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Journal articles on the topic 'Hyperbolic derivative'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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1

JIANG, MIAOHUA. "Differentiating potential functions of SRB measures on hyperbolic attractors." Ergodic Theory and Dynamical Systems 32, no. 4 (June 10, 2011): 1350–69. http://dx.doi.org/10.1017/s0143385711000241.

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AbstractThe derivation of Ruelle’s derivative formula of the SRB measure depends largely on the calculation of the derivative of the unstable Jacobian. Although Ruelle’s derivative formula is correct, the proofs in the original paper and its corrigendum are not complete. In this paper, we re-visit the differentiation process of the unstable Jacobian and provide a complete derivation of its derivative formula. Our approach is to extend the volume form provided by the SRB measure on local unstable manifolds to a system of Hölder continuous local Riemannian metrics on the manifold so that under this system of local metrics, the unstable Jacobian becomes differentiable with respect to the base point and its derivative with respect to the map can be obtained by the chain rule.

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2

Eriksson, Sirkka-Liisa, and Heikki Orelma. "Quaternionic k-Hyperbolic Derivative." Complex Analysis and Operator Theory 11, no. 5 (December 30, 2016): 1193–204. http://dx.doi.org/10.1007/s11785-016-0630-8.

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3

Ford, G. W., and R. F. O'Connell. "Derivative of the hyperbolic cotangent." Nature 380, no. 6570 (March 1996): 113–14. http://dx.doi.org/10.1038/380113b0.

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4

HOMBURG, ALE JAN. "Piecewise smooth interval maps with non-vanishing derivative." Ergodic Theory and Dynamical Systems 20, no. 3 (June 2000): 749–73. http://dx.doi.org/10.1017/s0143385700000407.

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We consider the dynamics of piecewise smooth interval maps $f$ with a nowhere vanishing derivative. We show that if $f$ is not infinitely renormalizable, then all its periodic orbits of sufficiently high period are hyperbolic repelling. If, in addition all periodic orbits of $f$ are hyperbolic, then $f$ has at most finitely many periodic attractors and there is a hyperbolic expansion outside the basins of these periodic attractors. In particular, if $f$ is not infinitely renormalizable and all its periodic orbits are hyperbolic repelling, then some iterate of $f$ is expanding. In this case, $f$ admits an absolutely continuous invariant probability measure.

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5

Kwon, E. G. "Fractional integration and the hyperbolic derivative." Bulletin of the Australian Mathematical Society 38, no. 3 (December 1988): 357–64. http://dx.doi.org/10.1017/s0004972700027714.

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We improve S. Yamashita's hyperbolic version of the well-known Hardy-Littlewood theorem. Let f be holomorphic and bounded by one in the unit disc D. If (f#)p has a harmonic mojorant in D for some p, p > 0, then so does σ(f)q for all q, 0 < q < ∞. Here

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6

Renegar, James. "Hyperbolic Programs, and Their Derivative Relaxations." Foundations of Computational Mathematics 6, no. 1 (January 5, 2006): 59–79. http://dx.doi.org/10.1007/s10208-004-0136-z.

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7

Birman, Graciela S., and Abraham A. Ungar. "The Hyperbolic Derivative in the Poincaré Ball Model of Hyperbolic Geometry." Journal of Mathematical Analysis and Applications 254, no. 1 (February 2001): 321–33. http://dx.doi.org/10.1006/jmaa.2000.7280.

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8

Wu, Nan. "On characteristic of bounded analytic functions involving hyperbolic derivative." Mathematica Slovaca 68, no. 4 (August 28, 2018): 811–22. http://dx.doi.org/10.1515/ms-2017-0147.

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Abstract In this article, we give the Nevanlinna type hyperbolic characteristics in simply connected domains and angular domains and the Tsuji type hyperbolic characteristics for bounded analytic functions for the first time. The first fundamental theorems are also established concerning hyperbolic derivative for bounded analytic functions in simply connected domains and angular domains. This is a continuous work of Makhmutov [3].

