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On a Class of Quasilinear Elliptic Systems Involving the $(p_{1},p_{2},\ldots ,p_{n})$-Laplacian Operators with Critical Case
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Abdelkrim Moussaoui
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Volume 3, Issue 4, 2011
pp.
45 - 57
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Received
04 April 2011,
Accepted
12 December 2011
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Abstract.
In this paper, we study the existence of solutions of nonlinear system involving quasilinear operators -?_{p_{i}}u_{i}=?m_{i}(x)?u_{j}^{a_{i,j}}+f_{i}, i=1,2,...,n, where the coefficients m_{i}(x) are bounded positive functions, ? is a positive parameter and f_{i} are given functions. Under suitable conditions on the exponents a_{i,j} (1=i,j=n) which is a critical case, we prove the existence of solutions defined on bounded and unbounded domains using the theory of nonlinear monotone operators.
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Keywords.
Nonlinear elliptic system; $p$-Laplacian; Monotone operators theory; Critical condition.
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A Remark on the Linearized Stability of Positive Solutions for the p_1,...,p_n-Laplacian Systems Involving Indefinite Weight Functions
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G.A. Afrouzi, S. Shakeri, S. Heidarkhani
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Volume 3, Issue 4, 2011
pp.
40 - 44
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Received
08 June 2011,
Accepted
03 December 2011
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Abstract.
In this note, we discuss stability and instability results of positive solutions for a class of equations involving indefinite weight functions driven by a $(p_1,\ldots,p_n)$-Laplacian.
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Keywords.
$(p_1,\ldots,p_n)$-Laplacian; Elliptic system; Linearized stability.
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Positive Solutions for Third-Order Generalized Sturm-Liouville Boundary Value Problems with p-Laplacian
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Nemat Niamoradi
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Volume 3, Issue 4, 2011
pp.
30 - 39
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Received
16 August 2011,
Accepted
27 November 2011
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Abstract.
In this paper, by employing the Leggett-Williams fixed point theorem, we study the existence of at least three positive solutions of boundary value problems.
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Keywords.
Positive solution; Third-order
ordinary differential equation; Fixed point theorem; p-Laplacian.
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A Chebyshev Approximate Method for Solving Pantograph Equations
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Gamze Yuksel, Mehmet Sezer
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Volume 3, Issue 4, 2011
pp.
14 - 29
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Received
13 April 2011,
Accepted
10 August 2011
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Abstract.
In this paper, a numerical method based on polynomial approximation, using Chebyshev polynomial basis, to obtain the approximate solution of generalized pantograph equations with variable coefficients is presented. The technique we have used is an improved Chebyshev collocation method. Some numerical examples, which consist of initial conditions, are given to illustrate the accuracy and efficiency of the method. Also, the results obtained are compared by the known results; the accuracy of solutions and the error analysis are performed.
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Keywords.
Pantograph equations; Chebyshev collocation method; Chebyshev polynomial.
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Approximate Solution of the Singular-Perturbation Problem on Chebyshev-Gauss Grid
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Mustafa Gulsu, Yalcin Ozturk, Mehmet Sezer
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Volume 3, Issue 4, 2011
pp.
1 - 13
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Received
09 May 2011,
Accepted
08 August 2011
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Abstract.
In this paper, a matrix method is presented for solving singularly perturbed two-point boundary value problems with a boundary layer at one end point. The main characteristic of this technique is that it reduces these problems to those of solving a system of algebraic equations, thus greatly simplifying the problem. The error analysis and convergence for the proposed method is introduced. Finally some experiments and their numerical solutions are given. The results reveal that this method is very effective.
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Keywords.
Singular perturbation problems; Two-point boundary value problems; Shifted Chebyshev polynomials; Chebyshev-Gauss grid; Approximation method; Matrix method.
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A Numerical Solution to a Modified Kawahara Equation
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M. Zarebnia, S. Jalili
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Volume 3, Issue 3, 2011
pp.
65 - 76
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Received
13 March 2011,
Accepted
06 July 2011
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Abstract.
In this paper, numerical solution of modified Kawahara equation by means of the Chebyshev spectral collocation method is considered. First, properties of the Chebyshev spectral collocation method required for our subsequent development are given and utilized to reduce the computation of modified Kawahara equation to some system of ordinary differential equations. Then, we use fourth-order Runge-Kutta formula for the numerical solution of the system of ordinary differential equations. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.
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Keywords.
Chebyshev polynomials; Spectral collocation method; Runge-Kutta; Modified Kawahara equation.
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On the Biultrahyperbolic Heat Kernel
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Wanchak Satsanit
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Volume 3, Issue 3, 2011
pp.
54 - 64
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Received
19 February 2011,
Accepted
23 June 2011
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Abstract.
In this paper, we study the heat equation in $n$ dimensional. We can find the unique solution by method of convolution and Fourier transform in Distribution theory and also obtain an interesting kernel related to the heat equation.
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Keywords.
Prime left; Fourier transform; Spectrum; Ultra-hyperbolic operator.
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Existence Result for a Strongly Coupled Problem with Heat Convection Term and Tresca's Law
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A. Ahmed Bensedik, M. Boukrouche
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Volume 3, Issue 3, 2011
pp.
33 - 53
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Received
03 March 2011,
Accepted
22 June 2011
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Abstract.
We study a problem describing the motion of an incompressible, non-isothermal and non-Newtonian fluid, taking into account the heat convection term. The novelty here is that fluid viscosity depends on the temperature, the velocity of the fluid, and also of the deformation tensor, but not explicitly. The boundary conditions take into account the slip phenomenon on a part of the boundary of the domain. By using the notion of pseudo-monotone operators and fixed point Theorem we prove an existence result of its weak solution.
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Keywords.
Heat convection; Non-Newtonian fluid; Non-isothermal fluid; Tresca fluid-solid conditions; Pseudo-monotone operators; Schauder fixed point theorem.
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A Numerical Approach for Solving Linear Differential Equation Systems
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Suayip Yuzbasi Niyazi Sahin, Mehmet Sezer
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Volume 3, Issue 3, 2011
pp.
8 - 32
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Received
08 March 2011,
Accepted
13 June 2011
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Abstract.
In this paper, an approximation method is developed for the solutions of the systems of high-order linear differential equations with variable coefficients under the mixed conditions. This system is usually difficult to solve analytically. Proposed approach consists of reducing the problem to a linear algebraic equation system by expanding the approximate solutions in terms of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are calculated using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. The results obtained are compared by the known results. We have performed all of the numerical computations on a computer by aid of a program written in MATLAB v7.6.0 (R2008a).
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Keywords.
System of differential equations; Bessel polynomials and series; Bessel polynomial solutions; Approximate solution; Collocation points; Collocation method.
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Derivation of Korteweg-de Vries Flow Equations from Third Order Nonlinear Schrödinger Equation
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Filiz Taşcan
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Volume 3, Issue 3, 2011
pp.
1 - 7
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Received
10 March 2011,
Accepted
10 June 2011
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Abstract.
The nonlinear Shrodinger equation with a third-order dispersive terms is considered. We perform a multiple scales analysis on the third order nonlinear Schrodinger equation and their corresponding Hamiltonian functions. We therefore derive, as amplitude equations, Korteweg-de Vries (KdV) flow equations with the corresponding Hamiltonian functions.
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Keywords.
KdV equation, NLS equation, Hamiltonian function, Multiple Scale Analysis
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