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  • Open Access


    Structural Interval Reliability Algorithm Based on Bernstein Polynomials and Evidence Theory

    Xu Zhang1, Jianchao Ni2, Juxi Hu3,*, Weisi Chen4

    Computer Systems Science and Engineering, Vol.46, No.2, pp. 1947-1960, 2023, DOI:10.32604/csse.2023.035118

    Abstract Structural reliability is an important method to measure the safety performance of structures under the influence of uncertain factors. Traditional structural reliability analysis methods often convert the limit state function to the polynomial form to measure whether the structure is invalid. The uncertain parameters mainly exist in the form of intervals. This method requires a lot of calculation and is often difficult to achieve efficiently. In order to solve this problem, this paper proposes an interval variable multivariate polynomial algorithm based on Bernstein polynomials and evidence theory to solve the structural reliability problem with cognitive uncertainty. Based on the non-probabilistic… More >

  • Open Access


    Algebraic Properties for Molecular Structure of Magnesium Iodide

    Ali N. A. Koam1, Ali Ahmad2,*, Muhammad Azeem3, Muhammad Kamran Siddiqui4

    CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.2, pp. 1131-1146, 2023, DOI:10.32604/cmes.2022.020884

    Abstract As an inorganic chemical, magnesium iodide has a significant crystalline structure. It is a complex and multi-functional substance that has the potential to be used in a wide range of medical advancements. Molecular graph theory, on the other hand, provides a sufficient and cost-effective method of investigating chemical structures and networks. M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory. It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function. We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this… More >

  • Open Access


    A Note on Bell-Based Bernoulli and Euler Polynomials of Complex Variable

    N. Alam1, W. A. Khan2,*, S. Obeidat1, G. Muhiuddin3, N. S. Diab1, H. N. Zaidi1, A. Altaleb1, L. Bachioua1

    CMES-Computer Modeling in Engineering & Sciences, Vol.135, No.1, pp. 187-209, 2023, DOI:10.32604/cmes.2022.021418

    Abstract In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials. Some fundamental properties of these functions are given. By using these generating functions and some identities, relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials, Stirling numbers are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are… More >

  • Open Access


    Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

    Kamal Shah1,2, Hafsa Naz2, Thabet Abdeljawad1,3,*, Aziz Khan1, Manar A. Alqudah4

    CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.2, pp. 941-955, 2023, DOI:10.32604/cmes.2022.021483

    Abstract In this manuscript, an algorithm for the computation of numerical solutions to some variable order fractional differential equations (FDEs) subject to the boundary and initial conditions is developed. We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices. Further, operational matrices are constructed using variable order differentiation and integration. We are finding the operational matrices of variable order differentiation and integration by omitting the discretization of data. With the help of aforesaid matrices, considered FDEs are converted to algebraic equations of Sylvester type. Finally, the algebraic equations we get are solved with the help of… More >

  • Open Access


    LaNets: Hybrid Lagrange Neural Networks for Solving Partial Differential Equations

    Ying Li1, Longxiang Xu1, Fangjun Mei1, Shihui Ying2,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.134, No.1, pp. 657-672, 2023, DOI:10.32604/cmes.2022.021277

    Abstract We propose new hybrid Lagrange neural networks called LaNets to predict the numerical solutions of partial differential equations. That is, we embed Lagrange interpolation and small sample learning into deep neural network frameworks. Concretely, we first perform Lagrange interpolation in front of the deep feedforward neural network. The Lagrange basis function has a neat structure and a strong expression ability, which is suitable to be a preprocessing tool for pre-fitting and feature extraction. Second, we introduce small sample learning into training, which is beneficial to guide the model to be corrected quickly. Taking advantages of the theoretical support of traditional… More >

  • Open Access


    Some Properties of Degenerate r-Dowling Polynomials and Numbers of the Second Kind

