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Автор: Grafiati

Опубліковано: 14 березня 2023

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1

Gama, Carmem, Mateus Gomes, and Liceia Pires. "Da teoria à prática: problematização e metodologias diferenciadas no Cálculo Numérico." Ensino em Re-vista 25, no. 1 (August 30, 2018): 234–55. http://dx.doi.org/10.14393/er-v25n1a2018-11.

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2

Hackbusch, Wolfgang. "Numerical tensor calculus." Acta Numerica 23 (May 2014): 651–742. http://dx.doi.org/10.1017/s0962492914000087.

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Анотація:

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

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3

Huber, B., F. Sottile, and B. Sturmfels. "Numerical Schubert Calculus." Journal of Symbolic Computation 26, no. 6 (December 1998): 767–88. http://dx.doi.org/10.1006/jsco.1998.0239.

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4

Hodyss, Daniel, Justin G. McLay, Jon Moskaitis, and Efren A. Serra. "Inducing Tropical Cyclones to Undergo Brownian Motion: A Comparison between Itô and Stratonovich in a Numerical Weather Prediction Model." Monthly Weather Review 142, no. 5 (April 30, 2014): 1982–96. http://dx.doi.org/10.1175/mwr-d-13-00299.1.

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Abstract Stochastic parameterization has become commonplace in numerical weather prediction (NWP) models used for probabilistic prediction. Here a specific stochastic parameterization will be related to the theory of stochastic differential equations and shown to be affected strongly by the choice of stochastic calculus. From an NWP perspective the focus will be on ameliorating a common trait of the ensemble distributions of tropical cyclone (TC) tracks (or position); namely, that they generally contain a bias and an underestimate of the variance. With this trait in mind the authors present a stochastic track variance inflation parameterization. This parameterization makes use of a properly constructed stochastic advection term that follows a TC and induces its position to undergo Brownian motion. A central characteristic of Brownian motion is that its variance increases with time, which allows for an effective inflation of an ensemble’s TC track variance. Using this stochastic parameterization the authors present a comparison of the behavior of TCs from the perspective of the stochastic calculi of Itô and Stratonovich within an operational NWP model. The central difference between these two perspectives as pertains to TCs is shown to be properly predicted by the stochastic calculus and the Itô correction. In the cases presented here these differences will manifest as overly intense TCs, which, depending on the strength of the forcing, could lead to problems with numerical stability and physical realism.

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5

Danca, Marius-F., and Michal Fečkan. "Chaos Suppression in a Gompertz-like Discrete System of Fractional Order." International Journal of Bifurcation and Chaos 30, no. 03 (March 15, 2020): 2050049. http://dx.doi.org/10.1142/s0218127420500492.

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Анотація:

In this paper, we introduce the fractional-order variant of a Gompertz-like discrete system. The chaotic behavior is suppressed with an impulsive control algorithm. The numerical integration and the Lyapunov exponent are obtained by means of the discrete fractional calculus. To verify numerically the obtained results, beside the Lyapunov exponent, the tools offered by the 0-1 test are used.

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6

Zhao, Yan Chun. "Design and Application of Digital Filter Based on Calculus Computing Concept." Applied Mechanics and Materials 513-517 (February 2014): 3151–55. http://dx.doi.org/10.4028/www.scientific.net/amm.513-517.3151.

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Анотація:

Calculus has been widely applied in engineering fields. The development of Integer order calculus theory is more mature in the project which can obtain fractional calculus theory through the promotion of integration order. It extends the flexibility of calculation and achieves the engineering analysis of multi-degree of freedom. According to fractional calculus features and the characteristics of fractional calculus, this paper treats the frequency domain as the object of study and gives the fractional calculus definition of the frequency characteristics. It also designs the mathematical model of fractional calculus digital filters using Fourier transform and Laplace transform. At last, this paper stimulates and analyzes numerical filtering of fractional calculus digital filter circuit using matlab general numerical analysis software and FDATool filter toolbox provided by matlab. It obtains the one-dimensional and two-dimensional filter curves of fractional calculus method which achieves the fractional Calculus filter of complex digital filter.

