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Journal articles on the topic 'Partial derivatives'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 April 2022

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1

Pandey, Shikha, Dragan Obradovic, Lakshmi Narayan Mishra, and Vishnu Narayan Mishra. "Second Order Partial Derivatives." JOURNAL OF ADVANCES IN MATHEMATICS 20 (September 8, 2021): 419–23. http://dx.doi.org/10.24297/jam.v20i.9097.

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The rules for calculating partial derivatives and differentials are the same as for calculating the derivative of a function of one variable, except that when finding partial derivatives per one variable, the other variables are considered as constants

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2

Yu, Chii Huei. "Using Maple to Study the Partial Differential Problems." Applied Mechanics and Materials 479-480 (December 2013): 800–804. http://dx.doi.org/10.4028/www.scientific.net/amm.479-480.800.

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This paper uses the mathematical software Maple for the auxiliary tool to study the partial differential problem of two types of multivariable functions. We can obtain the infinite series forms of any order partial derivatives of these two types of multivariable functions by using differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order partial derivative values. On the other hand, we propose two examples of multivariable functions to evaluate their any order partial derivatives, and some of their higher order partial derivative values practically. At the same time, we employ Maple to calculate the approximations of these higher order partial derivative values and their infinite series forms for verifying our answers.

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3

Attou, Samira, Ludovic Mignot, and Djelloul Ziadi. "Bottom-Up derivatives of tree expressions." RAIRO - Theoretical Informatics and Applications 55 (2021): 4. http://dx.doi.org/10.1051/ita/2021008.

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In this paper, we extend the notion of (word) derivatives and partial derivatives due to (respectively) Brzozowski and Antimirov to tree derivatives using already known inductive formulae of quotients. We define a new family of extended regular tree expressions (using negation or intersection operators), and we show how to compute a Brzozowski-like inductive tree automaton; the fixed point of this construction, when it exists, is the derivative tree automaton. Such a deterministic tree automaton can be used to solve the membership test efficiently: the whole structure is not necessarily computed, and the derivative computations can be performed in parallel. We also show how to solve the membership test using our (Bottom-Up) partial derivatives, without computing an automaton.

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4

Dalík, Josef. "Operators approximating partial derivatives at vertices of triangulations by averaging." Mathematica Bohemica 135, no. 4 (2010): 363–72. http://dx.doi.org/10.21136/mb.2010.140827.

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5

Progri, Ilir F. "Hypergeometric Function Partial Derivatives." Journal of Geolocation, Geo-information and Geo-intelligence 2016, no. 1 (2016): 53. http://dx.doi.org/10.18610/jg3.2016.071604.

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6

Ojha, Bhuwan Prasad. "Different Concepts of Derivatives." Journal of Advanced College of Engineering and Management 3 (January 10, 2018): 11. http://dx.doi.org/10.3126/jacem.v3i0.18809.

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<p>In this paper, different concept of derivatives with some properties has been introduced. In differential calculus, the partial derivative, directional derivative and total derivative are studied. Their generalization for Banach spaces are the Gateaux differential and Freshet derivative.</p><p><strong>Journal of Advanced College of Engineering and Management,</strong> Vol.3, 2017, Page: 11-14</p>

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7

Li, Changpin, Deliang Qian, and YangQuan Chen. "On Riemann-Liouville and Caputo Derivatives." Discrete Dynamics in Nature and Society 2011 (2011): 1–15. http://dx.doi.org/10.1155/2011/562494.

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Recently, many models are formulated in terms of fractional derivatives, such as in control processing, viscoelasticity, signal processing, and anomalous diffusion. In the present paper, we further study the important properties of the Riemann-Liouville (RL) derivative, one of mostly used fractional derivatives. Some important properties of the Caputo derivative which have not been discussed elsewhere are simultaneously mentioned. The partial fractional derivatives are also introduced. These discussions are beneficial in understanding fractional calculus and modeling fractional equations in science and engineering.

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8

Baksa, Vita, Andriy Bandura, and Oleh Skaskiv. "Analogs of Hayman’s Theorem and of logarithmic criterion for analytic vector-valued functions in the unit ball having bounded L-index in joint variables." Mathematica Slovaca 70, no. 5 (October 27, 2020): 1141–52. http://dx.doi.org/10.1515/ms-2017-0420.