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Aydın, Ömer Lütfü, Ozcan Bektas, Aydın Büyüksaraç, and Hüseyin Yılmaz. "3D Modeling and Tectonic Interpretation of the Erzincan Basin (Turkey) using Potential Field Data." Earth Sciences Research Journal 23, no. 1 (January 1, 2019): 57–66. http://dx.doi.org/10.15446/esrj.v23n1.71090.

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Erzincan Basin was investigated using gravity data within the scope of this study. It is also aimed to reveal the discontinuities in the work area as well as the buried discontinuities. Boundary determination filters and analysis of the structure of the data and its connection are revealed and clear information is obtained. Gravity anomalies were applied with an upward continuation method for 0.25, 0.50, 0.75 and 1 km levels. Total Horizontal Derivative (THD) filter, Analytical Signal (AS) filter, Tilt Angle Derivative (Tilt) filter, Total Horizontal Derivative (THDR) filter, Theta Angle Derivative (Cos ɵ) filter, Hyperbolic Tilt Angle Derivative (HTAD) were applied to upward continued data. The discontinuities in the region and the boundaries of the geological structure were revealed. Tilt and Theta Angle derivatives yield the best results from the applied derivative based filters. The obtained data were compared with the existing surface geology and the compatibility between the formations was checked. New discontinuities were found in addition to the discontinuities determined from surface observations in the light of the obtained results. Erzincan Basin was modeled in three dimensions using gravity data of the study area. As a result of modeling, Erzincan Basin has been determined to have an average thickness of 7 km.Total Horizontal Derivative (THD) filter, Analytical Signal (AS) filter, Tilt Angle Derivative (TAd) filter, Total Horizontal Derivative (THDR) filter, Teta Angle Derivative (Cos ɵ) filter, Hyperbolic Tilt Angle Derivative (HTAD) were applied to upward continued data. The discontinuities in the region and the boundaries of the geological structure were revealed. Tilt and Theta angle derivatives yield the best results from the applied derivative based filters. The obtained data were compared with the existing surface geology and the compatibility between the formations was checked. New discontinuities were found in addition to the discontinuities determined from surface observations in the light of the obtained results. Erzincan basin is modeled in three dimensions using gravity data of the study area. As a result of modeling, Erzincan Basin has been determined to have an average thickness of 7 km.

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10

Ding, Hengfei, and Changpin Li. "Numerical Algorithms for the Fractional Diffusion-Wave Equation with Reaction Term." Abstract and Applied Analysis 2013 (2013): 1–15. http://dx.doi.org/10.1155/2013/493406.

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Two numerical algorithms are derived to compute the fractional diffusion-wave equation with a reaction term. Firstly, using the relations between Caputo and Riemann-Liouville derivatives, we get two equivalent forms of the original equation, where we approximate Riemann-Liouville derivative by a second-order difference scheme. Secondly, for second-order derivative in space dimension, we construct a fourth-order difference scheme in terms of the hyperbolic-trigonometric spline function. The stability analysis of the derived numerical methods is given by means of the fractional Fourier method. Finally, an illustrative example is presented to show that the numerical results are in good agreement with the theoretical analysis.

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11

TÉLLEZ-SÁNCHEZ, GAMALIEL YAFTE, and JUAN BORY-REYES. "GENERALIZED ITERATED FUNCTION SYSTEMS ON HYPERBOLIC NUMBER PLANE." Fractals 27, no. 04 (June 2019): 1950045. http://dx.doi.org/10.1142/s0218348x19500452.

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Iterated function systems provide the most fundamental framework to create many fascinating fractal sets. They have been extensively studied when the functions are affine transformations of Euclidean spaces. This paper investigates the iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that the basics results of the classical Hutchinson–Barnsley theory can be carried over to construct fractal sets on hyperbolic number plane as its unique fixed point. We also discuss about the notion of hyperbolic derivative of an hyperbolic-valued function and then we use this notion to get some generalization of cookie-cutter Cantor sets in the real line to the hyperbolic number plane.

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12

Ghanem, Sari. "The global non-blow-up of the Yang–Mills curvature on curved space-times." Journal of Hyperbolic Differential Equations 13, no. 03 (September 2016): 603–31. http://dx.doi.org/10.1142/s0219891616500156.