    Hye Kyung Kim1,*, Dae Sik Lee2

    CMES-Computer Modeling in Engineering & Sciences, Vol.133, No.3, pp. 825-842, 2022, DOI:10.32604/cmes.2022.022103

    Abstract The generating functions of special numbers and polynomials have various applications in many fields as well as mathematics and physics. In recent years, some mathematicians have studied degenerate version of them and obtained many interesting results. With this in mind, in this paper, we introduce the degenerate r-Dowling polynomials and numbers associated with the degenerate r-Whitney numbers of the second kind. We derive many interesting properties and identities for them including generating functions, Dobinski-like formula, integral representations, recurrence relations, differential equation and various explicit expressions. In addition, we explore some expressions for them that can be derived from repeated applications… More >

  • Open Access


    Partial Bell Polynomials, Falling and Rising Factorials, Stirling Numbers, and Combinatorial Identities

    Siqintuya Jin1, Bai-Ni Guo2,*, Feng Qi3,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.132, No.3, pp. 781-799, 2022, DOI:10.32604/cmes.2022.019941

    Abstract In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, important, useful, and applicable in combinatorial number theory. More >

  • Open Access


    Some Identities of the Degenerate Poly-Cauchy and Unipoly Cauchy Polynomials of the Second Kind

    Ghulam Muhiuddin1,*, Waseem A. Khan2, Deena Al-Kadi3

    CMES-Computer Modeling in Engineering & Sciences, Vol.132, No.3, pp. 763-779, 2022, DOI:10.32604/cmes.2022.017272

    Abstract In this paper, we introduce modied degenerate polyexponential Cauchy (or poly-Cauchy) polynomials and numbers of the second kind and investigate some identities of these polynomials. We derive recurrence relations and the relationship between special polynomials and numbers. Also, we introduce modied degenerate unipolyCauchy polynomials of the second kind and derive some fruitful properties of these polynomials. In addition, positive associated beautiful zeros and graphical representations are displayed with the help of Mathematica. More >

  • Open Access


    Asymptotic Approximations of Apostol-Tangent Polynomials in Terms of Hyperbolic Functions

    Cristina B. Corcino1,2, Wilson D. Castañeda Jr.3, Roberto B. Corcino1,2,*

    CMES-Computer Modeling in Engineering & Sciences, Vol.132, No.1, pp. 133-151, 2022, DOI:10.32604/cmes.2022.019965

    Abstract The tangent polynomials Tn (z) are generalization of tangent numbers or the Euler zigzag numbers Tn. In particular, Tn (0) = Tn. These polynomials are closely related to Bernoulli, Euler and Genocchi polynomials. One of the extensions and analogues of special polynomials that attract the attention of several mathematicians is the Apostoltype polynomials. One of these Apostol-type polynomials is the Apostol-tangent polynomials Tn(z, λ). When λ = 1, Tn (z, 1) = Tn(z). The use of hyperbolic functions to derive asymptotic approximations of polynomials together with saddle point method was applied to the Bernoulli and Euler polynomials by Lopez and… More >

  • Open Access


    Analysis of Eigenvalues for Molecular Structures

    Muhammad Haroon Aftab1, Kamel Jebreen2,*, Mohammad Issa Sowaity3, Muhammad Hussain4

    CMC-Computers, Materials & Continua, Vol.73, No.1, pp. 1225-1236, 2022, DOI:10.32604/cmc.2022.029009

    Abstract In this article, we study different molecular structures such as Polythiophene network, for and , Orthosilicate (Nesosilicate) , Pyrosilicates (Sorosilicates) , Chain silicates (Pyroxenes), and Cyclic silicates (Ring Silicates) for their cardinalities, chromatic numbers, graph variations, eigenvalues obtained from the adjacency matrices which are square matrices in order and their corresponding characteristics polynomials. We convert the general structures of these chemical networks in to mathematical graphical structures. We transform the molecular structures of these chemical networks which are mentioned above, into a simple and undirected planar graph and sketch them with various techniques of mathematics. The matrices obtained from these… More >

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