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7

Passarino, Giampiero. "Structural Aspects of Numerical Loop Calculus." Nuclear Physics B - Proceedings Supplements 135 (October 2004): 265–69. http://dx.doi.org/10.1016/j.nuclphysbps.2004.09.026.

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8

Beylkin, G., and M. J. Mohlenkamp. "Numerical operator calculus in higher dimensions." Proceedings of the National Academy of Sciences 99, no. 16 (July 24, 2002): 10246–51. http://dx.doi.org/10.1073/pnas.112329799.

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9

Parker, G. Edgar. "TEACHING CALCULUS WITH A NUMERICAL EMPHASIS." PRIMUS 2, no. 1 (January 1992): 65–78. http://dx.doi.org/10.1080/10511979208965651.

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10

Freihat, Asad, and Shaher Momani. "Application of Multistep Generalized Differential Transform Method for the Solutions of the Fractional-Order Chua's System." Discrete Dynamics in Nature and Society 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/427393.

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Анотація:

We numerically investigate the dynamical behavior of the fractional-order Chua's system. By utilizing the multistep generalized differential transform method (MSGDTM), we find that the fractional-order Chua's system with “effective dimension” less than three can exhibit chaos as well as other nonlinear behavior. Numerical results are presented graphically and reveal that the multistep generalized differential transform method is an effective and convenient method to solve similar nonlinear problems in fractional calculus.

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11

Ubale, P. V. "Numerical Solution of Boole’s Rule in Numerical Integration by Using General Quadrature Formula." Bulletin of Society for Mathematical Services and Standards 2 (June 2012): 1–4. http://dx.doi.org/10.18052/www.scipress.com/bsmass.2.1.

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Анотація:

We have seen that definite integrals arise in many different areas and that the fundamental theorem of calculus is a powerful tool for evaluating definite integrals. This paper describes classical quadrature method for the numerical solution of Boole’s rule in numerical integration.

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12

Iliescu, Nicolae, Vasile Nastasescu, and Ghiță Barsan. "A Numerical and Experimental Aproach of Stress Waves Propagation in Short Tronconical Bars Under Axial Impact." International conference KNOWLEDGE-BASED ORGANIZATION 24, no. 3 (June 1, 2018): 101–6. http://dx.doi.org/10.1515/kbo-2018-0144.

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Abstract In the first part of the paper, using the numerical simulations with FEM and the results of some investigations made with different experimental techniques, a calculation methodology was developed for the study of the stress waves propagation in the short tronconical bars subjected at axial impact. Because a good agreement between data obtained from numerical analysis and experimental investigations was observed, the numerical model of calculus conceived for this study was considered validated. The calculus model established was used to investigate other aspects connected of stress wave propagation in the short tronconical bars. In the second part of the paper, using established calculus model and numerical analysis with Finite Element Method the influence of bar conicity on stress wave propagation and on stress distribution in different cross sections of the bar was analyzed

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13

French, A., J. P. Cullerne, and O. Kanchanasakdichai. "Numerical methods as an introduction to calculus." Physics Education 54, no. 4 (May 2, 2019): 045009. http://dx.doi.org/10.1088/1361-6552/aaefd1.

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14

Cai, Min, and Changpin Li. "Numerical Approaches to Fractional Integrals and Derivatives: A Review." Mathematics 8, no. 1 (January 1, 2020): 43. http://dx.doi.org/10.3390/math8010043.

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Анотація:

Fractional calculus, albeit a synonym of fractional integrals and derivatives which have two main characteristics—singularity and nonlocality—has attracted increasing interest due to its potential applications in the real world. This mathematical concept reveals underlying principles that govern the behavior of nature. The present paper focuses on numerical approximations to fractional integrals and derivatives. Almost all the results in this respect are included. Existing results, along with some remarks are summarized for the applied scientists and engineering community of fractional calculus.

