A. Grebennikov¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.15, No.3, pp. 85-90, 2010, DOI:10.3970/icces.2010.015.085

Abstract New General Ray (GR) method for solution of the Dirichlet boundary value problem for the class of elliptic Partial Differential Equations (PDE) is proposed. GR-method consists in application of the Radon transform directly to the PDE and in reduction PDE to assemblage of Ordinary Differential Equations (ODE). The class of the PDE includes the Laplace, Poisson and Helmgoltz equations. GR-method presents the solution of the Dirichlet boundary value problem for this type of equations by explicit analytical formulas that use the direct and inverse Radon transform. Proposed version of GR-method justified theoretically, realized by fast algorithms and MATLAB software, which… More >

N. Mai-Duy¹, T. Tran-Cong¹

The International Conference on Computational & Experimental Engineering and Sciences, Vol.3, No.3, pp. 127-132, 2007, DOI:10.3970/icces.2007.003.127

Abstract This lecture presents an overview of the Integral Collocation formulation for numerically solving partial differential equations (PDEs). However, due to space limitation, the paper only describes the latest development, namely schemes based only on one-dimensional (1D) integrated interpolation even in multi-dimensional problems. The proposed technique is examined with Chebyshev polynomials and radial basis functions (RBFs). The latter can be used in both regular and irregular domains. For both basis functions, the accuracy and convergence rates of the new technique are better than those of the differential formulation. More >

Hokwon A. Cho¹, M. A. Golberg², A. S. Muleshkov¹, Xin Li¹

CMC-Computers, Materials & Continua, Vol.1, No.1, pp. 1-38, 2004, DOI:10.3970/cmc.2004.001.001

Abstract In this paper we present a mesh-free approach to numerically solving a class of second order time dependent partial differential equations which include equations of parabolic, hyperbolic and parabolic-hyperbolic types. For numerical purposes, a variety of transformations is used to convert these equations to standard reaction-diffusion and wave equation forms. To solve initial boundary value problems for these equations, the time dependence is removed by either the Laplace or the Laguerre transform or time differencing, which converts the problem into one of solving a sequence of boundary value problems for inhomogeneous modified Helmholtz equations. These boundary value problems are then… More >

Timon Rabczuk^1,2,*, Huilong Ren³, Xiaoying Zhuang^4,5

CMC-Computers, Materials & Continua, Vol.59, No.1, pp. 31-55, 2019, DOI:10.32604/cmc.2019.04567

Abstract A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of… More >

Chein-Shan Liu¹, Chih-Wen Chang²

CMC-Computers, Materials & Continua, Vol.25, No.2, pp. 107-134, 2011, DOI:10.3970/cmc.2011.025.107

Abstract We consider two inverse problems for estimating radiative coefficients α(x) and α(x, y), respectively, in T_t(x, t) = T_xx(x, t)-α(x)T(x, t), and T_t(x, y, t) = T_xx(x, y, t) + T_yy(x, y, t)-α(x, y)T(x, y, t), where a are assumed to be continuous functions of space variables. A Lie-group adaptive method is developed, which can be used to find a at the spatially discretized points, where we only utilize the initial condition and boundary conditions, such as those for a typical direct problem. This point is quite different from other methods, which need the overspecified final time data. Three-fold advantages… More >

F. Bulut^1,2, Ö. Oruç³, A. Esen³

CMES-Computer Modeling in Engineering & Sciences, Vol.108, No.4, pp. 263-284, 2015, DOI:10.3970/cmes.2015.108.263

Abstract In this paper, time fractional one dimensional coupled KdV and coupled modified KdV equations are solved numerically by Haar wavelet method. Proposed method is new in the sense that it doesn’t use fractional order Haar operational matrices. In the proposed method L1 discretization formula is used for time discretization where fractional derivatives are Caputo derivative and spatial discretization is made by Haar wavelets. L₂ and L_∞ error norms for various initial and boundary conditions are used for testing accuracy of the proposed method when exact solutions are known. Numerical results which produced by the proposed method for the problems under… More >

J.R. Zabadal¹, B. Bodmann¹, V. G. Ribeiro², A. Silveira², S. Silveira²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.4, pp. 215-227, 2014, DOI:10.3970/cmes.2014.103.215

Abstract This work presents a new analytical method to transform exact solutions of linear diffusion equations into exact ones for nonlinear advection-diffusion models. The proposed formulation, based on Bäcklund transformations, is employed to obtain velocity fields for the unsteady two-dimensional Helmholtz equation, starting from analytical solutions of a heat conduction type model. More >

P.A. Trapper¹, P.Z. Bar-Yoseph²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.1, pp. 19-47, 2014, DOI:10.3970/cmes.2014.103.019

Abstract The space-time discontinuous Galerkin method for multi-dimensional nonlinear hyperbolic systems is enhanced with a generalized technique for splitting a flux vector that is not limited to the homogeneity property of the flux. This technique, based on the flux’s characteristic decomposition, extends the scope of the method’s applicability to a wider range of problems, including elastodynamics. The method is used for numerical solution of a number of representative problems based on models of vibrating string and vibrating rod that involve the propagation of a sharp front through the solution domain. More >

Xiaojing Liu¹, Jizeng Wang^1,2, Youhe Zhou^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.3, pp. 225-238, 2013, DOI:10.32604/cmes.2013.094.225

Abstract A wavelet method is proposed for solving a class of nonlinear timedependent partial differential equations. Following this method, the nonlinear equations are first transformed into a system of ordinary differential equations by using the modified wavelet Galerkin method recently developed by the authors. Then, the classical fourth-order explicit Runge-Kutta method is employed to solve the resulting system of ordinary differential equations. To justify the present method, the coupled viscous Burgers’ equations are solved as examples, results demonstrate that the proposed wavelet algorithm have a much better accuracy and efficiency than many existing numerical methods, and the order of convergence of… More >

Lifeng Wang¹, Zhijun Meng¹, Yunpeng Ma¹, Zeyan Wu²

CMES-Computer Modeling in Engineering & Sciences, Vol.91, No.4, pp. 269-287, 2013, DOI:10.3970/cmes.2013.091.269

Abstract In this paper, we present a computational method for solving a class of fractional partial differential equations which is based on Haar wavelets operational matrix of fractional order integration. We derive the Haar wavelets operational matrix of fractional order integration. Haar wavelets method is used because its computation is sample as it converts the original problem into Sylvester equation. Finally, some examples are included to show the implementation and accuracy of the approach. More >