Teaching and learning using mathematics software "The New Challenge"

Teaching and learning in mathematics curriculums in Universities by using mathematical software can be a difficult and demanding task, especially for novice learners. This paper presents efficient mathematical tool for teaching and learning of Linear Algebra courses. MAPLE software tool was used for teaching and learning of parts of the Linear Algebra course. Using MAPLE in teaching and learning mathematical concepts is a great challenge both from a didactical and a scientific point of view. In this paper we provide mathematical examples by using mathematical software and necessary steps as evidences that didactically it increases mathematical skills. By using interactive Maple worksheets and animated graphics, students can find the opportunity of numerous experiments that provide well understanding. Further the use of Maple provide conceptual and meaningful understanding for the student, several Maple can be designed to see, geometrical application of Linear Algebra topics. Indeed, utilizing ICT and particular the use of interactive facilities of Maple in teaching and learning which will provide a new challenge to both mathematics educators as well as students

An efficient approach for solving nonlinear troesch's and bratu's problems by wavelet analysis method

We introduce Chebyshev wavelet analysis method to solve the nonlinear Troesch and Bratu problems. Chebyshev wavelets expansions together with operational matrix of derivative are employed to reduce the computation of nonlinear problems to a system of algebraic equations. Several examples are given to validate the efficiency and accuracy of the proposed technique. We compare the results with those ones reported in the literature in order to demonstrate that the method converges rapidly and approximates the exact solution very accurately by using only a small number of Chebyshev wavelet basis functions. Convergence analysis is also included

On an unified reduction formula for Srivastava's triple hypergeometric series F(3)[x, y, z]

Very recently, by applying the so-called Beta integral method to the Henrici’s triple product formula for the generalized hypergeometric series, Choi, et al.[Commun. Korean Math. Soc. 28(2013), No.2, pp. 297-301] have obtained an interesting reduction formula for the Srivastava’s triple hypergeometric series F⁽ᶟ⁾[x,y,z]. The aim of this short note is to provide a unified reduction formula for the Srivastava’s triple hyper geometric series from which as many new reduction formulas (including the one obtained by Choi, etal.) as desired can be deduced. A few interesing special cases have also been given

Fourier operational matrices of differentiation . . .

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods

Effect of heat and mass transfer and rotation on peristaltic flow through a porous medium with compliant walls

Purpose: The purpose of this paper is to investigate the peristaltic flow of an incompressible Newtonian fluid in a channel with compliant walls. The effects of rotation and heat and mass transfer are also taken into account. The governing equations of two dimensional fluid have been simplified under long wavelength and low Reynolds number approximation. An exact solutions is presented for the stream function, temperature, concentration field, velocity and heat transfer coefficient. Design/methodology/approach: The effect of the concentration distribution, heat and mass transfer and rotation on the wave frame are analyzed theoretically and computed numerically. Numerical results are given and illustrated graphically in each case considered. Comparison was made with the results obtained in the presence and absence of rotation and heat and mass transfer. Findings: The results indicate that the effect of the permeability and rotation are very pronounced in the phenomena. Originality/value: The objective of the present analysis is to analyze the effects of rotation, heat and mass transfer and compliant walls on the peristaltic flow of a viscous fluid

Numerical solution of nonlinear fredholm integro-differential equations using spectral homotopy analysis method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions

Hadamard upper bound on optimum joint decoding capacity of Wyner Gaussian cellular MAC

This article presents an original analytical expression for an upper bound on the optimum joint decoding capacity of Wyner circular Gaussian cellular multiple access channel (C-GCMAC) for uniformly distributed mobile terminals (MTs). This upper bound is referred to as Hadamard upper bound (HUB) and is a novel application of the Hadamard inequality established by exploiting the Hadamard operation between the channel fading matrix G and the channel path gain matrix Ω. This article demonstrates that the actual capacity converges to the theoretical upper bound under the constraints like low signal-to-noise ratios and limiting channel path gain among the MTs and the respective base station of interest. In order to determine the usefulness of the HUB, the behavior of the theoretical upper bound is critically observed specially when the inter-cell and the intra-cell time sharing schemes are employed. In this context, we derive an analytical form of HUB by employing an approximation approach based on the estimation of probability density function of trace of Hadamard product of two matrices, i.e., G and Ω. A closed form of expression has been derived to capture the effect of the MT distribution on the optimum joint decoding capacity of C-GCMAC. This article demonstrates that the analytical HUB based on the proposed approximation approach converges to the theoretical upper bound results in the medium to high signal to noise ratio regime and shows a reasonably tighter bound on optimum joint decoding capacity of Wyner GCMAC