•Nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators is studied.•Numerical schemes are based on the fundamental theorem of fractional calculus and the ...Lagrange polynomial interpolation.•Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm.
We extended the nutrient–phytoplankton–zooplankton model involving variable-order fractional differential operators of Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu. Variable-order fractional operators permits model and describe accurately real world problems, for example, diffusion or spread of nutrients or species in different states. Particularly, we model the interaction of nutrient phytoplankton and its predator zooplankton. The variable-order fractional numerical scheme based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation was consider. Numerical simulation results are provided for illustrating the effectiveness and applicability of the algorithm to solve variable-order fractional differential equations.
Distributed computing is one of the powerful solutions for computational tasks that need the massive size of dataset. Lagrange coded computing (LCC), proposed by Yu et al. 15, realizes private and ...secure distributed computing under the existence of stragglers, malicious workers, and colluding workers by using an encoding polynomial. Since the encoding polynomial depends on a dataset, it must be updated every arrival of new dataset. Therefore, it is necessary to employ efficient algorithm to construct the encoding polynomial. In this paper, we propose Newton coded computing (NCC) which is based on Newton interpolation to construct the encoding polynomial. Let K, L, and T be the number of data, the length of each data, and the number of colluding workers, respectively. Then, the computational complexity for construction of an encoding polynomial is improved from O(L(K + T) log2 (K + T) log log(K + T)) for LCC to O(L(K + T) log(K + T)) for the proposed method. Furthermore, by applying the proposed method, the computational complexity for updating the encoding polynomial is improved from O(L(K + T) log2 (K + T) log log(K + T)) for LCC to O(L) for the proposed method.
Since most approximation functions in meshfree methods are rational functions which do not possess the Kronecker delta property, how to achieve exact integration and accurately impose the essential ...boundary conditions are two typical difficulties for meshfree methods. In this paper, a new stabilized Lagrange interpolation collocation method (SLICM) is proposed in which the Lagrange interpolation (LI) is employed for the approximation in a meshfree method. This method can satisfy the high order integration constraints which can conserves the high order consistency conditions in the integration form. This property leads to the exact integration in the subdomains and optimal convergence for the proposed method. Meanwhile, performing the integration in subdomains can also reduce the condition number of discrete matrix, which improves the stability of the algorithm. Since the Lagrange interpolation approximation has Kronecker delta property, the essential boundary conditions can be simply and exactly imposed like the finite element method, which further improves the accuracy of this method. Convergence studies present that the same convergence rate can be attained for utilizing the odd and even order LI shape functions, while the convergence rate is reduced if the odd order basis function is employed in the reproducing kernel (RK) approximation. Numerical examples validate the high accuracy and convergence as well as good stability of the presented method, which can outperform the direct collocation method and the stabilized collocation method based on RK approximation.
•A new stabilized Lagrange interpolation collocation method (SLICM) is proposed.•The SLICM can satisfy the high order integration constraints.•The essential boundary conditions can be simply and exactly imposed like the FEM.•The SLICM can achieve high accuracy, high efficiency and good stability.•The SLICM can outperform the DCM and the SCM based on RK approximation.
Artificial neural network has been several applications on fatigue crack, with prediction on fatigue crack growth life often serving as milestones. A typical difficulty in predicting the life curve ...of fatigue crack growth is the retardation of crack growth caused by the overload effect. Overload retardation, resulting in the lack of experiment data points in the retardation interval, is a long-term challenge for predicting fatigue crack growth. We improve the ability of back-propagation neural network to predict fatigue crack growth life by Lagrange interpolation, an interpolation for local data points. It combines the trend of whole data point to deal with the retardation interval, and interpolates multiple data points in the retardation interval when there is a lack of local data, and uses a neural network to construct the prediction formula of crack growth through self-optimization. In the study involving constant and variable amplitude fatigue crack growth experiments, the proposed method is proved to improve, with statistical significance, the predictive ability on the whole range of experiment data. The method is simple and accurate. Consequently, it is helpful to solve the problem of fatigue crack growth life of engineering structure.
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•ANN result of fatigue crack growth life is accurate under variable amplitude load.•The overload retardation effect with multiple overload ratios was taken into account.•ANN-based Lagrange interpolation with a back propagation training was proposed.•The proposed method can well predict fatigue crack growth with multiple overload.•The proposed method is easy to apply and has acceptable prediction accuracy.
