Wasfi Shatanawi^1,2,3, Ali Raza^4,5,*, Muhammad Shoaib Arif⁴, Kamaledin Abodayeh¹, Muhammad Rafiq⁶, Mairaj Bibi⁷

CMC-Computers, Materials & Continua, Vol.66, No.2, pp. 1121-1137, 2021, DOI:10.32604/cmc.2020.012070 - 26 November 2020

Abstract Nonlinear stochastic modeling plays a significant role in disciplines such as psychology, finance, physical sciences, engineering, econometrics, and biological sciences. Dynamical consistency, positivity, and boundedness are fundamental properties of stochastic modeling. A stochastic coronavirus model is studied with techniques of transition probabilities and parametric perturbation. Well-known explicit methods such as Euler Maruyama, stochastic Euler, and stochastic Runge–Kutta are investigated for the stochastic model. Regrettably, the above essential properties are not restored by existing methods. Hence, there is a need to construct essential properties preserving the computational method. The non-standard approach of finite difference is examined More >

Muhammad Shoaib Arif^1,*, Ali Raza^1,2, Kamaleldin Abodayeh³, Muhammad Rafiq⁴, Mairaj Bibi⁵, Amna Nazeer⁵

CMES-Computer Modeling in Engineering & Sciences, Vol.124, No.2, pp. 477-491, 2020, DOI:10.32604/cmes.2020.011121 - 20 July 2020

Abstract The essential features of the nonlinear stochastic models are positivity, dynamical consistency and boundedness. These features have a significant role in different fields of computational biology and many more. The aim of our paper, to achieve the comparison analysis of the stochastic susceptible, infected recovered epidemic model. The stochastic modelling is a realistic way to study the dynamics of compartmental modelling as compared to deterministic modelling. The effect of reproduction number has also observed in the stochastic susceptible, infected recovered epidemic model. For comparison analysis, we developed some explicit stochastic techniques, but they are the More >

Wasfi Shatanawi^{1, 2, 3}, Muhammad Shoaib Arif^{4, *}, Ali Raza⁴, Muhammad Rafiq⁵, Mairaj Bibi⁶, Javeria Nawaz Abbasi⁶

CMC-Computers, Materials & Continua, Vol.64, No.2, pp. 797-811, 2020, DOI:10.32604/cmc.2020.010759 - 10 June 2020

Abstract The structure-preserving features of the nonlinear stochastic models are positivity, dynamical consistency and boundedness. These features have a significant role in different fields of computational biology and many more. Unfortunately, the existing stochastic approaches in literature do not restore aforesaid structure-preserving features, particularly for the stochastic models. Therefore, these gaps should be occupied up in literature, by constructing the structure-preserving features preserving numerical approach. This writing aims to describe the structure-preserving dynamics of the stochastic model. We have analysed the effect of reproduction number in stochastic modelling the same as described in the literature for More >

Muhammad Shoaib Arif^{1, *}, Ali Raza¹, Muhammad Rafiq², Mairaj Bibi³, Javeria Nawaz Abbasi³, Amna Nazeer³, Umer Javed⁴

CMC-Computers, Materials & Continua, Vol.62, No.1, pp. 61-77, 2020, DOI:10.32604/cmc.2020.08595

Abstract We are presenting the numerical simulations for the stochastic computer virus propagation model in this manuscript. We are comparing the solutions of stochastic and deterministic computer virus models. Outcomes of a threshold number R₀ hold in stochastic computer virus model. If R₀ < 1 then in such a condition virus controlled in the computer population while R₀ > 1 shows virus rapidly spread in the computer population. Unfortunately, stochastic numerical techniques fail to cope with large step sizes of time. The suggested structure of the stochastic non-standard finite difference technique can never violate the dynamical properties. On 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 >

Jiaquan Xie^1,3,*, Fuqiang Zhao^1,3, Zhibin Yao^1,3, Jun Zhang^1,2

CMES-Computer Modeling in Engineering & Sciences, Vol.115, No.1, pp. 67-84, 2018, DOI:10.3970/cmes.2018.115.067

Abstract In this paper, the three-variable shifted Jacobi operational matrix of fractional derivatives is used together with the collocation method for numerical solution of three-dimensional multi-term fractional-order PDEs with variable coefficients. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The approximate solutions of nonlinear fractional PDEs with variable coefficients thus obtained by three-variable shifted Jacobi polynomials are compared with the exact solutions. Furthermore some theorems and lemmas are introduced to verify the convergence results of our algorithm. Lastly, More >

Jinsheng Wang¹, Liqing Liu², Yiming Chen², Lechun Liu², Dayan Liu³

CMES-Computer Modeling in Engineering & Sciences, Vol.104, No.1, pp. 69-85, 2015, DOI:10.3970/cmes.2015.104.069

Abstract The aim of this paper is to seek the numerical solution of a class of variable order fractional integral-differential equation in terms of Bernstein polynomials. The fractional derivative is described in the Caputo sense. Four kinds of operational matrixes of Bernstein polynomials are introduced and are utilized to reduce the initial equation to the solution of algebraic equations after dispersing the variable. By solving the algebraic equations, the numerical solutions are acquired. The method in general is easy to implement and yields good results. Numerical examples are provided to demonstrate the validity and applicability of More >

Yiming Chen¹, Xiaoning Han¹, Lechun Liu ¹

CMES-Computer Modeling in Engineering & Sciences, Vol.97, No.5, pp. 391-405, 2014, DOI:10.3970/cmes.2014.097.391

Abstract In this paper, a class of linear system of fractional differential equations is considered. It has been solved by operational matrix of Haar wavelet method which converts the problem into algebraic equations. Moreover the convergence of the method is studied, and three numerical examples are provided to demonstrate the accuracy and efficiency. More >

M. Ferronato¹, A. Mazzia¹, G. Pini¹

CMES-Computer Modeling in Engineering & Sciences, Vol.62, No.2, pp. 205-224, 2010, DOI:10.3970/cmes.2010.062.205

Abstract In the engineering practice meshing and re-meshing complex domains by Finite Elements (FE) is one of the most time-consuming efforts. Meshless methods avoid this task but are computationally more expensive than standard FE. A somewhat natural improvement can be attempted by combining the two techniques with the aim at emphasizing the respective merits. The present work describes a FE enrichment by the Meshless Local Petrov-Galerkin (MLPG) method. The basic idea is to add a limited number of moving MLPG points over a fixed coarse FE grid, in order to improve the solution accuracy in specific More >

Lorenzo Codecasa¹, Francesco Trevisan²

CMES-Computer Modeling in Engineering & Sciences, Vol.59, No.2, pp. 155-180, 2010, DOI:10.3970/cmes.2010.059.155

Abstract Electromagnetic problems spatially discretized by the so called Discrete Geometric Approach are considered, where Discrete Counterparts of Constitutive Relations are discretized within an Energetic Approach. Pairs of oriented dual grids are considered in which the primal grid is composed of (oblique) parallelepipeds, (oblique) triangular prisms and tetrahedra and the dual grid is obtained according to the barycentric subdivision. The focus of the work is the evaluation of the constants bounding the approximation error of the electromagnetic field; the novelty is that such constants will be expressed in terms of the geometrical details of oriented dual More >