Hakeem Ullah¹, Mehreen Fiza^1,*, Ilyas Khan^2,*, Abd Allah A. Mosa³, Saeed Islam¹, Abdullah Mohammed⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.136, No.1, pp. 277-291, 2023, DOI:10.32604/cmes.2023.022289 - 05 January 2023

Abstract In this paper, we present a new modification of the newly developed semi-analytical method named the Optimal Auxilary Function Method (OAFM) for fractional-order equations using the Caputo operator, which is named FOAFM. The mathematical theory of FOAFM is presented and the effectiveness of this method is proven by using it with well-known Fornberg-Whitham Equations (FWE). The FOAFM results are compared with other method results along with their exact solutions with the help of tables and plots to prove the validity of FOAFM. A rapidly convergent series solution is obtained from FOAFM and is validated by… More >

Ahmed M. Elshenhab^1,2,*, Xingtao Wang¹, Fatemah Mofarreh³, Omar Bazighifan^4,*

CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.2, pp. 927-940, 2023, DOI:10.32604/cmes.2022.021512 - 31 August 2022

Abstract We study nonhomogeneous systems of linear conformable fractional differential equations with pure delay. By using new conformable delayed matrix functions and the method of variation, we obtain a representation of their solutions. As an application, we derive a finite time stability result using the representation of solutions and a norm estimation of the conformable delayed matrix functions. The obtained results are new, and they extend and improve some existing ones. Finally, an example is presented to illustrate the validity of our theoretical results. More >

Nadeem Ahmad Sheikh^1,2, Dennis Ling Chuan Ching¹, Thabet Abdeljawad^3,4,5, Ilyas Khan^6,*, Muhammad Jamil^7,8, Kottakkaran Sooppy Nisar⁹

CMC-Computers, Materials & Continua, Vol.67, No.2, pp. 1385-1398, 2021, DOI:10.32604/cmc.2021.011986 - 05 February 2021

Abstract An emerging definition of the fractal-fractional operator has been used in this study for the modeling of Casson fluid flow. The magnetohydrodynamics flow of Casson fluid has cogent in a channel where the motion of the upper plate generates the flow while the lower plate is at a static position. The proposed model is non-dimensionalized using the Pi-Buckingham theorem to reduce the complexity in solving the model and computation time. The non-dimensional fractal-fractional model with the power-law kernel has been solved through the Laplace transform technique. The Mathcad software has been used for illustration of… More >

Muhammad Saqib¹, Ilyas Khan^2,*, Sharidan Shafie¹, Ahmad Qushairi Mohamad¹, El-Sayed M. Sherif^3,4

CMC-Computers, Materials & Continua, Vol.67, No.1, pp. 1069-1084, 2021, DOI:10.32604/cmc.2021.012000 - 12 January 2021

Abstract In recent times, scientists and engineers have been most attracted to electrically conducted nanofluids due to their numerous applications in various fields of science and engineering. For example, they are used in cancer treatment (hyperthermia), magnetic resonance imaging (MRI), drug-delivery, and magnetic refrigeration (MR). Bearing in mind the significance and importance of electrically conducted nanofluids, this article aims to study an electrically conducted water-based nanofluid containing carbon nanotubes (CNTs). CNTs are of two types, single-wall carbon nanotubes (SWCNTs) and multiple-wall carbon nanotubes (MWCNTs). The CNTs (SWCNTs and MWCNTs) have been dispersed in regular water as… More >

Kazunori Shinohara^{1, *}

CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.3, pp. 599-647, 2019, DOI:10.31614/cmes.2019.04472

Abstract According to the wave power rule, the second derivative of a function x(t) with respect to the variable t is equal to negative n times the function x(t) raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations… More >

Kazunori Shinohara¹

CMES-Computer Modeling in Engineering & Sciences, Vol.115, No.2, pp. 149-215, 2018, DOI:10.3970/cmes.2018.02179

Abstract Exact solutions of the cubic Duffing equation with the initial conditions are presented. These exact solutions are expressed in terms of leaf functions and trigonometric functions. The leaf function r=sleafn(t) or r=cleafn(t) satisfies the ordinary differential equation dx2/dt2=-nr2n-1. The second-order differential of the leaf function is equal to -n times the function raised to the (2n-1) power of the leaf function. By using the leaf functions, the exact solutions of the cubic Duffing equation can be derived under several conditions. These solutions are constructed using the integral functions of leaf functions sleaf2(t) and cleaf2(t) for More >

Ran Wang¹, Xuegang Yuan^1,2, Hongwu Zhang¹, Jing Zhang³, Na Lv^2,*

CMC-Computers, Materials & Continua, Vol.55, No.2, pp. 345-357, 2018, DOI:10.3970/cmc.2018.00233

Abstract In this paper, a nonlinear wave equation with variable coefficients is studied, interestingly, this equation can be used to describe the travelling waves propagating along the circular rod composed of a general compressible hyperelastic material with variable cross-sections and variable material densities. With the aid of Lou’s direct method1, the nonlinear wave equation with variable coefficients is reduced and two sets of symmetry transformations and exact solutions of the nonlinear wave equation are obtained. The corresponding numerical examples of exact solutions are presented by using different coefficients. Particularly, while the variable coefficients are taken as More >

M. C. Ray¹, L. Dong², S. N. Atluri³

CMES-Computer Modeling in Engineering & Sciences, Vol.111, No.5, pp. 437-471, 2016, DOI:10.3970/cmes.2016.111.437

Abstract This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams. The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements. The overall smart composite beam is composed of a laminated substrate conventional composite beam, and a piezoelectric layer attached at the top surface of the substrate beam. The piezoelectric layer acts as the actuator layer of the smart beam. Alternate finite element models of the beams, based on an "equivalent single… More >

Sen-Yung Lee¹, Shueei-Muh Lin^2,3, Kai-Ping Chang¹

CMC-Computers, Materials & Continua, Vol.51, No.1, pp. 1-19, 2016, DOI:10.3970/cmc.2016.051.001

Abstract The two coupled governing differential equations for the out-of-plane vibrations of non-uniform beams with variable curvature are derived via the Hamilton's principle. These equations are expressed in terms of flexural and torsional displacements simultaneously. In this study, the analytical method is proposed. Firstly, two physical parameters are introduced to simplify the analysis. One derives the explicit relations between the flexural and the torsional displacements which can also be used to reduce the difficulty in experimental measurements. Based on the relation, the two governing characteristic differential equations with variable coefficients can be uncoupled into a sixth-order More >

M. C. Ray¹, L. Dong², S. N. Atluri³

CMC-Computers, Materials & Continua, Vol.47, No.3, pp. 143-177, 2015, DOI:10.3970/cmc.2015.047.143

Abstract This paper is concerned with the development of new simple 4-noded locking-alleviated smart finite elements for modeling the smart composite beams. The exact solutions for the static responses of the overall smart composite beams are also derived for authenticating the new smart finite elements. The overall smart composite beam is composed of a laminated substrate conventional composite beam, and a piezoelectric layer attached at the top surface of the substrate beam. The piezoelectric layer acts as the actuator layer of the smart beam. Alternate finite element models of the beams, based on an “equivalent single… More >