Chein-Shan Liu¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.43, No.3, pp. 253-276, 2009, DOI:10.3970/cmes.2009.043.253

Abstract Since the works of Newton and Lagrange, interpolation had been a mature technique in the numerical mathematics. Among the many interpolation methods, global or piecewise, the polynomial interpolation p(x) = a₀ + a₁x + ... + a_nxⁿ expanded by the monomials is the simplest one, which is easy to handle mathematically. For higher accuracy, one always attempts to use a higher-order polynomial as an interpolant. But, Runge gave a counterexample, demonstrating that the polynomial interpolation problem may be ill-posed. Very high-order polynomial interpolation is very hard to realize by numerical computations. In this paper we propose a new polynomial interpolation… More >

M. Azreg-Aïnou¹

CMES-Computer Modeling in Engineering & Sciences, Vol.42, No.1, pp. 1-18, 2009, DOI:10.3970/cmes.2009.042.001

Abstract Adomian polynomials (AP's) are expressed in terms of new objects called reduced polynomials (RP's). These new objects, which carry two subscripts, are independent of the form of the nonlinear operator. Apart from the well-known two properties of AP's, curiously enough no further properties are discussed in the literature. We derive and discuss in full detail the properties of the RP's and AP's. We focus on the case where the nonlinear operator depends on one variable and construct the most general analytical expressions of the RP's for small values of the difference of their subscripts. It is shown that each RP… More >

Chia-Cheng Tsai¹

CMES-Computer Modeling in Engineering & Sciences, Vol.27, No.3, pp. 151-162, 2008, DOI:10.3970/cmes.2008.027.151

Abstract In this paper we develop analytical particular solutions for the polyharmonic and the products of Helmholtz-type partial differential operators with Chebyshev polynomials at right-hand side. Our solutions can be written explicitly in terms of either monomial or Chebyshev bases. By using these formulas, we can obtain the approximate particular solution when the right-hand side has been represented by a truncated series of Chebyshev polynomials. These formulas are further implemented to solve inhomogeneous partial differential equations (PDEs) in which the homogeneous solutions are complementarily solved by the method of fundamental solutions (MFS). Numerical experiments, which include eighth order PDEs and three-dimensional… More >

L. Parussini¹, V. Pediroda²

CMES-Computer Modeling in Engineering & Sciences, Vol.23, No.1, pp. 29-52, 2008, DOI:10.3970/cmes.2008.023.029

Abstract In this paper different Polynomial Chaos methods coupled to Fictitious Domain approach have been applied to one- and two- dimensional elliptic problems with multi uncertain variables in order to compare the accuracy and convergence of the methodologies. Both intrusive and non-intrusive methods have been considered, with particular attention to their employment for quantification of geometric uncertainties. A Fictitious Domain approach with Least-Squares Spectral Element approximation has been employed for the analysis of differential problems with uncertain boundary domains. Its main advantage lies in the fact that only a Cartesian mesh, that represents the enclosure, needs to be generated. Excellent accuracy… More >

L. Parussini¹, V. Pediroda²

CMES-Computer Modeling in Engineering & Sciences, Vol.22, No.1, pp. 41-64, 2007, DOI:10.3970/cmes.2007.022.041

Abstract In this paper the Non-Intrusive Polynomial Chaos Method coupled to a Fictitious Domain approach has been applied to one- and two-dimensional elliptic problems with geometric uncertainties, in order to demonstrate the accuracy and convergence of the methodology. The main advantage of non-intrusive formulation is that existing deterministic solvers can be used. A new Least-Squares Spectral Element method has been employed for the analysis of deterministic differential problems obtained by Non-Intrusive Polynomial Chaos. This algorithm employs a Fictitious Domain approach and for this reason its main advantage lies in the fact that only a Cartesian mesh needs to be generated. Excellent… More >

Shang Zhang^{1, *}, Seyedmehdi Karimi², Shahaboddin Shamshirband^{3, 4, *}, Amir Mosavi^5,6

CMC-Computers, Materials & Continua, Vol.58, No.2, pp. 567-583, 2019, DOI:10.32604/cmc.2019.02795

Abstract In the process of eliminating variables in a symbolic polynomial system, the extraneous factors are referred to the unwanted parameters of resulting polynomial. This paper aims at reducing the number of these factors via optimizing the size of Dixon matrix. An optimal configuration of Dixon matrix would lead to the enhancement of the process of computing the resultant which uses for solving polynomial systems. To do so, an optimization algorithm along with a number of new polynomials is introduced to replace the polynomials and implement a complexity analysis. Moreover, the monomial multipliers are optimally positioned to multiply each of the… More >

Y. A. Amer¹, A. M. S. Mahdy^{1, 2, *}, E. S. M. Youssef¹

CMC-Computers, Materials & Continua, Vol.54, No.2, pp. 161-180, 2018, DOI:10.3970/cmc.2018.054.161

Abstract In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method. The fractional derivatives are described in the Caputo sense. The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations. More >

Xiaoming Zhang¹, Jiangong Yu^1,2, Min Zhang¹, Dengpan Zhang¹

CMC-Computers, Materials & Continua, Vol.48, No.3, pp. 163-179, 2015, DOI:10.3970/cmc.2015.048.163

Abstract The characteristics of the guided waves propagation in functionally graded rods with rectangular cross-section (finite width and height) under initial stress are investigated in this paper based on Biot’s theory of incremental deformation. An extended orthogonal polynomial approach is present to solve the coupled wave equations with variable coefficients. By comparisons with the available results of a rectangular aluminum rod, the validity of the present approach is illustrated. The dispersion curves and displacement profiles of various rectangular functionally graded rods are calculated to reveal the wave characteristics, and the effects of different width to height ratios and initial stress and… More >

Yuchun Duan¹, Xiaoming Zhang^2,3, Yuqing Wang², Jiangong Yu²

CMC-Computers, Materials & Continua, Vol.43, No.3, pp. 153-174, 2014, DOI:10.3970/cmc.2014.043.153

Abstract The ring ultrasonic transducers are widely used in the ocean engineering and medical fields. This paper proposes a double orthogonal polynomial series approach to solve the wave propagation problem in a functionally graded piezoelectric-piezomagnetic (FGPP) ring with a rectangular cross-section. Through numerical comparison with the available reference results for a pure elastic homogeneous rectangular bar, the validity of the proposed approach is illustrated. The dispersion curves and displacement distributions of various FGPP rectangular bars are calculated to reveal their wave characteristics. The results can be used for the design and optimization of the ring FGPP transducers. More >