Yasmin Alaa Hassan^1,*, Abdul Monem S. Rahma²

CMC-Computers, Materials & Continua, Vol.78, No.1, pp. 1423-1442, 2024, DOI:10.32604/cmc.2023.046149

Abstract Video watermarking plays a crucial role in protecting intellectual property rights and ensuring content authenticity. This study delves into the integration of Galois Field (GF) multiplication tables, especially GF(2⁴), and their interaction with distinct irreducible polynomials. The primary aim is to enhance watermarking techniques for achieving imperceptibility, robustness, and efficient execution time. The research employs scene selection and adaptive thresholding techniques to streamline the watermarking process. Scene selection is used strategically to embed watermarks in the most vital frames of the video, while adaptive thresholding methods ensure that the watermarking process adheres to imperceptibility criteria, maintaining the video’s visual quality.… More >

Shina D. Oloniiju^† , Sicelo P. Goqo, Precious Sibanda

Frontiers in Heat and Mass Transfer, Vol.13, pp. 1-8, 2019, DOI:10.5098/hmt.13.19

Abstract In some recent studies, it has been suggested that non–Newtonian fluid flow can be modeled by a spatially non–local velocity, whose dynamics are described by a fractional derivative. In this study, we use the space fractional constitutive relation to model heat and mass transfer in a nanofluid. We present a numerically accurate algorithm for approximating solutions of the system of fractional ordinary differential equations describing the nanofluid flow. We present numerically stable differentiation matrices for both integer and fractional order derivatives defined by the one–sided Caputo derivative. The differentiation matrices are based on the series expansion of the unknown functions… More >

Xu Zhang¹, Jianchao Ni², Juxi Hu^3,*, 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 >

Ali N. A. Koam¹, Ali Ahmad^2,*, 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 >

N. Alam¹, W. A. Khan^2,*, 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 >

Kamal Shah^1,2, Hafsa Naz², Thabet Abdeljawad^1,3,*, 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 >

Ying Li¹, Longxiang Xu¹, Fangjun Mei¹, Shihui Ying^2,*

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 >

Hye Kyung Kim^1,*, 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 >

Siqintuya Jin¹, Bai-Ni Guo^2,*, Feng Qi^3,*

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 >

Ghulam Muhiuddin^1,*, 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 >