Dolat Khan¹, Gohar Ali¹, Arshad Khan², Ilyas Khan^{3, *}, Yu-Ming Chu^{4, 5}, Kottakkaran Sooppy Nisar⁶

CMC-Computers, Materials & Continua, Vol.65, No.2, pp. 1237-1251, 2020, DOI:10.32604/cmc.2020.011492 - 20 August 2020

Abstract Nowadays some new ideas of fractional derivatives have been used successfully in the present research community to study different types of mathematical models. Amongst them, the significant models of fluids and heat or mass transfer are on priority. Most recently a new idea of fractal-fractional derivative is introduced; however, it is not used for heat transfer in channel flow. In this article, we have studied this new idea of fractal fractional operators with power-law kernel for heat transfer in a fluid flow problem. More exactly, we have considered the free convection heat transfer for a… More >

Anees Imtiaz¹, Oi-Mean Foong², Aamina Aamina¹, Nabeel Khan¹, Farhad Ali^{3, 4, *}, Ilyas Khan⁵

CMC-Computers, Materials & Continua, Vol.65, No.1, pp. 171-192, 2020, DOI:10.32604/cmc.2020.011397 - 23 July 2020

Abstract Gold metallic nanoparticles are generally used within a lab as a tracer, to uncover on the presence of specific proteins or DNA in a sample, as well as for the recognition of various antibiotics. They are bio companionable and have properties to carry thermal energy to tumor cells by utilizing different clinical approaches. As the cancer cells are very smaller so for the infiltration, the properly sized nanoparticles have been injected in the blood. For this reason, gold nanoparticles are very effective. Keeping in mind the above applications, in the present work a generalized model… More >

Şuayip Yüzbaşı^{1, *}, Murat Karaçayır¹

CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 941-956, 2020, DOI:10.32604/cmes.2020.08938 - 28 May 2020

Abstract In this study, we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2. In this approach, the approximate solution is assumed to have the form of a polynomial in the variable t = x^α , where α is a positive real parameter of our choice. The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation. After taking inner product of this vector with nonnegative integer powers of t up to a More >

Mahdi Saedshoar Heris¹, Mohammad Javidi¹, Bashir Ahmad^2,*

CMES-Computer Modeling in Engineering & Sciences, Vol.121, No.1, pp. 249-272, 2019, DOI:10.32604/cmes.2019.08080

Abstract In this paper, we propose numerical methods for the Riesz space fractional advection-dispersion equations with delay (RFADED). We utilize the fractional backward differential formulas method of second order (FBDF2) and weighted shifted Grünwald difference (WSGD) operators to approximate the Riesz fractional derivative and present the finite difference method for the RFADED. Firstly, the FBDF2 and the shifted Grünwald methods are introduced. Secondly, based on the FBDF2 method and the WSGD operators, the finite difference method is applied to the problem. We also show that our numerical schemes are conditionally stable and convergent with the accuracy More >

Yanxin Wang^{1, *}, Li Zhu¹, Zhi Wang¹

CMES-Computer Modeling in Engineering & Sciences, Vol.118, No.2, pp. 339-350, 2019, DOI:10.31614/cmes.2019.04575

Abstract An Euler wavelets method is proposed to solve a class of nonlinear variable order fractional differential equations in this paper. The properties of Euler wavelets and their operational matrix together with a family of piecewise functions are first presented. Then they are utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. And the convergence of the Euler wavelets basis is given. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy. More >

Yaqing Liu^a,*, Jinyu Ma^b

Frontiers in Heat and Mass Transfer, Vol.6, pp. 1-5, 2015, DOI:10.5098/hmt.6.17

Abstract This paper presents an exact solution for the magnetohydrodynamic (MHD) flow of an incompressible generalized Oldroyd-B fluid due to an infinite accelerating plate. The fractional calculus approach is introduced to establish the constitutive relationship of the Oldroyd-B fluid. The solutions in terms of Fox H-function are obtained by using the Laplace transform. When N = 0 the solutions corresponds to the generalized Oldroyd-B fluids, while θ → 0 and λ → 0 describes the Maxwell fluid and the generalized second fluid, as limiting cases of our general results, respectively. More >

M. Dehghan¹, M. Abbaszadeh², A. Mohebbi³

CMES-Computer Modeling in Engineering & Sciences, Vol.107, No.6, pp. 481-516, 2015, DOI:10.3970/cmes.2015.107.481

Abstract In the current paper the two-dimensional time fractional Klein-Kramers equation which describes the subdiffusion in the presence of an external force field in phase space has been considered. The numerical solution of fractional Klein-Kramers equation is investigated. The proposed method is based on using finite difference scheme in time variable for obtaining a semi-discrete scheme. Also, to achieve a full discretization scheme, the Kansa's approach and meshless local Petrov-Galerkin technique are used to approximate the spatial derivatives. The meshless method has already proved successful in solving classic and fractional differential equations as well as for… More >

X.S. He¹, Q.X. Liu¹, X.C. Huang², Y.M. Chen^1,3

CMES-Computer Modeling in Engineering & Sciences, Vol.108, No.3, pp. 159-169, 2015, DOI:10.3970/cmes.2015.108.159

Abstract This paper presented dynamic response analysis for an MEMS viscometer. The responses are governed by a set of differential equations containing fractional derivatives. The memory-free Yuan-Agrawal’s approach was extended to solve fractional differential equations containing arbitrary fractional order derivative and then a simple yet efficient numerical scheme was constructed. Numerical examples show that the proposed method can provide very accurate results and computational efforts can be significantly saved. Moreover, the numerical scheme was extended to solve problems with a nonlinear spring. The influences of the nonlinear parameters on the dynamic responses were also efficiently analyzed. More >

S. Saha Ray¹

CMES-Computer Modeling in Engineering & Sciences, Vol.105, No.3, pp. 231-249, 2015, DOI:10.3970/cmes.2015.105.231

Abstract In the present article, a relatively very new technique viz. Coupled Fractional Reduced Differential Transform, has been executed to attain the approximate numerical solution of the predator-prey dynamical system. The fractional derivatives are defined in the Caputo sense. Utilizing the present method we can solve many linear and nonlinear coupled fractional differential equations. The results thus obtained are compared with those of other available methods. Numerical solutions are presented graphically to show the simplicity and authenticity of the method. More >

Diptiranjan Behera¹, S. Chakraverty²

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.3, pp. 211-225, 2015, DOI:10.3970/cmes.2015.104.211

Abstract This paper presents the numerical solution of a viscoelastic continuous beam whose damping behaviours are defined in term of fractional derivatives of arbitrary order. Homotopy Perturbation Method (HPM) is used to obtain the dynamic response with respect to unit impulse load. Obtained results are depicted in term of plots. Comparisons are made with the analytic solutions obtained by Zu-feng and Xiao-yan (2007) to show the effectiveness and validation of the present method. More >