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15

Kadak, Uğur, and Muharrem Özlük. "Generalized Runge-Kutta Method with respect to the Non-Newtonian Calculus." Abstract and Applied Analysis 2015 (2015): 1–10. http://dx.doi.org/10.1155/2015/594685.

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Theory and applications of non-Newtonian calculus have been evolving rapidly over the recent years. As numerical methods have a wide range of applications in science and engineering, the idea of the design of such numerical methods based on non-Newtonian calculus is self-evident. In this paper, the well-known Runge-Kutta method for ordinary differential equations is developed in the frameworks of non-Newtonian calculus given in generalized form and then tested for different generating functions. The efficiency of the proposed non-Newtonian Euler and Runge-Kutta methods is exposed by examples, and the results are compared with the exact solutions.

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16

Azizi, Tahmineh. "Application of the Fractional Calculus in Pharmacokinetic Compartmental Modeling." Communication in Biomathematical Sciences 5, no. 1 (August 2, 2022): 63–77. http://dx.doi.org/10.5614/cbms.2022.5.1.4.

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In this study, we present the application of fractional calculus (FC) in biomedicine. We present three different integer order pharmacokinetic models which are widely used in cancer therapy with two and three compartments and we solve them numerically and analytically to demonstrate the absorption, distribution, metabolism, and excretion (ADME) of drug in different tissues. Since tumor cells interactions are systems with memory, the fractional-order framework is a better approach to model the cancer phenomena rather than ordinary and delay differential equations. Therefore, the nonstandard finite difference analysis or NSFD method following the Grunwald-Letinkov discretization may be applied to discretize the model and obtain the fractional-order form to describe the fractal processes of drug movement in body. It will be of great significance to implement a simple and efficient numerical method to solve these fractional-order models. Therefore, numerical methods using finite difference scheme has been carried out to derive the numerical solution of fractional-order two and tri-compartmental pharmacokinetic models for oral drug administration. This study shows that the fractional-order modeling extends the capabilities of the integer order model into the generalized domain of fractional calculus. In addition, the fractional-order modeling gives more power to control the dynamical behaviors of (ADME) process in different tissues because the order of fractional derivative may be used as a new control parameter to extract the variety of governing classes on the non local behaviors of a model, however, the integer order operator only deals with the local and integer order domain. As a matter of fact, NSFD may be used as an effective and very easy method to implement for this type application, and it provides a convenient framework for solving the proposed fractional-order models.

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17

Li, T. Y., Xiaoshen Wang, and Mengnien Wu. "Numerical Schubert Calculus by the Pieri Homotopy Algorithm." SIAM Journal on Numerical Analysis 40, no. 2 (January 2002): 578–600. http://dx.doi.org/10.1137/s003614290139175x.

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18

Năstăsescu, Vasile, Ghiță Bârsan, and Silvia Marzavan. "On the Calculus of Functionally Graded Plates." International conference KNOWLEDGE-BASED ORGANIZATION 28, no. 3 (June 1, 2022): 71–85. http://dx.doi.org/10.2478/kbo-2022-0090.

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Abstract This paper presents some of our results regarding calculus of the functionally graded plates (FGPs). Such plates are made of Functionally Graded Materials (FGMs), which represent a new material class belonging to the composite materials. Our paper presents some material laws in a comparative way. But the main purpose of this paper is to provide calculus concepts and calculus methodologies, based on the means available in scientific research of mechanical engineering field, for the calculation of plates made of FGMs. Thus, the authors use the concepts of multilayer plate and equivalent plate, using both analytical and numerical calculus. The numerical method used are the Finite Element Method (FEM).The calculation aims both to determine the displacements and the stresses of the plates under statical loads, as well as to determine the free vibration frequencies. The research methodology is based on combining analytical calculation, where and when possible, with numerical simulation. The validation of our results is done by comparison with the analytical solution and the comparative analysis of the methods.

