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  • 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


    A Hybrid Particle Swarm Optimization to Forecast Implied Volatility Risk

    Kais Tissaoui1,2,*, Sahbi Boubaker3,4, Waleed Saud Alghassab1, Taha Zaghdoudi1,5, Jamel Azibi6

    CMC-Computers, Materials & Continua, Vol.73, No.2, pp. 4291-4309, 2022, DOI:10.32604/cmc.2022.028830

    Abstract The application of optimization methods to prediction issues is a continually exploring field. In line with this, this paper investigates the connectedness between the infected cases of COVID-19 and US fear index from a forecasting perspective. The complex characteristics of implied volatility risk index such as non-linearity structure, time-varying and non-stationarity motivate us to apply a nonlinear polynomial Hammerstein model with known structure and unknown parameters. We use the Hybrid Particle Swarm Optimization (HPSO) tool to identify the model parameters of nonlinear polynomial Hammerstein model. Findings indicate that, following a nonlinear polynomial behaviour cascaded to an autoregressive with exogenous input… 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


    A Mathematical Model for COVID-19 Image Enhancement based on Mittag-Leffler-Chebyshev Shift

    Ibtisam Aldawish1, Hamid A. Jalab2,*

    CMC-Computers, Materials & Continua, Vol.73, No.1, pp. 1307-1316, 2022, DOI:10.32604/cmc.2022.029445

    Abstract The lungs CT scan is used to visualize the spread of the disease across the lungs to obtain better knowledge of the state of the COVID-19 infection. Accurately diagnosing of COVID-19 disease is a complex challenge that medical system face during the pandemic time. To address this problem, this paper proposes a COVID-19 image enhancement based on Mittag-Leffler-Chebyshev polynomial as pre-processing step for COVID-19 detection and segmentation. The proposed approach comprises the Mittag-Leffler sum convoluted with Chebyshev polynomial. The idea for using the proposed image enhancement model is that it improves images with low gray-level changes by estimating the probability… 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 >

  • Open Access


    Protected Fair Secret Sharing Based Bivariate Asymmetric Polynomials in Satellite Network

    Yanyan Han1,2, Jiangping Yu3, Guangyu Hu4, Chenglei Pan4, Dingbang Xie5, Chao Guo1,2,6,*, Abdul Waheed7

    CMC-Computers, Materials & Continua, Vol.72, No.3, pp. 4789-4802, 2022, DOI:10.32604/cmc.2022.027496

    Abstract Verifiable secret sharing mainly solves the cheating behavior between malicious participants and the ground control center in the satellite network. The verification stage can verify the effectiveness of secret shares issued by the ground control center to each participant and verify the effectiveness of secret shares shown by participants. We use a lot of difficult assumptions based on mathematical problems in the verification stage, such as solving the difficult problem of the discrete logarithm, large integer prime factorization, and so on. Compared with other verifiable secret sharing schemes designed for difficult problems under the same security, the verifiable secret sharing… More >

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