An efficient collocation scheme for new type of variable-order fractional Lane–Emden equation by H. Azin, A. Habibirad, E. Hesameddini, M.H. Heydari

“…The operational matrix of the V-O fractional derivative is obtained for the Vt-L polynomials. In the established approach, these polynomials are applied to transform the main problem into an algebraic equations system. …”
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An Efficient Polynomial Chaos Method for Stiffness Analysis of Air Spring Considering Uncertainties by Feng Kong, Penghao Si, Shengwen Yin

“…To model the uncertainties in the air spring, the interval/random variables models are introduced. For response analysis of the interval/random variables models of the air spring system, a new unified orthogonal polynomial expansion method, named as sparse quadrature-based interval and random moment arbitrary polynomial chaos method (SQ-IRMAPC), is proposed. …”
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An Efficient Petrov–Galerkin Scheme for the Euler–Bernoulli Beam Equation via Second-Kind Chebyshev Polynomials by Youssri Hassan Youssri, Waleed Mohamed Abd-Elhameed, Amr Ahmed Elmasry, Ahmed Gamal Atta

“…Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. …”
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An Operational Matrix Technique for Solving Variable Order Fractional Differential-Integral Equation Based on the Second Kind of Chebyshev Polynomials by Jianping Liu, Xia Li, Limeng Wu

“…An operational matrix technique is proposed to solve variable order fractional differential-integral equation based on the second kind of Chebyshev polynomials in this paper. …”
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An Operational Matrix Technique Based on Chebyshev Polynomials for Solving Mixed Volterra-Fredholm Delay Integro-Differential Equations of Variable-Order by Kamal R. Raslan, Khalid K. Ali, Emad M. H. Mohamed, Jihad A. Younis, Mohamed A. Abd El salam

“…In this work, an algorithm for finding numerical solutions of linear fractional delay-integro-differential equations (LFDIDEs) of variable-order (VO) is introduced. The operational matrices are used as discretization technique based on shifted Chebyshev polynomials (SCPs) of the first kind with the spectral collocation method. …”
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Efficient Numerical Schemes for a Heterogeneous Reaction–Diffusion System with Applications by Samima Akhter, Md. Ariful Islam Arif, Rubayyi T. Alqahtani, Samir Kumar Bhowmik

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Advanced hybrid intelligence evaluation of nanomedicine delivery to various organs using machine learning and adaptive tree structured Parzen estimator by Wael A. Mahdi, Adel Alhowyan, Ahmad J. Obaidullah

“…Abstract We have examined the efficiency of drug delivery for targeted therapy by theoretical models. …”
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Prediction of Maximum Precipitation in Turkey with Polynomial Regression by Fatih Dikbaş, Orhan Koç

“…Understanding the expected variability in precipitation events and predicting the location and yearly periods of probable extreme precipitation are important for the efficient prevention of potential natural catastrophes like floods and dam failures. …”
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Efficient Probabilistic Evaluation and Sensitivity Analysis of Load Supply Capability for Renewable-Energy-Based Power Systems by Jie Zhang, Kaixiang Fu, Weizhi Huang, Yilin Zhang, Qing Sun, Yuan Chi, Junjie Tang

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Polynomial Chaos Expansion Approach to Interest Rate Models by Luca Di Persio, Gregorio Pellegrini, Michele Bonollo

“…The Polynomial Chaos Expansion (PCE) technique allows us to recover a finite second-order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity ξ, hence acting as a kind of random basis. …”
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Polynomial and Differential Networks for End-to-End Autonomous Driving by Youngseong Cho, Kyoungil Lim

“…This study introduces a novel model for predicting control variables in end-to-end autonomous driving by leveraging polynomial and differential networks. …”
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An Innovative Online Adaptive High-Efficiency Controller for Micro Gas Turbine: Design and Simulation Validation by Rui Yang, Yongbao Liu, Xing He, Zhimeng Liu

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Bernoulli Polynomials Method for Solving Integral Equations with Singular Kernel by Muna M. Mustafa, Heba A. Abd-Alrazak

“…Also, assuming that the denominator of the kernel will never be zero or have an imaginary value due to the selected nodes of the unique two kernel variables. With the 4th and 8th-degree Bernoulli polynomials as an example, the current approach provides a solution very close to the exact solution in the test examples. …”
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Indirect Method for Optimal Control Problem Using Boubaker Polynomial by Baghdad Science Journal

“…In this paper, a computational method for solving optimal problem is presented, using indirect method (spectral methodtechnique) which is based on Boubaker polynomial. By this method the state and the adjoint variables are approximated by Boubaker polynomial with unknown coefficients, thus an optimal control problem is transformed to algebraic equations which can be solved easily, and then the numerical value of the performance index is obtained. …”
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Constructing the Lyapunov Function through Solving Positive Dimensional Polynomial System by Zhenyi Ji, Wenyuan Wu, Yong Feng, Guofeng Zhang

“…Then, the positive polynomial system is converted into an equation system by adding some variables. …”
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Local Polynomial Regression Solution for Partial Differential Equations with Initial and Boundary Values by Liyun Su, Tianshun Yan, Yanyong Zhao, Fenglan Li

“…Local polynomial regression (LPR) is applied to solve the partial differential equations (PDEs). …”
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Numerical solutions for fractional optimal control problems using Mü‎‎‎‎‎‎‎ntz-Legendre polynomials by Mohammad Sahabi, Allahbakhsh Yazdani Cherati

“…Utilizing the unique properties of Müntz-Legendre polynomials when dealing with fractional operators, these polynomials are used to approximate the state and control variables in the considered problems. …”
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Tunnel composting Optimisation using polynomial models with moisture and density control for electrical conductivity stabilisation by Tiago Santos, Isabel Bentes, Carlos Afonso Teixeira

“…These conditions minimise the need for corrective aeration or irrigation, enhancing process efficiency. Future studies may incorporate variables such as the carbon-to-nitrogen (C/N) ratio, microbial activity, and germination index to expand the model’s robustness and applicability. …”
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Uncertainty evaluation of surface profile measurement error based on adaptive sparse grid polynomial chaos expansion by Ke Zhang, Xinya Zheng, Ruiyu Zhang

“…To address the challenge of balancing accuracy and computational efficiency in evaluating the measurement error and uncertainty of surface profile errors on complex free-form surfaces, this paper proposes an evaluation method combining adaptive sparse grids and polynomial chaos expansion. …”
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Chebyshev Polynomials of Sixth Kind for Solving Nonlinear Fractional PDEs with Proportional Delay and Its Convergence Analysis by Khadijeh Sadri, Hossein Aminikhah

“…For this, a numerical scheme is implemented that applies operational matrices to convert the main problem into a system of algebraic equations; then, solving the resultant system leads to an approximate solution. The two-variable Chebyshev polynomials of the sixth kind, as basis functions in the proposed method, are constructed by the one-variable ones, and their operational matrices are derived. …”
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