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19

Persechino, A. "An introduction to fractional calculus." Advanced Electromagnetics 9, no. 1 (February 19, 2020): 19–30. http://dx.doi.org/10.7716/aem.v9i1.1192.

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The aim of this work is to introduce the main concepts of Fractional Calculus, followed by one of its application to classical electrodynamics, illustrating how non-locality can be interpreted naturally in a fractional scenario. In particular, a result relating fractional dynamics to high frequency dielectric response is used as motivation. In addition to the theoretical discussion, a comprehensive review of two numerical procedures for fractional integration is carried out, allowing one immediately to build numerical models applied to high frequency electromagnetics and correlated fields.

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20

Zine, Houssine, and Delfim F. M. Torres. "A Stochastic Fractional Calculus with Applications to Variational Principles." Fractal and Fractional 4, no. 3 (August 1, 2020): 38. http://dx.doi.org/10.3390/fractalfract4030038.

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We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler–Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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21

Kang, Zhi Qiang, Ying Chun Wang, Feng Nan, Shi Lin Yuan, and Jiu Lin Yang. "Research on Numerical Analysis of Landslide Cataclysm Mechanism Coal Mine." Advanced Materials Research 383-390 (November 2011): 7697–701. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.7697.

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This article according to the ShengLi Open-pit Coal Mine landslide and the slope project special details, used the FLAC numerical calculus analysis software to conduct the research to the landslide cataclysm mechanism, has carried on the optimized analysis to the reinforcement plan.Has obtained the pre-stressed anchor rope frame beam + high pressure splitting grouting reinforcement plan government landslide most superior processing plan through the numerical calculus, thus active control ShengLi Open-pit Coal Mine slope distortion destruction.

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22

Risser, H. Smith. "Activities for Students: Exploring Numerical Derivatives." Mathematics Teacher 102, no. 3 (October 2008): 224–30. http://dx.doi.org/10.5951/mt.102.3.0224.

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Several years ago I gave a test on finding derivatives to students in my AP Calculus class. The test covered basic derivative rules and equations of tangent lines. During the unit, I had also taught students how to find numerical derivatives with their calculators.

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23

Risser, H. Smith. "Activities for Students: Exploring Numerical Derivatives." Mathematics Teacher 102, no. 3 (October 2008): 224–30. http://dx.doi.org/10.5951/mt.102.3.0224.

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Анотація:

Several years ago I gave a test on finding derivatives to students in my AP Calculus class. The test covered basic derivative rules and equations of tangent lines. During the unit, I had also taught students how to find numerical derivatives with their calculators.

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24

Shaimardan, S., and N. S. Tokmagambetov. "On the solutions of some fractional q-differential equations with the Riemann-Liouville fractional q-derivative." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, no. 4 (December 30, 2021): 130–41. http://dx.doi.org/10.31489/2021m4/130-141.

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Анотація:

This paper is devoted to explicit and numerical solutions to linear fractional q-difference equations and the Cauchy type problem associated with the Riemann-Liouville fractional q-derivative in q-calculus. The approaches based on the reduction to Volterra q-integral equations, on compositional relations, and on operational calculus are presented to give explicit solutions to linear q-difference equations. For simplicity, we give results involving fractional q-difference equations of real order a > 0 and given real numbers in q-calculus. Numerical treatment of fractional q-difference equations is also investigated. Finally, some examples are provided to illustrate our main results in each subsection.

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25

Scherger, Nicole. "Technology Tips: Using Maple to Enhance Students' Understanding of Numerical Integration." Mathematics Teacher 103, no. 1 (August 2009): 76–80. http://dx.doi.org/10.5951/mt.103.1.0076.

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Анотація:

Typically, calculus students are introduced to the simplest numerical approximations of the definite integral through the process of finding the areas of rectangles. Students are initially shown how to use the endpoints of each subinterval to find lower and upper sums, a process that gives them a bound on the actual area. They are then shown, sometimes through a series of labor-intensive computations or through visualization with graphs, that as the number of rectangles, or partitions, increases, the approximations become more and more accurate. Somewhere in this process students are probably also shown how to use midpoints to obtain slightly more accurate numerical approximations. At this point, most calculus courses lead students toward the fundamental theorem of calculus, at which time they learn that they can evaluate a definite integral by finding the antiderivative and evaluating between the limits of integration.

