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

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 comparison with other results. The… 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

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

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 the influence of various parameters,… 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

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 base fluid to form water-CNTs… 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 are presented. In the… 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 the phase of a trigonometric… 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 layer high order shear deformation… 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 >

Ching Kong Chao^1,2, Chin Kun Chen¹, Fu Mo Chen³

CMES-Computer Modeling in Engineering & Sciences, Vol.47, No.3, pp. 283-298, 2009, DOI:10.3970/cmes.2009.047.283

Abstract Analytical exact solutions of a fundamental heat conduction problem for a three-phase elliptical composite under a remote uniform heat flow are provided in this paper. The steady-state temperature and heat flux fields in each phase of an elliptical composite are analyzed in detail. Investigations on the present heat conduction problem are tedious due to the presence of material inhomogeneities and geometric discontinuities. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the temperature and heat flux are derived explicitly in a closed form. Some numerical results… More >

Sen Yung Lee¹, Jyh Shyang Wu²

CMES-Computer Modeling in Engineering & Sciences, Vol.40, No.2, pp. 133-154, 2009, DOI:10.3970/cmes.2009.040.133

Abstract The three coupled governing differential equations for the in-plane vibrations of curved non-uniform Timoshenko beams are derived via the Hamilton's principle. Three physical parameters are introduced to simplify the analysis. By eliminating all the terms with the axial displacement parameter, then reducing the order of differential operator acting on the flexural displacement parameter, one uncouples the three governing characteristic differential equations with variable coefficients and reduces them into a sixth-order ordinary differential equation with variable coefficients in term of the angle of the rotation due to bending for the first time. The explicit relations between the axial and the flexural… More >