Muhammad Shoaib Arif^1,2,*, Kamaleldin Abodayeh¹, Yasir Nawaz²

CMES-Computer Modeling in Engineering & Sciences, Vol.141, No.2, pp. 1213-1241, 2024, DOI:10.32604/cmes.2024.054819 - 27 September 2024

Abstract Fractal time-dependent issues in fluid dynamics provide a distinct difficulty in numerical analysis due to their complex characteristics, necessitating specialized computing techniques for precise and economical solutions. This study presents an innovative computational approach to tackle these difficulties. The main focus is applying the Fractal Runge-Kutta Method to model the time-dependent magnetohydrodynamic (MHD) Newtonian fluid with rescaled viscosity flow on Riga plates. An efficient computational scheme is proposed for handling fractal time-dependent problems in flow phenomena. The scheme is comprised of three stages and constructed using three different time levels. The stability of the scheme… More >

Sergiy Reutskiy¹, Yuhui Zhang^2,*, Jun Lu^3,*, Ciren Pubu⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.2, pp. 1583-1612, 2024, DOI:10.32604/cmes.2023.044878 - 29 January 2024

Abstract This paper presents an efficient numerical technique for solving multi-term linear systems of fractional ordinary differential equations (FODEs) which have been widely used in modeling various phenomena in engineering and science. An approximate solution of the system is sought in the form of the finite series over the Müntz polynomials. By using the collocation procedure in the time interval, one gets the linear algebraic system for the coefficient of the expansion which can be easily solved numerically by a standard procedure. This technique also serves as the basis for solving the time-fractional partial differential equations More >

M. B. Almatrafi, Abdulghani Alharbi^*

CMES-Computer Modeling in Engineering & Sciences, Vol.137, No.1, pp. 827-841, 2023, DOI:10.32604/cmes.2023.027344 - 23 April 2023

Abstract The extraction of traveling wave solutions for nonlinear evolution equations is a challenge in various mathematics, physics, and engineering disciplines. This article intends to analyze several traveling wave solutions for the modified regularized long-wave (MRLW) equation using several approaches, namely, the generalized algebraic method, the Jacobian elliptic functions technique, and the improved Q-expansion strategy. We successfully obtain analytical solutions consisting of rational, trigonometric, and hyperbolic structures. The adaptive moving mesh technique is applied to approximate the numerical solution of the proposed equation. The adaptive moving mesh method evenly distributes the points on the high error areas. More >

Tohid Adibi¹, Shams Forruque Ahmed^2,*, Seyed Esmail Razavi³, Omid Adibi⁴, Irfan Anjum Badruddin⁵, Syed Javed⁵

CMC-Computers, Materials & Continua, Vol.74, No.3, pp. 5123-5139, 2023, DOI:10.32604/cmc.2023.034008 - 28 December 2022

Abstract The numerical solution of compressible flows has become more prevalent than that of incompressible flows. With the help of the artificial compressibility approach, incompressible flows can be solved numerically using the same methods as compressible ones. The artificial compressibility scheme is thus widely used to numerically solve incompressible Navier-Stokes equations. Any numerical method highly depends on its accuracy and speed of convergence. Although the artificial compressibility approach is utilized in several numerical simulations, the effect of the compressibility factor on the accuracy of results and convergence speed has not been investigated for nanofluid flows in… More >

Dumitru Baleanu^1,2,3, Mehran Namjoo⁴, Ali Mohebbian⁴, Amin Jajarmi^5,*

CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 1147-1163, 2023, DOI:10.32604/cmes.2022.022403 - 27 October 2022

Abstract In the present paper, the numerical solution of Itô type stochastic parabolic equation with a time white noise process is imparted based on a stochastic finite difference scheme. At the beginning, an implicit stochastic finite difference scheme is presented for this equation. Some mathematical analyses of the scheme are then discussed. Lastly, to ascertain the efficacy and accuracy of the suggested technique, the numerical results are discussed and compared with the exact solution. More >

Abdulghani Ragaa Alharbi^*

CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.3, pp. 2193-2209, 2023, DOI:10.32604/cmes.2022.018445 - 20 September 2022

Abstract The Riemann wave system has a fundamental role in describing waves in various nonlinear natural phenomena, for instance, tsunamis in the oceans. This paper focuses on executing the generalized exponential rational function approach and some numerical methods to obtain a distinct range of traveling wave structures and numerical results of the two-dimensional Riemann problems. The stability of obtained traveling wave solutions is analyzed by satisfying the constraint conditions of the Hamiltonian system. Numerical simulations are investigated via the finite difference method to verify the accuracy of the obtained results. To extract the approximation solutions to More >

Kamal Shah^1,2, Hafsa Naz², Thabet Abdeljawad^1,3,*, Aziz Khan¹, Manar A. Alqudah⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.2, pp. 941-955, 2023, DOI:10.32604/cmes.2022.021483 - 31 August 2022

Abstract In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are More >

Muneerah Al Nuwairan^1,*, Zulqurnain Sabir², Muhammad Asif Zahoor Raja³, Maryam Alnami¹, Hanan Almuslem¹

CMC-Computers, Materials & Continua, Vol.73, No.2, pp. 4441-4454, 2022, DOI:10.32604/cmc.2022.029432 - 16 June 2022

Abstract In this paper, a fractional order model based on the management of waste plastic in the ocean (FO-MWPO) is numerically investigated. The mathematical form of the FO-MWPO model is categorized into three components, waste plastic, Marine debris, and recycling. The stochastic numerical solvers using the Levenberg-Marquardt backpropagation neural networks (LMQBP-NNs) have been applied to present the numerical solutions of the FO-MWPO system. The competency of the method is tested by taking three variants of the FO-MWPO model based on the fractional order derivatives. The data ratio is provided for training, testing and authorization is 77%, More >

Ayman Abd-Elhamed^1,2,*, Mohamed Fathy³, Khaled M. Abdelgaber¹

CMC-Computers, Materials & Continua, Vol.71, No.2, pp. 2175-2190, 2022, DOI:10.32604/cmc.2022.021899 - 07 December 2021

Abstract The capability of piles to withstand horizontal loads is a major design issue. The current research work aims to investigate numerically the responses of laterally loaded piles at working load employing the concept of a beam-on-Winkler-foundation model. The governing differential equation for a laterally loaded pile on elastic subgrade is derived. Based on Legendre-Galerkin method and Runge-Kutta formulas of order four and five, the flexural equation of long piles embedded in homogeneous sandy soils with modulus of subgrade reaction linearly variable with depth is solved for both free- and fixed-headed piles. Mathematica, as one of More >

Radwan Abu-Gdairi¹, Shatha Hasan², Shrideh Al-Omari^3,*, Mohammad Al-Smadi^2,4, Shaher Momani^4,5

CMES-Computer Modeling in Engineering & Sciences, Vol.130, No.1, pp. 299-313, 2022, DOI:10.32604/cmes.2022.017010 - 29 November 2021

Abstract In this paper, an efficient multi-step scheme is presented based on reproducing kernel Hilbert space (RKHS) theory for solving ordinary stiff differential systems. The solution methodology depends on reproducing kernel functions to obtain analytic solutions in a uniform form for a rapidly convergent series in the posed Sobolev space. Using the Gram-Schmidt orthogonality process, complete orthogonal essential functions are obtained in a compact field to encompass Fourier series expansion with the help of kernel properties reproduction. Consequently, by applying the standard RKHS method to each subinterval, approximate solutions that converge uniformly to the exact solutions More >