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26

Scherger, Nicole. "Technology Tips: Using Maple to Enhance Students' Understanding of Numerical Integration." Mathematics Teacher 103, no. 1 (August 2009): 76–80. http://dx.doi.org/10.5951/mt.103.1.0076.

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Анотація:

Typically, calculus students are introduced to the simplest numerical approximations of the definite integral through the process of finding the areas of rectangles. Students are initially shown how to use the endpoints of each subinterval to find lower and upper sums, a process that gives them a bound on the actual area. They are then shown, sometimes through a series of labor-intensive computations or through visualization with graphs, that as the number of rectangles, or partitions, increases, the approximations become more and more accurate. Somewhere in this process students are probably also shown how to use midpoints to obtain slightly more accurate numerical approximations. At this point, most calculus courses lead students toward the fundamental theorem of calculus, at which time they learn that they can evaluate a definite integral by finding the antiderivative and evaluating between the limits of integration.

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27

TALL, DAVID. "A Versatile Approach to Calculus and Numerical Methods." Teaching Mathematics and its Applications 9, no. 3 (1990): 124–31. http://dx.doi.org/10.1093/teamat/9.3.124.

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28

Razzaghi, M. "A numerical scheme for problems in fractional calculus." ITM Web of Conferences 20 (2018): 02001. http://dx.doi.org/10.1051/itmconf/20182002001.

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In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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29

Aguila, F. del, and R. Pittau. "Recursive numerical calculus of one-loop tensor integrals." Journal of High Energy Physics 2004, no. 07 (July 13, 2004): 017. http://dx.doi.org/10.1088/1126-6708/2004/07/017.

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30

Drury, S. W. "Symbolic calculus of operators with unit numerical radius." Linear Algebra and its Applications 428, no. 8-9 (April 2008): 2061–69. http://dx.doi.org/10.1016/j.laa.2007.11.007.

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31

Carpinteri, A., B. Chiaia, and P. Cornetti. "Numerical modelization of disordered media via fractional calculus." Computational Materials Science 30, no. 1-2 (May 2004): 155–62. http://dx.doi.org/10.1016/j.commatsci.2004.01.023.

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32

Gentle, Adrian P. "Regge Calculus: A Unique Tool for Numerical Relativity." General Relativity and Gravitation 34, no. 10 (October 2002): 1701–18. http://dx.doi.org/10.1023/a:1020128425143.

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33

Zheng, Xie, Ye Zheng, and Ma Yu-Jie. "Numerical Simulation of Antennae by Discrete Exterior Calculus." Communications in Theoretical Physics 52, no. 6 (December 2009): 1067–70. http://dx.doi.org/10.1088/0253-6102/52/6/17.

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34

Di Rocco, Sandra, David Eklund, and Chris Peterson. "Numerical polar calculus and cohomology of line bundles." Advances in Applied Mathematics 100 (September 2018): 148–62. http://dx.doi.org/10.1016/j.aam.2018.06.002.

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35

Crouzeix, Michel. "Numerical range and functional calculus in Hilbert space." Journal of Functional Analysis 244, no. 2 (March 2007): 668–90. http://dx.doi.org/10.1016/j.jfa.2006.10.013.

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36

Benson, David A., Mark M. Meerschaert, and Jordan Revielle. "Fractional calculus in hydrologic modeling: A numerical perspective." Advances in Water Resources 51 (January 2013): 479–97. http://dx.doi.org/10.1016/j.advwatres.2012.04.005.

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37

Năstăsescu, Vasile, and Gheorghe Bârsan. "SPH method in numerical calculus of detonation parameters." Journal of Engineering Sciences and Innovation 4, no. 1 (March 5, 2019): 1–16. http://dx.doi.org/10.56958/jesi.2019.4.1.1.

