Ali Ahmad¹, Roslan Hasni^2,*, Nahid Akhter³, Kashif Elahi⁴

CMC-Computers, Materials & Continua, Vol.70, No.2, pp. 2895-2904, 2022, DOI:10.32604/cmc.2022.015644

Abstract Chemical compounds are modeled as graphs. The atoms of molecules represent the graph vertices while chemical bonds between the atoms express the edges. The topological indices representing the molecular graph corresponds to the different chemical properties of compounds. Let be are two positive integers, and be the zero-divisor graph of the commutative ring . In this article some direct questions have been answered that can be utilized latterly in different applications. This study starts with simple computations, leading to a quite complex ring theoretic problems to prove certain properties. The theory of finite commutative rings is useful due to its… More >

Lee-Chae Jang¹, Dae San Kim², Hanyoung Kim³, Taekyun Kim^3,*, Hyunseok Lee³

CMES-Computer Modeling in Engineering & Sciences, Vol.129, No.1, pp. 393-408, 2021, DOI:10.32604/cmes.2021.016917

Abstract Recently, degenerate poly-Bernoulli polynomials are defined in terms of degenerate polyexponential functions by Kim-Kim-Kwon-Lee. The aim of this paper is to further examine some properties of the degenerate poly-Bernoulli polynomials by using three formulas from the recently developed ‘λ-umbral calculus.’ In more detail, we represent the degenerate poly-Bernoulli polynomials by Carlitz Bernoulli polynomials and degenerate Stirling numbers of the first kind, by fully degenerate Bell polynomials and degenerate Stirling numbers of the first kind, and by higherorder degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind. More >

Taekyun Kim^1,*, Dae San Kim², Dmitry V. Dolgy³, Si-Hyeon Lee¹, Jongkyum Kwon^4,*

CMES-Computer Modeling in Engineering & Sciences, Vol.128, No.3, pp. 1121-1132, 2021, DOI:10.32604/cmes.2021.016532

Abstract We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind. In this paper, we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates. More >

Naila Rafiq¹, Saima Akram², Mudassir Shams^3,*, Nazir Ahmad Mir¹

CMC-Computers, Materials & Continua, Vol.69, No.2, pp. 2635-2651, 2021, DOI:10.32604/cmc.2021.018955

Abstract In this research article, we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation. We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab. Lower bound of convergence of simultaneous schemes is calculated using Mathematica. Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented. Some real life engineering applications along with some higher degree polynomials… More >

W. M. Abd-Elhameed^1,2,*, Asmaa M. Alkenedri²

CMES-Computer Modeling in Engineering & Sciences, Vol.126, No.3, pp. 955-989, 2021, DOI:10.32604/cmes.2021.013603

Abstract This paper is dedicated to implementing and presenting numerical algorithms for solving some linear and nonlinear even-order two-point boundary value problems. For this purpose, we establish new explicit formulas for the high-order derivatives of certain two classes of Jacobi polynomials in terms of their corresponding Jacobi polynomials. These two classes generalize the two celebrated non-symmetric classes of polynomials, namely, Chebyshev polynomials of third- and fourth-kinds. The idea of the derivation of such formulas is essentially based on making use of the power series representations and inversion formulas of these classes of polynomials. The derived formulas serve in converting the even-order… More >

Ali N. A. Koam¹, Ali Ahmad^{2, *}

CMC-Computers, Materials & Continua, Vol.65, No.2, pp. 1271-1282, 2020, DOI:10.32604/cmc.2020.011736

Abstract In order to study the behavior and interconnection of network devices, graphs structures are used to formulate the properties in terms of mathematical models. Mesh network (meshnet) is a LAN topology in which devices are connected either directly or through some intermediate devices. These terminating and intermediate devices are considered as vertices of graph whereas wired or wireless connections among these devices are shown as edges of graph. Topological indices are used to reflect structural property of graphs in form of one real number. This structural invariant has revolutionized the field of chemistry to identify molecular descriptors of chemical compounds.… More >

Jun Ye^a, Xianlin Zhou^b, Zheng Xu^c, Yong Ding^d

Intelligent Automation & Soft Computing, Vol.24, No.1, pp. 41-46, 2018, DOI:10.1080/10798587.2016.1267239

Abstract In big data era, people cannot afford more and more complex computation work due to the constrained computation resources. The high reliability, strong processing capacity, large storage space of cloud computing makes the resource-constrained clients remotely operate the heavy computation task with the help of cloud server. In this paper, a new algorithm for secure outsourcing of high degree polynomials is proposed. We introduce a camouflage technique, which the real polynomial will be disguised to the untrusted cloud server. In addition, the input and output will not be revealed in the computation process and the clients can easily verify the… More >

Gökçe Yıldız¹, Gültekin Tınaztepe^{2, *}, Mehmet Sezer¹

CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 973-993, 2020, DOI:10.32604/cmes.2020.09329

Abstract In this article, we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials. Using collocation points and treating the solution as a linear combination of Bell polynomials, the problem is reduced to linear system of equations whose unknown variables are Bell coefficients. The solution to this algebraic system determines the approximate solution. Error estimation of approximate solution is done. Some examples are provided to illustrate the performance of the method. The numerical results are compared with the collocation method based on Legendre polynomials and the other two methods… More >

M. H. T. Alshbool¹, W. Shatanawi^{2, 3, 4, *}, I. Hashim⁵, M. Sarr¹

CMC-Computers, Materials & Continua, Vol.64, No.1, pp. 63-80, 2020, DOI:10.32604/cmc.2020.09431

Abstract One of the most attractive subjects in applied sciences is to obtain exact or approximate solutions for different types of linear and nonlinear systems. Systems of ordinary differential equations like systems of second-order boundary value problems (BVPs), Brusselator system and stiff system are significant in science and engineering. One of the most challenge problems in applied science is to construct methods to approximate solutions of such systems of differential equations which pose great challenges for numerical simulations. Bernstein polynomials method with residual correction procedure is used to treat those challenges. The aim of this paper is to present a technique… More >

W. M. Abd-Elhameed^1,2,*, Y. H. Youssri²

CMES-Computer Modeling in Engineering & Sciences, Vol.121, No.3, pp. 1029-1049, 2019, DOI:10.32604/cmes.2019.08378

Abstract This paper is confined to analyzing and implementing new spectral solutions of the fractional Riccati differential equation based on the application of the spectral tau method. A new explicit formula for approximating the fractional derivatives of shifted Chebyshev polynomials of the second kind in terms of their original polynomials is established. This formula is expressed in terms of a certain terminating hypergeometric function of the type ₄F₃(1). This hypergeometric function is reduced in case of the integer case into a certain terminating hypergeometric function of the type ₃F₂(1) which can be summed with the aid of Watson’s identity. Six illustrative… More >