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AbstractIn this paper, we present necessary and sufficient conditions of boundedness of L-index in joint variables for vector-valued functions analytic in the unit ball $\begin{array}{} \mathbb{B}^2\! = \!\{z\!\in\!\mathbb{C}^2: |z|\! = \!\small\sqrt{|z_1|^2+|z_2|^2}\! \lt \! 1\}, \end{array} $ where L = (l1, l2): 𝔹2 → $\begin{array}{} \mathbb{R}^2_+ \end{array} $ is a positive continuous vector-valued function.Particularly, we deduce analog of Hayman’s theorem for this class of functions. The theorem shows that in the definition of boundedness of L-index in joint variables for vector-valued functions we can replace estimate of norms of all partial derivatives by the estimate of norm of (p + 1)-th order partial derivative. This form of criteria could be convenient to investigate analytic vector-valued solutions of system of partial differential equations because it allow to estimate higher-order partial derivatives by partial derivatives of lesser order. Also, we obtain sufficient conditions for index boundedness in terms of estimate of modulus of logarithmic derivative in each variable for every component of vector-valued function outside some exceptional set by the vector-valued function L(z).

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9

Wu, Wang, Su, and Zhang. "A MATLAB Package for Calculating Partial Derivatives of Surface-Wave Dispersion Curves by a Reduced Delta Matrix Method." Applied Sciences 9, no. 23 (November 30, 2019): 5214. http://dx.doi.org/10.3390/app9235214.

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Various surface-wave exploration methods have become increasingly important tools in investigating the properties of subsurface structures. Inversion of the experimental dispersion curves is generally an indispensable component of these methods. Accurate and reliable calculation of partial derivatives of surface-wave dispersion curves with respect to parameters of subsurface layers is critical to the success of these approaches if the linearized inversion strategies are adopted. Here we present an open-source MATLAB package, named SWPD (Surface Wave Partial Derivative), for modeling surface-wave (both Rayleigh- and Love-wave) dispersion curves (both phase and group velocity) and particularly for computing their partial derivatives with high precision. The package is able to compute partial derivatives of phase velocity and of Love-wave group velocity analytically based on the combined use of the reduced delta matrix theory and the implicit function theorem. For partial derivatives of Rayleigh-wave group velocity, a hemi-analytical method is presented, which analytically calculates all the first-order partial differentiations and approximates the mixed second-order partial differentiation term with a central difference scheme. We provide examples to demonstrate the effectiveness of this package, and demo scripts are also provided for users to reproduce all results of this paper and thus to become familiar with the package as quickly as possible.

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10

Champarnaud, J. M., and D. Ziadi. "Canonical derivatives, partial derivatives and finite automaton constructions." Theoretical Computer Science 289, no. 1 (October 2002): 137–63. http://dx.doi.org/10.1016/s0304-3975(01)00267-5.

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11

Sulzmann, Martin, and Peter Thiemann. "Derivatives and partial derivatives for regular shuffle expressions." Journal of Computer and System Sciences 104 (September 2019): 323–41. http://dx.doi.org/10.1016/j.jcss.2016.11.010.

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12

Michelsen, M. L., and Jørgen Mollerup. "Partial derivatives of thermodynamic properties." AIChE Journal 32, no. 8 (August 1986): 1389–92. http://dx.doi.org/10.1002/aic.690320818.

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13

Yudin, V. A. "RIESZ TRANSFORMS AND PARTIAL DERIVATIVES." Mathematics of the USSR-Sbornik 69, no. 2 (February 28, 1991): 445–51. http://dx.doi.org/10.1070/sm1991v069n02abeh002115.

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14

Hassenpflug, W. C. "Revised notation for partial derivatives." Computers & Mathematics with Applications 26, no. 3 (August 1993): 95–105. http://dx.doi.org/10.1016/0898-1221(93)90111-8.

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15

Spitzmüller, Klaus. "Partial derivatives of Bézier surfaces." Computer-Aided Design 28, no. 1 (January 1996): 67–72. http://dx.doi.org/10.1016/0010-4485(95)00044-5.