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We give a proof of the non-blow-up of the Yang–Mills curvature on arbitrary curved space-times using the Klainerman–Rodnianski parametrix combined with suitable Grönwall type inequalities. While the Chruściel–Shatah argument requires a control on two derivatives of the Yang–Mills curvature, we can get away by controlling only one derivative instead, and we propose a new gauge-independent proof on sufficiently smooth, globally hyperbolic, curved 4-dimensional Lorentzian manifolds.

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13

Mercer, Peter R. "Schwarz-Pick-type estimates for the hyperbolic derivative." Journal of Inequalities and Applications 2006 (2006): 1–6. http://dx.doi.org/10.1155/jia/2006/96368.

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14

Cooke, Charlie H., and Tze-Jang Chen. "A weak-derivative form for linear hyperbolic systems." Numerical Methods for Partial Differential Equations 20, no. 6 (2004): 933–47. http://dx.doi.org/10.1002/num.20017.

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15

Ford, G. W., and R. F. O'Connell. "Note on the derivative of the hyperbolic cotangent." Journal of Physics A: Mathematical and General 35, no. 18 (April 26, 2002): 4183–86. http://dx.doi.org/10.1088/0305-4470/35/18/313.

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16

Fajardo, José, and Aquiles Farias. "Derivative pricing using multivariate affine generalized hyperbolic distributions." Journal of Banking & Finance 34, no. 7 (July 2010): 1607–17. http://dx.doi.org/10.1016/j.jbankfin.2010.03.007.

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17

Fajardo, José, and Aquiles Farias. "Generalized Hyperbolic Distributions and Brazilian Data." Brazilian Review of Econometrics 24, no. 2 (November 2, 2004): 249. http://dx.doi.org/10.12660/bre.v24n22004.2712.

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The aim of this paper is to discuss the use of the Generalized Hyperbolic Distributions to fit Brazilian assets returns. Selected subclasses are compared regarding goodness of fit statistics and distances. Empirical results show that these distributions fit data well. Then we show how to use these distributions in value at risk estimation and derivative price computation.

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18

Kehaili, Abdelkader, Ali Hakem, and Abdelkader Benali. "Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients." Global Journal of Pure and Applied Sciences 26, no. 1 (June 1, 2020): 35–55. http://dx.doi.org/10.4314/gjpas.v26i1.6.

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In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation.

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19

Bose, Christopher, and Rua Murray. "First hyperbolic times for intermittent maps with unbounded derivative." Dynamical Systems 29, no. 3 (April 10, 2014): 352–68. http://dx.doi.org/10.1080/14689367.2014.902038.

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20

McCartin, Brian J., and Matthew F. Causley. "Angled derivative approximation of the hyperbolic heat conduction equations." Applied Mathematics and Computation 182, no. 2 (November 2006): 1581–607. http://dx.doi.org/10.1016/j.amc.2006.05.045.

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21

Ali, Khalid K., and RahmatullahIbrahim Nuruddeen. "Analytical treatment for the conformable space-time fractional Benney-Luke equation via two reliable methods." International Journal of Physical Research 5, no. 2 (November 8, 2017): 109. http://dx.doi.org/10.14419/ijpr.v5i2.8403.

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In this study, with help of the Mathematica software, we employ the Kudryashov method and the modified extended tanh expansion method with the Riccati differential equation to analytically treat the Benney-Luke equation. The Benney-Luke equation considered in this study features fractional derivatives in both the spatial and the temporal variables of the newly introduced conformable fractional derivative. We extensively examine the equation via the two methods, and we construct various structures such as the exponential functions, trigonometric functions and hyperbolic functions. Finally, we depict the graphs of all solutions.

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22

Mishra, Suchana, Rabindra Kishore Mishra, and Srikanta Patnaik. "Discrete (G'/G )-expansion: a Method Used to Get Exact Solution of Fdde (Fractional Differential-difference Equation) Linked With Nltl (Non-linear Transmission Line)." International Journal of Circuits, Systems and Signal Processing 15 (May 18, 2021): 453–60. http://dx.doi.org/10.46300/9106.2021.15.49.

