite element approximation in the L^{∞}-norm of system of parabolic quasi variational inequalities

Mohamed el Amine BENCHEIKH LE HOCINE

Tamanghasset University Centre, Algeria

ABSTRACT

This paper deals with the numerical analysis of system of parabolic quasi variational inequalities related to stochastic control problems. An optimal L^{∞}-convergence of a piecewise linear finite element method is established using the concept of subsolution and discrete regularity.