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Анотація:

This paper presents some of our results in using of the Smoothed Particles Hydrodynamics (SPH) method for the numerical modelling of the detonation phenomena. The study of the detonation is very important for evaluation of different explosives and even for their design. The paper also presents, in a synthetically way, some fundamentals of the detonation. The numerical modelling of the detonation can be made by Finite Element Method (FEM), but using of the SPH method brings some important advantages because the large deformations, large distortions occur. If the FEM can be used in Arbitrary Lagrangian-Eulerian (ALE) formulation, the SPH method can easily works and next to it, some specific parameters (density variation, specific energy etc.) can be obtained and analyzed by post-processing. The numerical results (by FEM and by SPH method) are compared with theoretical results. The numerical study allowed us to analyse the influence of some explosive characteristics and of the circumstances (non confined and confined explosive) upon detonation parameters.

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38

WANG, TIANSHU, and XINGYUAN WANG. "GENERALIZED SYNCHRONIZATION OF FRACTIONAL ORDER HYPERCHAOTIC LORENZ SYSTEM." Modern Physics Letters B 23, no. 17 (July 10, 2009): 2167–78. http://dx.doi.org/10.1142/s021798490902031x.

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Анотація:

In this paper, a type of new fractional order hyperchaotic Lorenz system is proposed. Based on the fractional calculus predictor-corrector algorithm, the fractional order hyperchaotic Lorenz system is investigated numerically, and the simulation results show that the lowest orders for hyperchaos in hyperchaotic Lorenz system is 3.884. According to the stability theory of fractional order system, an improved state-observer is designed, and the response system of generalized synchronization is obtained analytically, whose feasibility is proved theoretically. The synchronization method is adopted to realize the generalized synchronization of 3.884-order hyperchaotic Lorenz system, and the numerical simulation results verify the effectiveness.

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39

Hare, Warren, and Gabriel Jarry-Bolduc. "About the Performance of a Calculus-Based Approach to Building Model Functions in a Derivative-Free Trust-Region Algorithm." Algorithms 16, no. 2 (February 3, 2023): 84. http://dx.doi.org/10.3390/a16020084.

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Анотація:

This paper examines a calculus-based approach to building model functions in a derivative-free algorithm. This calculus-based approach can be used when the objective function considered is defined via more than one blackbox. Two versions of a derivative-free trust-region method are implemented. The first version builds model functions by using a calculus-based approach, and the second version builds model functions by directly considering the objective function. The numerical experiments demonstrate that the calculus-based approach provides better results in most situations and significantly better results in specific situations.

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40

Li, Song, Bo Fang, Tianzhi Yang, and Wenhu Huang. "A Fractional Calculus for Nonlinear Energy Sink Used in Vibration Absorption System." Noise & Vibration Worldwide 42, no. 10 (November 2011): 62–67. http://dx.doi.org/10.1260/0957-4565.42.10.62.

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Анотація:

The phenomenon of energy pumping, in which vibratory energy is transferred irreversibly within a nonlinear, multi-degree-of-freedom system with the goal of reducing the transient response of the primary substructure, has recently been investigated analytically and through numerical simulations. The dynamics of single degree of freedom linear subsystem with attached nonlinear energy sink is investigated. The response of a linear oscillator attached to nonlinear energy sink with relatively small mass under external forcing in a vicinity of main resonance is studied analytically and numerically. It is possible that targeted energy could transfer from linear oscillators to the nonlinear energy sink in this system. Analytical model is verified numerically and a fairly good correspondence is observed. Fractional calculus offers a powerful tool to describe the dynamic behavior of real vibration absorption. A version of the fractional derivative models is presented and investigated in this paper for analyzing vibration absorption behavior of nonlinear energy sink. It is shown that the fractional-order system is in a stronger position than the traditional nonlinear energy sink coupled to the linear oscillator.