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16

Maslyuchenko, V. K. "A property of partial derivatives." Ukrainian Mathematical Journal 39, no. 4 (1988): 431–33. http://dx.doi.org/10.1007/bf01060782.

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17

Vilseck, Jonah Z., Julian Tirado-Rives, and William L. Jorgensen. "Determination of partial molar volumes from free energy perturbation theory." Physical Chemistry Chemical Physics 17, no. 13 (2015): 8407–15. http://dx.doi.org/10.1039/c4cp05304d.

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Free Energy Perturbation calculations are employed to determine free energies of solvation (ΔG_solv) for benzene and benzene-derivatives at elevated pressures. Absolute and relative partial molar volumes are determined as the pressure derivative of ΔG_solv.

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18

Özkan, Ozan, and Ali Kurt. "Exact solutions of fractional partial differential equation systems with conformable derivative." Filomat 33, no. 5 (2019): 1313–22. http://dx.doi.org/10.2298/fil1905313o.

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Main goal of this paper is to have the new exact solutions of some fractional partial differential equation systems (FPDES) in conformable sense. The definition of conformable fractional derivative (CFD) is similar to the limit based definition of known derivative. This derivative obeys both rules which other popular derivatives do not satisfy such as derivative of the quotient of two functions, the derivative product of two functions, chain rule and etc. By using conformable derivative it is seen that the solution procedure for (PDES) is simpler and more efficient.

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19

Kolyada, V. I., and F. J. Pérez Lázaro. "Inequalities for Partial Moduli of Continuity and Partial Derivatives." Constructive Approximation 34, no. 1 (February 9, 2010): 23–59. http://dx.doi.org/10.1007/s00365-010-9088-5.

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20

Kojima, Masakazu. "EFFICIENT EVALUATION OF POLYNOMIALS AND THEIR PARTIAL DERIVATIVES IN HOMOTOPY CONTINUATION METHODS." Journal of the Operations Research Society of Japan 51, no. 1 (2008): 29–54. http://dx.doi.org/10.15807/jorsj.51.29.

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21

Zainal, Nor Hafizah, and Adem Kılıçman. "Solving Fractional Partial Differential Equations with Corrected Fourier Series Method." Abstract and Applied Analysis 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/958931.

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The corrected Fourier series (CFS) is proposed for solving partial differential equations (PDEs) with fractional time derivative on a finite domain. In the previous work, we have been solving partial differential equations by using corrected Fourier series. The fractional derivatives are described in Riemann sense. Some numerical examples are presented to show the solutions.

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22

Hu, Langhua, Duan Chen, and Guo-Wei Wei. "High-order fractional partial differential equation transform for molecular surface construction." Computational and Mathematical Biophysics 1 (December 20, 2012): 1–25. http://dx.doi.org/10.2478/mlbmb-2012-0001.

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AbstractFractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.

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23

Dzagnidze, O. "A Radial Derivative with Boundary Values of the Spherical Poisson Integral." gmj 6, no. 1 (February 1999): 19–32. http://dx.doi.org/10.1515/gmj.1999.19.

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Abstract A formula of a radial derivative is obtained with the aid of derivatives with respect to θ and to φ of the functions closely connected with the spherical Poisson integral 𝑢f (r, θ, φ) and the boundary values are determined for . The boundary values are also found for partial derivatives with respect to the Cartesian coordinates , and .

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24

Almgren, Robert. "Financial Derivatives and Partial Differential Equations." American Mathematical Monthly 109, no. 1 (January 2002): 1. http://dx.doi.org/10.2307/2695763.

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25

Kamont, Z. "Functional differential inequalities with partial derivatives." Bulletin of the Belgian Mathematical Society - Simon Stevin 21, no. 1 (February 2014): 127–46. http://dx.doi.org/10.36045/bbms/1394544299.

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26

Almgren, Robert. "Financial Derivatives and Partial Differential Equations." American Mathematical Monthly 109, no. 1 (January 2002): 1–12. http://dx.doi.org/10.1080/00029890.2002.11919834.