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Here, we have used the discrete (G'/G)-expansion procedure with the derivative operator MR-L (modified Riemann-Liouville) and FCT (fractional complex transform) to find the exact/analytical solution of an electrical transmission line which is non-linear. Results include solutions for integer and fractional DDE. We consider two special cases of solutions: hyperbolic and trigonometric. Hyperbolic solutions indicate propagation of singular wave on the transmission line. Trigonometric solutions show propagation of complex wave.

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23

Yuan, Wenfa, Dongli Chen, and Pingan Wang. "A strengthened Schwarz-pick inequality for derivatives of the hyperbolic metric." Tamkang Journal of Mathematics 37, no. 2 (June 30, 2006): 131–34. http://dx.doi.org/10.5556/j.tkjm.37.2006.157.

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This paper is to investigate the Schwarz-Pick inequality for the hyperbolic derivative. Our result is not only a contraction but also a contraction minus a positive constant and this improves Beardon's theorem greatly.

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24

Dong, Qixiang, Guangxian Wu, and Lanping Zhu. "Existence and Continuous Dependence for Fractional Partial Hyperbolic Differential Equations." Journal of Function Spaces 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/875170.

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This paper is concerned with a class of fractional hyperbolic partial differential equations with the Caputo derivative. Existence and continuous dependence results of solutions are obtained under the hypothesis of the Lipschitz condition without any restriction on the Lipschitz constant. Examples are discussed to illustrate the results.

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25

Biler, Piotr. "A singular perturbation problem for nonlinear damped hyperbolic equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 1-2 (1989): 21–31. http://dx.doi.org/10.1017/s0308210500024987.

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SynopsisWe consider damped nonlinear hyperbolic equations utt + Aut + αAu + βA2u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter β tends to zero.

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26

Mayer, Volker, and Mariusz Urbański. "Ergodic properties of semi-hyperbolic functions with polynomial Schwarzian derivative." Proceedings of the Edinburgh Mathematical Society 53, no. 2 (April 30, 2010): 471–502. http://dx.doi.org/10.1017/s0013091507001332.

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AbstractThe ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative are investigated under the condition that the function is semi-hyperbolic, i.e. the asymptotic values of the Fatou set are in attracting components and the asymptotic values in the Julia set are boundedly non-recurrent. We first show the existence, uniqueness, conservativity and ergodicity of a conformal measure m with minimal exponent h; furthermore, we show weak metrical exactness of this measure. Then we prove the existence of a σ-finite invariant measure μ absolutely continuous with respect to m. Our main result states that μ is finite if and only if the order ρ of the function f satisfies the condition h > 3ρ/(ρ+1). When finite, this measure is shown to be metrically exact. We also establish a version of Bowen's Formula, showing that the exponent h equals the Hausdorff dimension of the Julia set of f.

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27

Vatsala, Aghalaya S., and Yunxiang Bai. "One Dimensional Sub-hyperbolic Equation via Sequential Caputo Fractional Derivative." Journal of Combinatorics, Information & System Sciences 44, no. 1-4 (December 30, 2020): 91–102. http://dx.doi.org/10.32381/jciss.2019.44.1-4.6.

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28

Cassisa, C., P. E. Ricci, and I. Tavkhelidze. "Operational Identities for Circular and Hyperbolic Functions and Their Generalizations." Georgian Mathematical Journal 10, no. 1 (March 2003): 45–56. http://dx.doi.org/10.1515/gmj.2003.45.

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Abstract Starting from the exponential, some classes of analytic functions of the derivative operator are studied, including pseudo-hyperbolic and pseudo-circular functions. Some formulas related to operational calculus are deduced, and the important role played in such a context by Hermite–Kampé de Fériet polynomials is underlined.

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29

AGOSHKOV, V. I. "SOME IMBEDDING THEOREMS AND THE SOLVABILITY OF THE SYSTEM OF HYPERBOLIC-PARABOLIC EQUATIONS." Mathematical Models and Methods in Applied Sciences 03, no. 01 (February 1993): 95–108. http://dx.doi.org/10.1142/s0218202593000060.

