Zuhur Alqahtani¹, Ahmed Eissa Hagag²

Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, Vol.39, No.4, pp. 1-15, 2023, DOI:10.23967/j.rimni.2023.10.003 - 25 October 2023

Abstract This article introduces and illustrates a novel approximation to the compound KdV-Burgers equation. For such a challenge, the q-homotopy analysis transform technique (q-HATM) is a potent approach. The suggested procedure avoids the complexity seen in many other methods and provides an approximation that is extremely near to the exact solution. The uniqueness theorem and convergence analysis of the expected problem are explored with the aid of Banach's fixed-point theory. Through a difference in the fractional derivative, the normal frequency for the fractional solution to this issue changes. All of the discovered solutions are illustrated in More >

Areej Almuneef¹, Ahmed Eissa Hagag²

Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, Vol.39, No.4, pp. 1-8, 2023, DOI:10.23967/j.rimni.2024.01.001 - 29 December 2023

Abstract The analysis of nonlinear events related to physical phenomena is a popular issue in the modern-day. The essential purpose of this work is to discover a novel approximate solution to the fractional nonlinear Benjamin Bona Mahony Peregrine Burgers equation (BBMPB) utilizing the natural decomposition method (NDM) of fractional order. The suggested approach provides analytical solutions that are extremely near to the exact solution whereas obviating the complexities associated with many other approaches. The expected issue’s uniqueness theorem and convergence analysis are explored using Banach’s fixed-point theory. The reliability and accuracy of the recommended method were More >

K. Sayevand¹, Yasir Khan², E. Moradi³, M. Fardi⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.5, pp. 361-373, 2015, DOI:10.3970/cmes.2015.105.361

Abstract In this study, the solitary wave solutions for third order equal-width wave-Burgers (EW-Burgers) equation, the second order Bratu and sinh-Bratu type equations will be discussed. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative effects and furthermore the Bratu type problem appears a simplification of the solid fuel ignition model in thermal combustion theory. Our methodology, is investigated by using (G'/G)- expansion method. The obtained results can be extended to the other models. More >

C.M.T. Tien¹, N. Thai-Quang¹, N. Mai-Duy¹, C.-D. Tran¹, T. Tran-Cong¹

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.6, pp. 425-469, 2015, DOI:10.3970/cmes.2015.104.425

Abstract In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra… More >

S. Saha Ray¹, A. K. Gupta²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.5, pp. 315-341, 2014, DOI:10.3970/cmes.2014.103.315

Abstract In this paper, an efficient numerical schemes based on the Haar wavelet method are applied for finding numerical solution of nonlinear third-order modified Korteweg-de Vries (mKdV) equation as well as modified Burgers' equations. The numerical results are then compared with the exact solutions. The accuracy of the obtained solutions is quite high even if the number of calculation points is small. More >

M.D. Campos¹, E.C. Romão²

CMES-Computer Modeling in Engineering & Sciences, Vol.103, No.3, pp. 139-154, 2014, DOI:10.3970/cmes.2014.103.139

Abstract The objective of this paper aims to present a numerical solution of high accuracy and low computational cost for the three-dimensional Burgers equations. It is a well-known problem and studied the form for one and two-dimensional, but still little explored numerically for three-dimensional problems. Here, by using the High-Order Finite Difference Method for spatial discretization, the Crank-Nicolson method for time discretization and an efficient linearization technique with low computational cost, two numerical applications are used to validate the proposed formulation. In order to analyze the numerical error of the proposed formulation, an unpublished exact solution More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.15, No.3, pp. 221-240, 2010, DOI:10.3970/cmc.2010.015.221

Abstract We propose a simple numerical scheme for solving the space- and time-fractional derivative Burgers equations: D_t^αu + εuu_x = vu_xx + ηD_x^βu, 0 < α, β ≤ 1, and u_t + D_*^β(D_*^1-βu)²/2 = vu_xx, 0 < β ≤ 1. The time-fractional derivative D_t^αu and space-fractional derivative D_x^βu are defined in the Caputo sense, while D_*^βu is the Riemann-Liouville space-fractional derivative. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ)^γu(x,t) =: v(x,t,τ), where 0 < γ ≤ 1 is a parameter, such that the original equation is written as a new functional-differential type partial differential equation More >

Chein-Shan Liu¹

CMC-Computers, Materials & Continua, Vol.9, No.3, pp. 229-252, 2009, DOI:10.3970/cmc.2009.009.229

Abstract When the given input data are corrupted by an intensive noise, most numerical methods may fail to produce acceptable numerical solutions. Here, we propose a new numerical scheme for solving the Burgers equation forward in time and backward in time. A fictitious time τ is used to transform the dependent variable u(x,t) into a new one by (1+τ )u(x,t) =: v(x,t,τ), such that the original Burgers equation is written as a new parabolic type partial differential equation in the space of (x,t,τ). A fictitious damping coefficient can be used to strengthen the stability in the numerical integration… More >

N. Mai-Duy^1,2, T. Tran-Cong², R.I. Tanner³

CMES-Computer Modeling in Engineering & Sciences, Vol.16, No.3, pp. 177-186, 2006, DOI:10.3970/cmes.2006.016.177

Abstract This paper presents a new high-order time-kernel boundary-integral-equation method (BIEM) for numerically solving transient problems governed by the Burgers equation. Instead of using high-order Lagrange polynomials such as quadratic and quartic interpolation functions, the proposed method employs integrated radial-basis-function networks (IRBFNs) to represent the unknown functions in boundary and volume integrals. Numerical implementations of ordinary and double integrals involving time in the presence of IRBFNs are discussed in detail. The proposed method is verified through the solution of diffusion and convection-diffusion problems. A comparison of the present results and those obtained by low-order BIEMs and More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.3, pp. 197-212, 2006, DOI:10.3970/cmes.2006.012.197

Abstract In this paper we numerically solve the Burgers equation by semi-discretizing it at the n interior spatial grid points into a set of ordinary differential equations: u^· = f(u,t), u ∈ Rⁿ. Then, we take the dissipative behavior of Burgers equation into account by considering the magnitude ||u|| as another component; hence, an augmented quasilinear differential equations system X^˙ = AX with X := (u^T,||u||)^T ∈ Mⁿ⁺¹ is derived. According to a Lie algebra property of A∈so(n,1) we thus develop a new numerical scheme with the transformation matrix G∈SO_o(n,1) being an element of the proper orthochronous Lorentz group. More >