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27

Aksoy, Asuman, and Mario Martelli. "Mixed Partial Derivatives and Fubini's Theorem." College Mathematics Journal 33, no. 2 (March 2002): 126–30. http://dx.doi.org/10.1080/07468342.2002.11921930.

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28

Wang, Guo-Jin, Thomas W. Sederberg, and Takafumi Saito. "Partial derivatives of rational Bézier surfaces." Computer Aided Geometric Design 14, no. 4 (May 1997): 377–81. http://dx.doi.org/10.1016/s0167-8396(96)00033-7.

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29

Aksoy, Asuman, and Mario Martelli. "Mixed Partial Derivatives and Fubini's Theorem." College Mathematics Journal 33, no. 2 (March 2002): 126. http://dx.doi.org/10.2307/1558995.

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30

Golubitsky, Oleg. "Construction of rankings on partial derivatives." ACM Communications in Computer Algebra 40, no. 2 (June 2006): 38–42. http://dx.doi.org/10.1145/1182553.1182557.

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31

Lyons, Russell, and Kevin Zumbrun. "Homogeneous partial derivatives of radial functions." Proceedings of the American Mathematical Society 121, no. 1 (January 1, 1994): 315. http://dx.doi.org/10.1090/s0002-9939-1994-1227524-9.

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32

Brešar, Boštjan, Paul Dorbec, Sandi Klavžar, and Michel Mollard. "Hamming polynomials and their partial derivatives." European Journal of Combinatorics 28, no. 4 (May 2007): 1156–62. http://dx.doi.org/10.1016/j.ejc.2006.03.001.

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33

Favorite, Jeffrey A. "Adjoint-based constant-mass partial derivatives." Annals of Nuclear Energy 110 (December 2017): 1052–59. http://dx.doi.org/10.1016/j.anucene.2017.08.015.

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34

Yu, Chii-Huei. "Partial Derivatives of Three Variables Functions." Universal Journal of Computational Mathematics 2, no. 2 (February 2014): 23–27. http://dx.doi.org/10.13189/ujcmj.2014.020201.

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35

Dračínský, Martin, Simona Hybelbauerová, Jan Sejbal, and Miloš Buděšínský. "Preparation and Conformational Study of B-Ring Substituted Lupane Derivatives." Collection of Czechoslovak Chemical Communications 71, no. 8 (2006): 1131–60. http://dx.doi.org/10.1135/cccc20061131.

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New lupane-type triterpenoids with 5(6) double bond were prepared using the method of partial demethylation on carbon C-4. Hydroboration of the double bond led to 6α-hydroxy derivative. By the oxidation and following reduction of 6α-hydroxy derivative the 6-oxo and 6β-hydroxy derivatives were prepared. A new method for selective oxidation of secondary hydroxy group in the presence of primary hydroxy group was performed. The conformation of ring A of new lupane-type 3-oxo derivatives with a substituent on ring B was elucidated on the bases of 1H and 13C NMR spectra and molecular modelling.

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36

Babajide, Ayeni O., and Idowu K. Oluwatobi. "ON THE ELZAKI SUBSTITUTION AND HOMOTOPY PERTUBATION METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATION INVOLVING MIXED PARTIAL DERIVATIVES." FUDMA JOURNAL OF SCIENCES 5, no. 3 (November 2, 2021): 159–68. http://dx.doi.org/10.33003/fjs-2021-0503-668.

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This paper investigated new methods of solving partial differential equations involving mixed partial derivatives that were initially solved by Sujit and Karande in their usual notation by making use of Laplace substitution method. The methods investigated in this paper are Elzaki Substitution and Homotopy perturbation Methods of solving partial differential equations with mixed partial derivatives. Finally, the results obtained showed that Elzaki Substitution method and Homotopy Perturbation method are accurate and efficient method to solve partial differential equations involving mixed partial derivatives

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37

Popov, Igor. "Scalar and vector division and derivatives vectors." Applied Mathematics and Control Sciences, no. 2 (June 29, 2018): 43–55. http://dx.doi.org/10.15593/2499-9873/2018.2.03.