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A number of physical problems is described by a system of first order hyperbolic equations in a subregion of the physical domain, and of parabolic equations in the complementary domain. To state these problems correctly an investigation of the existence of the trace for a function which possesses a weak derivative in some direction is needed. This problem is solved in the paper. Also we formulate the problem for hyperbolic-parabolic equations (including the description of the transmission condition on the internal boundary), then we prove its solvability.

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30

Helal, Mohamed. "Fractional order differential inclusions on an unbounded domain with infinite delay." MATHEMATICA 62 (85), no. 2 (November 15, 2020): 167–78. http://dx.doi.org/10.24193/mathcluj.2020.2.06.

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We provide sufficient conditions for the existence of solutions to initial value problems, for partial hyperbolic differential inclusions of fractional order involving Caputo fractional derivative with infinite delay by applying the nonlinear alternative of Frigon type for multivalued admissible contraction in Frechet spaces.

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31

Touchent, Kamal Ait, Zakia Hammouch, Toufik Mekkaoui, and Canan Unlu. "A Boiti-Leon Pimpinelli equations with time-conformable derivative." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 9, no. 3 (September 13, 2019): 95–101. http://dx.doi.org/10.11121/ijocta.01.2019.00766.

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In this paper, we derive some new soliton solutions to $(2+1)$-Boiti-Leon Pempinelli equations with conformable derivative by using an expansion technique based on the Sinh-Gordon equation. The obtained solutions have different expression such as trigonometric, complex and hyperbolic functions. This powerful and simple technique can be used to investigate solutions of other nonlinear partial differential equations.

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32

KOLB, MARTIN. "ON THE STRONG UNIQUENESS OF SOME FINITE DIMENSIONAL DIRICHLET OPERATORS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 11, no. 02 (June 2008): 279–93. http://dx.doi.org/10.1142/s0219025708003117.

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We prove essential self-adjointness of a class of Dirichlet operators in ℝn using the hyperbolic equation approach. This method allows one to prove essential self-adjointness under minimal conditions on the logarithmic derivative of the density and a condition of Muckenhoupt type on the density itself.

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33

Bin-Mohsin, Bandar. "Some new solutions of the conformable extended Zakharov-Kuznetsov equation using Atangana-Baleanu conformable derivative." Thermal Science 23, Suppl. 6 (2019): 2127–37. http://dx.doi.org/10.2298/tsci190303402b.

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The generalized Riccati equation mapping method, coupled with Atangana?s conformable derivative is implemented to solve non-linear extended Zakharov-Kuznetsov equation which results in producing hyperbolic, trigonometric and the rational solutions. The obtained results are new and are of great importance in engineering and applied sciences.

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34

OHWA, HIROKI. "THE SHOCK CURVE APPROACH TO THE RIEMANN PROBLEM FOR 2 × 2 HYPERBOLIC SYSTEMS OF CONSERVATION LAWS." Journal of Hyperbolic Differential Equations 07, no. 02 (June 2010): 339–64. http://dx.doi.org/10.1142/s0219891610002128.

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We consider the Riemann problem for 2 × 2 hyperbolic systems of conservation laws in one space variable. Our main assumptions are that the product of non-diagonal elements within the Fréchet derivative (Jacobian) of the flux is positive, and that the system is genuinely nonlinear. The first assumption implies that the system is strictly hyperbolic, but we do not require a convexity-like condition such as the Smoller–Johnson condition. By using the shock curve approach, we show that those two assumptions are sufficient to establish the uniqueness of self-similar solutions satisfying the Lax entropy conditions at discontinuities.

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35

Kong, De-Xing, Qi Liu, and Chang-Ming Song. "Classical solutions to a dissipative hyperbolic geometry flow in two space variables." Journal of Hyperbolic Differential Equations 16, no. 02 (June 2019): 223–43. http://dx.doi.org/10.1142/s0219891619500085.

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We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.

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36

Bock, Igor. "Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle." Mathematics and Mechanics of Solids 23, no. 3 (July 23, 2017): 411–19. http://dx.doi.org/10.1177/1081286517719937.

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We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

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Orlov, Y. V. "Sliding Mode Observer-Based Synthesis of State Derivative-Free Model Reference Adaptive Control of Distributed Parameter Systems." Journal of Dynamic Systems, Measurement, and Control 122, no. 4 (January 20, 2000): 725–31. http://dx.doi.org/10.1115/1.1320447.