Since the classical asymptotic theorems of Voronovskaya-type for positive and linear operators are in fact based on the Taylor’s formula which is a very particular case of Lagrange-Hermite ...interpolation formula, in the recent paper Gal 3, I have obtained semi-discrete quantitative Voronovskaya-type theorems based on other Lagrange-Hermite interpolation formulas, like Lagrange interpolation on two and three simple knots and Hermite interpolation on two knots, one simple and the other one double. In the present paper we obtain a semi-discrete quantitative Voronovskaya-type theorem based on Lagrange interpolation on arbitrary p + 1 simple distinct knots.
The monkeypox virus (MPXV) is what causes monkeypox (MPX) disease, which is comparable to both smallpox and cowpox. Using classical, fractional-order and complex order differential equations, we ...offer a deterministic mathematical model of the monkeypox virus in this study to research its possible breakouts in United States. The complex order derivative makes the fractional order derivative and the integer order derivative more common when the imaginary part of the complex order equals zero and when the real part in complex order derivatives is zero in this case, the new behaviour appears that doesn't appear in integer and fractional order derivatives. Eight nonlinear complex order differential equations make up this model consisting of humans and rodents population sizes. The population of humans Nh is divided into five different classes. The population of rodent Nr is divided into three classes, and the derivatives are described in the Atangana-Baleanu-Caputo sense and Mittag-Leffler kernels are employed. Numerical methods to simulate complex order systems such as The standard and nonstandard two-step Lagrange interpolation methods are utilised to fit the model. The basic reproduction number of the model is given. The stability of the suggested model's disease-free equilibrium point is shown. Finally, we study the duration and monkeypox outbreak's transmission pattern in the United States from June 13 to Sep 16, 2022, numerical simulations to illustrate our findings are presented. The results show that keeping diseased people apart from the general population decreases the spread of disease.
In this paper, we derive a variant of the Taylor theorem to obtain a new minimized remainder. For a given function f defined on the interval a,b, this formula is derived by introducing a linear ...combination of f′ computed at n+1 equally spaced points in a,b, together with f′′(a) and f′′(b). We then consider two classical applications of this Taylor-like expansion: the interpolation error and the numerical quadrature formula. We show that using this approach improves both the Lagrange P2- interpolation error estimate and the error bound of the Simpson rule in numerical integration.
Frost damage during flowering is recognized as one of the most serious agro-meteorological disasters affecting apple production in Shaanxi province, a typical apple producing area in China. ...Quantitative assessments of flowering frost damage to apple yield are critical for the development of strategies on mitigating yield losses, but have been rarely conducted. For the first time, our study used the process-based STICS model and statistical methods based on Lagrange interpolation method to assess the impacts of frost damage during flowering on apple yield at five study sites over the past three decades. Four canopy temperature thresholds were set as 0 °C, − 2 °C, − 5 °C, and − 35 °C for 0, 10%, 90%, and 100% frost damage on fruit number, respectively. The study results showed that STICS could effectively simulate the phenology and yield of apple, with the simulation errors less than 15%. Simulated yield losses caused by frost damage during flowering by STICS model were consistent with that estimated by statistical methods. Average yield loss due to frost damage during flowering estimated by STICS model was 6.35–29.27% in frost years during past three decades at five study sites. In general, both the frost occurrence days and intensities were the highest in the recent ten years. Our study provides a method to quantitatively assess the impacts of frost damage during flowering on apple yield for the prevention and mitigation of frost disasters in apple production.
•Crop model and statistical method were cross-validated to assess frost damage.•STICS could effectively simulate the flowering frost damage to apple yield.•Flowering frost was most serious in recent 10 years during past three decades.
Differential evolution (DE) is a simple yet powerful evolutionary algorithm that has been used to solve various complex optimization problems in numerous engineering fields. However, DE has some ...problems, such as premature convergence and sensitivity to parameter settings. To improve the performance of DE and extend its application, an adaptive differential evolution with the Lagrange interpolation argument algorithm (ADELI) is proposed in this paper. To accelerate the convergence speed of DE, a local search with Lagrange interpolation (LSLI) is introduced into DE. LSLI performs a local search in the neighborhood of the best individual in the current generation to enhance the exploitation capability of DE. Meanwhile, an adaptive argument strategy is presented to adaptively determine whether to use LSLI in terms of its performance in the previous generation, which can balance the global exploration capability and the local exploitation capability of ADELI. To verify the feasibility and effectiveness of ADELI, 30 test functions in the CEC 2014 benchmark sets with different dimensions were simulated. Moreover, a path synthesis problem was also optimized. The results demonstrated that ADELI considerably outperforms other EAs in most functions and obtains the most accurate solution among the compared algorithms in the application of path generation.