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41

Akter, Sayeda Irin, Md Shahriar Mahmud, Md Kamrujjaman, and Hazrat Ali. "Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations." GANIT: Journal of Bangladesh Mathematical Society 40, no. 1 (July 14, 2020): 28–42. http://dx.doi.org/10.3329/ganit.v40i1.48193.

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Numerical analysis is the area of mathematics that creates, analyzes, and implements algorithms for solving numerically the problems from real-world applications of algebra, geometry, and calculus, and they involve variables which vary continuously. Till now, numerous numerical methods have been introduced. Spectral method is one of those techniques used in applied mathematics and scientific computing to numerically solve certain differential equations, potentially involving the use of the Fast Fourier Transform (FFT). This study presents some of the fundamental ideas of spectral method. Orthogonal basis are used to establish computational algorithm. The accuracy and efficiency of proposed model are discussed. This manuscript estimates for the error between the exact and approximated discrete solutions. This paper shows that, grid points for polynomial spectral methods should lie approximately in a minimal energy configuration associated with inverse linear repulsion between points. The wave equation, linear and non-linear boundary value problems are solved using spectral method on the platform of MATLAB language. GANIT J. Bangladesh Math. Soc.Vol. 40 (2020) 28-42

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42

Ma, Ling, and Noel J. Walkington. "On Algorithms for Nonconvex Optimization in the Calculus of Variations." SIAM Journal on Numerical Analysis 32, no. 3 (June 1995): 900–923. http://dx.doi.org/10.1137/0732042.

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43

Schüssler, Rudolf. "Equi-Probability Prior to 1650." Early Science and Medicine 21, no. 1 (February 22, 2016): 54–74. http://dx.doi.org/10.1163/15733823-00211p03.

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The assumption that two probabilities can be equal is a conceptual prerequisite for the development of a numerical probability calculus. Such a calculus first emerged in the seventeenth century. Several accounts have been proposed to explain the delayed development of numerical probability, yet it has thus far not been noted that the concept of equi-probability was virtually absent from medieval thought. This article argues that its rise began in the early sixteenth century, a fact that contributes to a better understanding of the preconditions which facilitated the modern mathematization of probability.

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44

Li, Shengjie, Jean-Paul Penot, and Xiaowei Xue. "Codifferential Calculus." Set-Valued and Variational Analysis 19, no. 4 (December 30, 2010): 505–36. http://dx.doi.org/10.1007/s11228-010-0171-7.

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45

Behr, Nicolas, Giuseppe Dattoli, Ambra Lattanzi, and Silvia Licciardi. "Dual Numbers and Operational Umbral Methods." Axioms 8, no. 3 (July 2, 2019): 77. http://dx.doi.org/10.3390/axioms8030077.

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Анотація:

Dual numbers and their higher-order version are important tools for numerical computations, and in particular for finite difference calculus. Based on the relevant algebraic rules and matrix realizations of dual numbers, we present a novel point of view, embedding dual numbers within a formalism reminiscent of operational umbral calculus.

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46

Gremaud, P. A. "Numerical optimization and quasiconvexity." European Journal of Applied Mathematics 6, no. 1 (February 1995): 69–82. http://dx.doi.org/10.1017/s0956792500001674.

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In the Calculus of Variations, several notions of convexity have emerged, corresponding to different properties of the functionals to be minimized. The relations between these various notions are not yet fully understood. In this context, we present a numerical study of quasiconvexity for some functions of the type f(ξ) = g(|ξ|2, det ξ), where Ξ is a 2×2-matrix. The corresponding global optimization problems are solved using a simulated annealing-like algorithm. The computations strongly indicate that the considered functions are quasiconvex if and only if they are rank-one convex. The relation to Morrey's conjecture, various applications and implementation problems are discussed.

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47

Zhang, Xiao-Li, Wei Zhang, Yu-Lan Wang, and Ting-Ting Ban. "The space spectral interpolation collocation method for reaction-diffusion systems." Thermal Science 25, no. 2 Part B (2021): 1269–75. http://dx.doi.org/10.2298/tsci200402022z.

