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On Convergence of the Modified Noor Iterations for a Family of Three Maps
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Mogbademu Adesanmi Alao
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Volume 4, Issue 2, 2012
pp.
54 - 62
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Received
28 December 2011,
Accepted
13 April 2012
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Abstract.
In this paper, we prove the convergence of a modified Noor iterations for solving nonlinear equations under some mild conditions. An example is given to illustrate the efficiency of the Noor iterations. Our results represent an improvement of previously known results.
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Keywords.
The Noor iteration; Strongly accretive mappings; Strongly pseudocontractive mappings; Strongly $\Phi$- pseudocontractive operators; Banach spaces.
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Cubic Transformational High Dimensional Model Representation and Hermite-Padé Approximation
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Erdoğan Şen, Kamil Oruçoğlu
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Volume 4, Issue 2, 2012
pp.
42 - 53
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Received
06 January 2012,
Accepted
30 March 2012
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Abstract.
This work focuses on the development of a multivariate function approximating method by using cubic transformational high dimensional model representation (THDMR). The method uses the target function’s image under a cubic transformation for HDMR instead of the function’s itself. Also we compare this method with Hermite–Padé approximation.
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Keywords.
Cubic Equation; Transformational High Dimensional Model Representation; Multivariate Functions; Approximation; Hermite–Padé approximation.
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Separation Axioms for $g^*p$-Closed Sets in Bitopological Spaces
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A. Vadivel, A. Swaminathan
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Volume 4, Issue 2, 2012
pp.
36 - 41
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Received
13 February 2012,
Accepted
29 March 2012
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Abstract.
The object of the present paper is to present some more characterizations and properties of the concepts of pairwise $g^*p$-$T_k$, pairwise $g^*p$-regular and pairwise $g^*p$-normal spaces, $k=0,~1,~2$.
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Keywords.
Pairwise $g^*p$-$T_k$; Pairwise $g^*p$-regular and pairwise $g^*p$-normal spaces.
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Generalized Differential Transform Based Analytic Algorithm for Fractional Advection-Dispersion Equation
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Ram K. Pandey, Vipul K. Baranwal, Manoj P. Tripathi, Om P. Singh
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Volume 4, Issue 2, 2012
pp.
14 - 35
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Received
25 September 2011,
Accepted
16 February 2012
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Abstract.
Fractional advection-dispersion equation (FADE) is a generalization of the classical ADE in which the first order time derivative and first and second order space derivatives are replaced by Caputo derivatives of orders $(0<\alpha\leq1)$, $(0<\beta\leq1)$ and $(0<\gamma\leq1)$, respectively. We use Caputo definition to avoid (i) mass balance error (ii) hyper-singular improper integral (iii) non-zero derivative of constant and (iv) fractional derivative involved in the initial condition which is often ill-defined. We present a generalized differential transform method (GDTM) to solve FADE. The proposed method converges to the numerical/exact solution of the FADE as the fractional orders $\alpha,\beta,\gamma$ tend to their integral values. Numerical examples are given to illustrate the proposed method. Example 4 describes the intermediate processes between advection and dispersion via Caputo fractional derivative.
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Keywords.
Differential transform method; Fractional advection-dispersion equation; Caputo derivative; Intermediate physical process.
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An Off-Step Discretization for the Solution of 1-D Non-linear Wave Equations with Variable Coefficients
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R.K. Mohanty, Venu Gopal
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Volume 4, Issue 2, 2012
pp.
1 - 13
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Received
13 September 2011,
Accepted
26 January 2012
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Abstract.
In this paper, we propose a new three-level implicit scheme of based on off-step discretization for the solution of 1-D non-linear hyperbolic partial differential equation of the form subject to appropriate initial and Dirichlet boundary conditions, where k>0 and h>0 are mesh sizes in time and space dimensions respectively. We require only 9-grid points and a single computational cell. The proposed method is directly applicable to hyperbolic equations in polar coordinates, which is main advantages of our work. We do not require any special technique to handle singular coefficients of the differential equation. We describe the derivation procedure in details. The proposed method when applied to a linear hyperbolic equation is shown to be unconditionally stable. Numerical results illustrate the fourth order convergence of the proposed method.
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Keywords.
Non-linear hyperbolic equation; Off-step discretization; Variable coefficients; Singular coefficients; Wave equation in polar coordinates; Vander Pol equation; Telegraphic equation; Maximum absolute errors.
