Rashid Jan¹, Hassan Khan^2,3, Poom Kumam^4,5,*, Fairouz Tchier⁶, Rasool Shah², Haifa Bin Jebreen⁶

CMC-Computers, Materials & Continua, Vol.68, No.3, pp. 3185-3201, 2021, DOI:10.32604/cmc.2021.015252 - 06 May 2021

Abstract It is eminent that partial differential equations are extensively meaningful in physics, mathematics and engineering. Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior. In the present research, mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives. First, the Helmholtz equations are presented in Caputo’s fractional derivative. Then Natural transformation, along with the decomposition method, is used to attain the series form solutions of the suggested problems. For justification of the proposed technique, it is More >

Miaomiao Yang, Wentao Ma, Yongbin Ge^*

CMES-Computer Modeling in Engineering & Sciences, Vol.126, No.1, pp. 25-54, 2021, DOI:10.32604/cmes.2021.012575 - 22 December 2020

Abstract In this paper, Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation. First of all, the interpolation basis function is applied to treat the spatial variables and their partial derivatives, and the collocation method for solving the second order differential equations is established. Secondly, the differential equations on a given test node. Finally, based on three kinds of test nodes, numerical experiments show that the present scheme can not only calculate the high wave numbers problems, but also calculate the variable wave numbers problems. More >

Kue-Hong Chen^{1, *}, Cheng-Tsung Chen^{2, 3}

CMC-Computers, Materials & Continua, Vol.64, No.1, pp. 145-160, 2020, DOI:10.32604/cmc.2020.08864 - 20 May 2020

Abstract In this study, we applied a defined auxiliary problem in a novel error estimation technique to estimate the numerical error in the method of fundamental solutions (MFS) for solving the Helmholtz equation. The defined auxiliary problem is substituted for the real problem, and its analytical solution is generated using the complementary solution set of the governing equation. By solving the auxiliary problem and comparing the solution with the quasianalytical solution, an error curve of the MFS versus the source location parameters can be obtained. Thus, the optimal location parameter can be identified. The convergent numerical More >

Jingen Xiao^1,*, Chengyu Ku^1,2

The International Conference on Computational & Experimental Engineering and Sciences, Vol.22, No.4, pp. 177-177, 2019, DOI:10.32604/icces.2019.05068

Abstract This paper presents a novel boundary-type meshless method for solving the two-dimensional modified Helmholtz equation in multiply connected regions. Numerical approximation is obtained by the superposition principle of the non-singular basis functions satisfied the governing equation. The advantage of the proposed method is that the locations of the source points are not sensitive to the results. The novel concept may resolve the major issue for the method of fundamental solutions (MFS). In contrast to the collocation Trefftz method (CTM), the Trefftz order of the non-singular basis functions can be reduced since the multiple source points More >

J. E. Sebold¹, L. A. Lacerda², J. A. M. Carrer³

CMES-Computer Modeling in Engineering & Sciences, Vol.106, No.2, pp. 127-145, 2015, DOI:10.3970/cmes.2015.106.127

Abstract This paper deals with a Finite Element Method (FEM) for the approximation of the Helmholtz equation for two dimensional problems. The acoustic boundary conditions are weakly posed and an auxiliary problem with homogeneous boundary conditions is defined. This auxiliary approach allows for the formulation of a general solution method. Second order finite elements are used along with a discretization parameter based on the fixed wave vector and the imposed error tolerance. An explicit formula is defined for the mesh size control parameter based on Padé approximant. A parametric analysis is conducted to validate the rectangular 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 >

Chia-Cheng Tsai^1,2, D. L. Young³

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.2, pp. 175-205, 2013, DOI:10.3970/cmes.2013.094.175

Abstract It is well known that the method of fundamental solutions (MFS) is a numerical method of exponential convergence. In this study, the exponential convergence of the MFS is demonstrated by obtaining the eigensolutions of the Helmholtz equation. In the solution procedure, the sought solution is approximated by a superposition of the Helmholtz fundamental solutions and a system matrix is resulted after imposing the boundary condition. A golden section determinant search method is applied to the matrix for finding exponentially convergent eigenfrequencies. In addition, the least-squares method of fundamental solutions is applied for solving the corresponding More >

N Pimprikar¹, J Teresa², D Roy^1,3, R M Vasu⁴, K Rajan⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.92, No.1, pp. 33-61, 2013, DOI:10.3970/cmes.2013.092.033

Abstract Nearly pollution-free solutions of the Helmholtz equation for k-values corresponding to visible light are demonstrated and verified through experimentally measured forward scattered intensity from an optical fiber. Numerically accurate solutions are, in particular, obtained through a novel reformulation of the H1 optimal Petrov-Galerkin weak form of the Helmholtz equation. Specifically, within a globally smooth polynomial reproducing framework, the compact and smooth test functions are so designed that their normal derivatives are zero everywhere on the local boundaries of their compact supports. This circumvents the need for a priori knowledge of the true solution on the More >

J.Y. Zhang¹, F.Z. Wang^2,3

CMES-Computer Modeling in Engineering & Sciences, Vol.88, No.2, pp. 141-154, 2012, DOI:10.3970/cmes.2012.088.141

Abstract The boundary knot method (BKM) is a kind of boundary-type meshless method, only boundary points are needed in the solution process. Since the BKM is mathematically simple and easy to implement, it is superior in dealing with Helmholtz problems with high wavenumbers and high dimensional problems. In this paper, we give an overview of the traditional BKM with collocation approach and provide three novel approaches for the BKM, as far as they are relevant for the other boundary-type techniques. The promising research directions are expected from an improved BKM aspect. More >

V. Mallardo¹, M.H. Aliabadi²

CMES-Computer Modeling in Engineering & Sciences, Vol.87, No.2, pp. 77-96, 2012, DOI:10.3970/cmes.2012.087.077

Abstract The present paper intends to couple the Fast Multipole Method (FMM) with the Boundary Element Method (BEM) in the 2D scalar wave propagation. The procedure is aimed at speeding the computation of the integrals involved in the governing Boundary Integral Equations (BIEs) on the basis of the distance between source point and integration element. There are three main contributions. First, the approach is of adaptive type in order to reduce the number of floating-point operations. Second, most integrals are evaluated analytically: the diagonal and off-diagonal terms of the H and G matrices by consolidated techniques, More >