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The work is devoted to the operations of differentiation in the space of vector fields and smooth functions. In mechanics, it is widely used derivative of a scalar function of the vector. To some extent, like it is determined by the derivative of the vector to another vector. However, formally interpreting the derivative as division differentials are entered in consideration of scalar and vector derived vector on another vector, which may have application to the solution of problems of mechanics. The definition of a derivative of a scalar vector field on another vector field. We prove a theorem on the representation of the scalar derivative in the form of a combination of partial derivatives. As a typical particular case is considered a scalar derivative in the radius vector, generating formalism linking it with the operator nabla. It is noted that in solving some problems in the mechanics to simplify the calculation coordinate system is chosen so that at least some vectors direction coincides with one of the coordinate axes. If it concerns the vector for derivation to be performed, in such cases, the formula for the three-dimensional case can not be used because some of this vector differentials are equal to zero. This circumstance makes it necessary to prove two theorems for the two-dimensional and one-dimensional case. The definition of a vector derivative of a vector field on another vector field. We prove a theorem on the representation of the derivative vector as a combination of partial derivatives. As a typical particular case considered vector derivative of the radius vector, generating formalism linking it with the operator nabla. We prove similar theorems for two-dimensional and one-dimensional case. We give examples of applications of these results to problems of mechanics.

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38

Fornberg, Bengt, and David M. Sloan. "A review of pseudospectral methods for solving partial differential equations." Acta Numerica 3 (January 1994): 203–67. http://dx.doi.org/10.1017/s0962492900002440.

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Finite Difference (FD) methods approximate derivatives of a function by local arguments (such as du(x) / dx ≈ (u(x + h) − u(x − h))/2h, where h is a small grid spacing) – these methods are typically designed to be exact for polynomials of low orders. This approach is very reasonable: since the derivative is a local property of a function, it makes little sense (and is costly) to invoke many function values far away from the point of interest.

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39

Sanwale, Jitu, and Dhan Jeet Singh. "Aerodynamic Parameters Estimation Using Radial Basis Function Neural Partial Differentiation Method." Defence Science Journal 68, no. 3 (April 16, 2018): 241. http://dx.doi.org/10.14429/dsj.68.11843.

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Aerodynamic parameter estimation involves modelling of force and moment coefficients and computation of stability and control derivatives from recorded flight data. This problem is extensively studied in the past using classical approaches such as output error, filter error and equation error methods. An alternative approach to these model based methods is the machine learning such as artificial neural network. In this paper, radial basis function neural network (RBF NN) is used to model the lateral-directional force and moment coefficients. The RBF NN is trained using k-means clustering algorithm for finding the centers of radial basis function and extended Kalman filter for obtaining the weights in the output layer. Then, a new method is proposed to obtain the stability and control derivatives. The first order partial differentiation is performed analytically on the radial basis function neural network approximated output. The stability and control derivatives are computed at each training data point, thus reducing the post training time and computational efforts compared to hitherto delta method and its variants. The efficacy of the identified model and proposed neural derivative method is demonstrated using real time flight data of ATTAS aircraft. The results from the proposed approach compare well with those from the other.

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40

Elbeleze, Asma Ali, Adem Kılıçman, and Bachok M. Taib. "Fractional Variational Iteration Method and Its Application to Fractional Partial Differential Equation." Mathematical Problems in Engineering 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/543848.

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We use the fractional variational iteration method (FVIM) with modified Riemann-Liouville derivative to solve some equations in fluid mechanics and in financial models. The fractional derivatives are described in Riemann-Liouville sense. To show the efficiency of the considered method, some examples that include the fractional Klein-Gordon equation, fractional Burgers equation, and fractional Black-Scholes equation are investigated.

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41

Van Brunt, B., D. Pidgeon, M. Vlieg-Hulstman, and W. D. Halford. "Conservation laws for second-order parabolic partial differential equations." ANZIAM Journal 45, no. 3 (January 2004): 333–48. http://dx.doi.org/10.1017/s1446181100013407.