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On the Contiguous Relations of Appell's Hypergeometric Functions
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Hossam A Ghany, Mohamed Saad Mohamed
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Volume 4, Issue 1, 2012
pp.
52 - 66
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Received
19 October 2011,
Accepted
25 January 2012
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Abstract.
This paper is devoted to add some formulas for the known collection of differential recursion formulas for Appell’s function $F_1$, the two-dimensional cases which generalize Gauss’s function of one variable. A new method for deriving contiguous relations for hypergeometric series is given. Finally, we will give some consequence of contiguous relations of Appell's hypergeometric function $F_1$.
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Keywords.
Appell's hypergeometric function; Contiguous relations.
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Approximate Solution of Linear Generalized Pantograph Equations with Variable Coefficients on Chebyshev-Gauss Grid
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Yalcin Ozturk, Mustafa Gulsu
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Volume 4, Issue 1, 2012
pp.
36 - 51
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Received
01 August 2011,
Accepted
23 January 2012
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Abstract.
An approximation method is developed for the solution of linear generalized pantograph equations under the mixed conditions. The approach is based on the second kind Chebyshev polynomials. Using the zeroes of the second kind Chebyshev polynomials, the error analysis and convergence for the proposed method is also introduced. Finally some experiments and their numerical solutions are given.
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Keywords.
Pantograph equations; Chebyshev polynomials; Approximation method; Chebyshev-Gauss grid; Collocation method.
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Strictly Convex Quadratic Underestimator using Halton Sequence and Rand Function
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Akram Taati, Maziar Salahi
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Volume 4, Issue 1, 2012
pp.
24 - 35
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Received
18 August 2011,
Accepted
21 January 2012
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Abstract.
Strictly convex quadratic underestimators are widely used in finding global minimum of energy functions in computational biology. An important step of the developed algorithms is the refinement process which picks a number of points in the current search region, constructs a strictly convex quadratic underestimator, and finds its global minimum that is an approximation to the global minimum of energy function. After that it is used to determine the new search region. In this paper, we use Halton sequence to pick points in the current search region instead of MATLAB's rand function. The motivation is due to the fact that Halton sequence results to more uniformly distributed points in the search region than MATLAB's rand function. Our computational experiments on two known classes of test problems show the efficiency of Halton sequence.
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Keywords.
Strictly convex quadratic underestimation; Energy functions; Semidefinite optimization; Halton sequence.
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Finite Difference Solution for Absorbing Boundary Condition over Dielectric Surfaces
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Biswajeet Mukherjee
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Volume 4, Issue 1, 2012
pp.
13 - 23
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Received
03 August 2011,
Accepted
09 January 2012
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Abstract.
The Absorbing boundary condition has been an active area of research, ever since the evolution of computational techniques to analyze complex and vivid structures. The conventional idea of simulation in any solver implements the component under analysis in a box like structure and applying the Absorbing Boundary Condition on the sides of the box. The various ABCs used for this purpose are Mur ABC, PML, UPML, etc. In this paper, a novel ABC which can terminate on a dielectric surface, has been implemented. The derivation of this mathematical model opens up the space for a more accurate EM field analysis of the different structures to be simulated enclosed in a dielectric material with comparable accuracy. The results have been verified against a simple EM problem by calculating the transmission coefficient when an EM wave impinges perpendicularly on a lossless dielectric interface for a One Dimensional EM wave.
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Keywords.
Finite difference time domain (FDTD); One dimensional wave; Transmission coefficient; Absorbing boundary condition (ABC).
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Solution of Lane-Emden Type Equations Using Chebyshev Wavelet Operational Matrix
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Rajesh K. Pandey, Abhinav Bhardwaj, Narayan Kumar
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Volume 4, Issue 1, 2012
pp.
1 - 12
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Received
01 August 2011,
Accepted
10 November 2011
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Abstract.
In this paper, a new numerical method is presented for linear and non-linear Lane- Emden type equations using Chebyshev wavelet operational matrix. The proposed approach is different from other numerical techniques as it is based on integration matrix of Chebyshev wavelets. Some illustrative examples are given to demonstrate the efficiency and validity of the proposed algorithm.
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Keywords.
Lane-Emden equations; Chebyshev wavelet operational matrix; Isothermal gas spheres equation.
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