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lagrange’s spectral Collocation Method for Numerical Approximations of Two-Dimensional space Fractional Diffusion Equation

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American Journal of Computational Mathematics 2018年第2期8卷 121-136页

作者： Hasib Uddin Molla Mushfika Hossain Nova Department of Mathematics University of Dhaka Dhaka Bangladesh Institute of Natural sciences United International University Dhaka Bangladesh

Due to the ability to model various complex phenomena where classical calculus failed, fractional calculus is getting enormous attention recently. There are several approaches available for numerical approximations of various types of fractional differential equations. For fractional diffusion equations spectral collocation is one of the efficient and most popular ap-proximation techniques. In this research, we introduce spectral collocation method based on lagrange’s basis polynomials for numerical approximations of two-dimensional (2D) space fractional diffusion equations where spatial fractional derivative is described in Riemann-Liouville sense. We consider four different types of nodes to generate lagrange’s basis polynomials and as collocation points in the proposed spectral collocation technique. spectral collocation method converts the diffusion equation into a system of ordinary differential equations (ODE) for time variable and we use 4th order Runge-Kutta method to solve the resulting system of ODE. Two examples are considered to verify the efficiency of different types of nodes in the proposed method. We compare approximated solution with exact solution and find that lagrange’s spectral collocation method gives very high accuracy approximation. Among the four types of nodes, nodes from Jacobi polynomial give highest accuracy and nodes from Chebyshev polynomials of 1st kind give lowest accuracy in the proposed method.

关键词： lagrange’s spectral Method 2D Fractional Diffusion Equation Collocation Method

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Comparison of Numerical Approximations of One-Dimensional space Fractional Diffusion Equation Using Different Types of Collocation Points in spectral Method Based on lagrange’s Basis Polynomials

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American Journal of Computational Mathematics 2017年第4期7卷 469-480页

作者： Mushfika Hossain Nova Hasib Uddin Molla sajeda Banu Institute of Natural sciences United International University Dhaka Bangladesh Department of Mathematics University of Dhaka Dhaka Bangladesh

Recently many research works have been conducted and published regarding fractional order differential equations. There are several approaches available for numerical approximations of the solution of fractional order diffusion equations. spectral collocation method based on lagrange’s basis polynomials to approximate numerical solutions of one-dimensional (1D) space fractional diffusion equations are introduced in this research paper. The proposed form of approximate solution satisfies non-zero Dirichlet’s boundary conditions on both boundaries. Collocation scheme produce a system of first order Ordinary Differential Equations (ODE) from the fractional diffusion equation. We applied this method with four different sets of collocation points to compare their performance.

关键词： Fractional Diffusion Equation spectral Method Collocation Method lagrange’s Basis Polynomial

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Multivariate Ratio Estimator of the Population Total under stratified Random sampling

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Open Journal of statistics 2012年第3期2卷 300-304页

作者： Oscar O. Ngesa G. O. Orwa R. O. Otieno H. M. Murray Department of statistics Jomo Kenyatta University of Agriculture and Technology Nairobi Ministry of state for Planning National Development and V2030 Kenya

Olkin [1] proposed a ratio estimator considering p auxiliary variables under simple random sampling. As is expected, simple Random sampling comes with relatively low levels of precision especially with regard to the fact that its variance is greatest amongst all the sampling schemes. We extend this to stratified random sampling and we consider a case where the strata have varying weights. We have proposed a Multivariate Ratio Estimator for the population mean in the presence of two auxiliary variables under stratified Random sampling with L strata. Based on an empirical study with simulations in R statistical software, the proposed estimator was found to have a smaller bias as compared to Olkin’s estimator.

关键词： Ratio Estimator stratification Auxiliary Variables lagrange’s Multiplier

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A negative answer to Problem 10 of P. Turan

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science China Mathematics 1996年第10期39卷 1054-1057页

作者：史应光 Institute of Computational Mathematics and scientific/Engineering Computing Chinese Academy of Sciences Beijing 100080 China

A weight w(x) is provided with the properties: w(x)>(1-x2) l/2 and lagrange's interpolation based on the zeros of orthogonal polynomials with respect to w diverges in Lp (p>6) for some [-1, 1], This gives a negative a... 详细信息

关键词： mean convergence lagrange’s interpolation weight functions.

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GENERALIZED CONTINUED FRACTIONs TO FIND ALL COMPLEX ROOTs OF ALGEBRAIC EQUATIONs sIMULTANEOUsLY

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Chinese science Bulletin 1988年第15期33卷 1235-1237页

作者：王兴华郑士明 Hangzhou University Hangzhou

Let f（z）be a monic polynomial of degree n with complex coefficients and n complex numbers Z1, ..., Zn be different from each other. Constructing

关键词： continued fraction complex root algebraic equation lagrange’s interpolation polynomial

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