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A space spectral interpolation collocation method is proposed to study non-linear reaction-diffusion systems with complex dynamics characters. A detailed solution process is elucidated, and some pattern formations are given. The numerical results have a good agreement with theoretical ones. The method can be extended to fractional calculus and fractal calculus.

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48

Huang, Meihua, Pongsakorn Sunthrayuth, Amjad Ali Pasha, and Muhammad Altaf Khan. "Numerical solution of stochastic and fractional competition model in Caputo derivative using Newton method." AIMS Mathematics 7, no. 5 (2022): 8933–52. http://dx.doi.org/10.3934/math.2022498.

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<abstract><p>Many useful numerical algorithms of the numerical solution are proposed due to the increasing interest of the researchers in fractional calculus. A new discretization of the competition model for the real statistical data of banking finance for the years 2004–2014 is presented. We use a novel numerical method that is more reliable and accurate which is introduced recently for the solution of ordinary differential equations numerically. We apply this approach to solve our model for the case of Caputo derivative. We apply the Caputo derivative on the competition system and obtain its numerical results. For the numerical solution of the competition model, we use the Newton polynomial approach and present in detail a novel numerical procedure. We utilize the numerical procedure and present various numerical results in the form of graphics. A comparison of the present method versus the predictor corrector method is presented, which shows the same solution behavior to the Newton Polynomial approach. We also suggest that the real data versus model provide good fitting for both the data for the fractional-order parameter value $ \rho = 0.7 $. Some more values of $ \rho $ are used to obtain graphical results. We also check the model in the stochastic version and show the model behaves well when fitting to the data.</p></abstract>

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49

Liang, Guishu, and Yulan Yang. "Fractional calculus-based analysis of soil electrical properties." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 39, no. 2 (November 28, 2019): 279–95. http://dx.doi.org/10.1108/compel-05-2019-0179.

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Purpose This paper aims to analyze soil electrical properties based on fractional calculus theory due to the fact that the frequency dependence of soil electrical parameters at high frequencies exhibits a fractional effect. In addition, for the fractional-order formulation, this paper aims to provide a more accurate numerical algorithm for solving the fractional differential equations. Design/methodology/approach This paper analyzes the frequency-dependence of soil electrical properties based on fractional calculus theory. A collocation method based on the Puiseux series is proposed to solve fractional differential equations. Findings The algorithm proposed in this paper can be used to solve fractional differential equations of arbitrary order, especially for 0.5th-order equations, obtaining accurate numerical solutions. Calculating the impact response of the grounding electrode based on the fractional calculus theory can obtain a more accurate result. Originality/value This paper proposes an algorithm for solving fractional differential equations of arbitrary order, especially for 0.5th-order equations. Using fractional calculus theory to analyze the frequency-dependent effect of soil electrical properties, provides a new idea for ground-related transient calculation.

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50

Alsulami, Hamed, M. Syed Ali, M. Hymavathi, Tareq Saeed, Bashir Ahmad та Ahmed Alsaedi. "Mixed ℋ ∞ and Passivity Analysis of Delayed Fractional-Order Complex Dynamical Networks with Hybrid Coupling". Mathematical Problems in Engineering 2022 (20 жовтня 2022): 1–15. http://dx.doi.org/10.1155/2022/6327922.

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In this article, global asymptotic stability analysis, and mixed ℋ ∞ and passive control for a class of control fractional-order systems is investigated. Based on the fractional-order Lyapunov stability theorem and some properties of fractional calculus, we propose sufficient conditions to ensure the mixed ℋ ∞ and passivity performance. More relaxed conditions by employing the new type of augmented matrices by using Kronecker product terms can be handled, which can be introduced. The derived criteria are expressed in terms of linear matrix inequalities that which can be checked numerically using toolbox MATLAB. Finally, two numerical examples are provided to demonstrate the correctness of the proposed results.

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