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AbstractConservation laws for partial differential equations can be characterised by an operator, the characteristic and a condition involving the adjoint of the Fréchet derivatives of this operator and the operator defining the partial differential equation. This approach was developed by Anco and Bluman and we exploit it to derive conditions for second-order parabolic partial differential equations to admit conservation laws. We show that such partial differential equations admit conservation laws only if the time derivative appears in one of two ways. The adjoint condition, however, is a biconditional, and we use this to prove necessary and sufficient conditions for a certain class of partial differential equations to admit a conservation law.

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42

Gurefe, Yusuf. "The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative." Revista Mexicana de Física 66, no. 6 Nov-Dec (November 5, 2020): 771. http://dx.doi.org/10.31349/revmexfis.66.771.

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In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

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43

Topuria, S. "Generalized Derivatives of an Arbitrary Order and the Boundary Properties of Differentiated Poisson Integrals for the Half-Space." Georgian Mathematical Journal 7, no. 2 (June 2000): 387–400. http://dx.doi.org/10.1515/gmj.2000.387.

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Abstract The notions of a generalized differential and a generalized spherical derivative of an arbitrary order are introduced for a function of several variables and Fatou type theorems are proved on the boundary properties of partial derivatives of an arbitrary order of the Poisson integral for the half-space, when the integral density has a generalized differential or a generalized spherical derivative.

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44

Turov, M. M., V. E. Fedorov, and B. T. Kien. "Linear Inverse Problems for Multi-term Equations with Riemann — Liouville Derivatives." Bulletin of Irkutsk State University. Series Mathematics 38 (2021): 36–53. http://dx.doi.org/10.26516/1997-7670.2021.38.36.

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The issues of well-posedness of linear inverse coefficient problems for multi-term equations in Banach spaces with fractional Riemann – Liouville derivatives and with bounded operators at them are considered. Well-posedness criteria are obtained both for the equation resolved with respect to the highest fractional derivative, and in the case of a degenerate operator at the highest derivative in the equation. Two essentially different cases are investigated in the degenerate problem: when the fractional part of the order of the second-oldest derivative is equal to or different from the fractional part of the order of the highest fractional derivative. Abstract results are applied in the study of inverse problems for partial differential equations with polynomials from a self-adjoint elliptic differential operator with respect to spatial variables and with Riemann – Liouville derivatives in time.

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45

Dalík, Josef. "Approximations of the partial derivatives by averaging." Central European Journal of Mathematics 10, no. 1 (October 24, 2011): 44–54. http://dx.doi.org/10.2478/s11533-011-0107-y.

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46

Patthy, Á., L. F. Szabó, B. Podányi, and P. Tétényi. "DIASTEREOSELECTIVE PARTIAL SYNTHESIS OF INDOLIC SECOLOGANIN DERIVATIVES." Acta Horticulturae, no. 306 (May 1992): 129–32. http://dx.doi.org/10.17660/actahortic.1992.306.12.

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47

Veretennikov, A. Yu, and E. V. Veretennikova. "On partial derivatives of multivariate Bernstein polynomials." Siberian Advances in Mathematics 26, no. 4 (October 2016): 294–305. http://dx.doi.org/10.3103/s1055134416040039.

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48

Thanh Duc, Dinh, Nguyen Du Vi Nhan, and Nguyen Tong Xuan. "Inequalities for Partial Derivatives and their Applications." Canadian Mathematical Bulletin 58, no. 3 (September 1, 2015): 486–96. http://dx.doi.org/10.4153/cmb-2015-020-6.

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AbstractWe present various weighted integral inequalities for partial derivatives acting on products and compositions of functions that are applied in order to establish some new Opial-type inequalities involving functions of several independent variables. We also demonstrate the usefulness of our results in the field of partial differential equations.

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49

Chen, Xi. "Partial Derivatives in Arithmetic Complexity and Beyond." Foundations and Trends® in Theoretical Computer Science 6, no. 1-2 (2010): 1–138. http://dx.doi.org/10.1561/0400000043.

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Estevez, Gentil A., Kai Yang, and Basah B. Dasgupta. "Thermodynamic partial derivatives and experimentally measurable quantities." Journal of Chemical Education 66, no. 11 (November 1989): 890. http://dx.doi.org/10.1021/ed066p890.

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