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This paper presents control laws for distributed parameter systems of parabolic and hyperbolic types which, on the one hand ensure robustness with respect to small dynamic uncertainties and disturbances, and on the other hand, permit on-line plant parameter estimation. The novelty of the algorithms proposed is (a) in the construction of a sliding mode-based state derivative observer and (b) in the inclusion of this observer into a model reference adaptive controller which thereby regularizes the ill-posed identification problem itself. Apart from this, the controllers constructed do not suffer from on-line computation of spatial derivatives of the measurement data, and hence they are of reduced sensitivity with respect to the measurement noise. [S0022-0434(00)02104-3]

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38

Carles, Rémi, and Clément Gallo. "WKB analysis of generalized derivative nonlinear Schrödinger equations without hyperbolicity." Mathematical Models and Methods in Applied Sciences 27, no. 09 (May 23, 2017): 1727–42. http://dx.doi.org/10.1142/s0218202517500300.

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We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity.

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39

KUMAR, P., and K. N. RAI. "FRACTIONAL MODELING OF HYPERBOLIC BIOHEAT TRANSFER EQUATION DURING THERMAL THERAPY." Journal of Mechanics in Medicine and Biology 17, no. 03 (December 29, 2016): 1750058. http://dx.doi.org/10.1142/s0219519417500580.

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In this paper, we have developed a fractional hyperbolic bioheat transfer (FHBHT) model by applying fractional Taylor series formula to the single-phase-lag constitutive relation. A new hybrid numerical scheme that combines the multi-resolution and multi-scale computational property of Legendre wavelets based on fractional operational matrix has been used to find the numerical solution of the present problem. This study demonstrates that FHBHT model can provide a unified approach for analyzing heat transfer within living biological tissues, as standard hyperbolic bioheat transfer (SHBHT) and Pennes models are particular cases of FHBHT model. The effect of phase lag time and order of fractional derivative on temperature distribution within living biological tissues for both SHBHT and FHBHT models have been studied and shown graphically. It has been observed that thermal signal propagates more easily with larger values of order of fractional derivative within living biological tissues. The time interval for achieving temperature range of thermal treatment for different models have been studied and compared. It is least for Pennes model, highest for FHBHT model and in between them for SHBHT model. The whole analysis is presented in dimensionless form.

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40

Shakeel, Muhammad, Qazi Mahmood Ul-Hassan, and Jamshad Ahmad. "Applications of the Novel (G′/G)-Expansion Method for a Time Fractional Simplified Modified Camassa-Holm (MCH) Equation." Abstract and Applied Analysis 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/601961.

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We use the fractional derivatives in modified Riemann-Liouville derivative sense to construct exact solutions of time fractional simplified modified Camassa-Holm (MCH) equation. A generalized fractional complex transform is properly used to convert this equation to ordinary differential equation and, as a result, many exact analytical solutions are obtained with more free parameters. When these free parameters are taken as particular values, the traveling wave solutions are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. Moreover, the numerical presentations of some of the solutions have been demonstrated with the aid of commercial software Maple. The recital of the method is trustworthy and useful and gives more new general exact solutions.

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41

Marano, Salvatore. "Classical solutions of hyperbolic partial differential equations with implicit mixed derivative." Annales Polonici Mathematici 56, no. 2 (1992): 163–78. http://dx.doi.org/10.4064/ap-56-2-163-178.

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42

Asanova, A. T. "Multipoint Problem for a System of Hyperbolic Equations with Mixed Derivative." Journal of Mathematical Sciences 212, no. 3 (December 19, 2015): 213–33. http://dx.doi.org/10.1007/s10958-015-2660-6.

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43

Az-Zo’bi, Emad A., Wael A. Alzoubi, Lanre Akinyemi, Mehmet Şenol, and Basem S. Masaedeh. "A variety of wave amplitudes for the conformable fractional (2 + 1)-dimensional Ito equation." Modern Physics Letters B 35, no. 15 (March 12, 2021): 2150254. http://dx.doi.org/10.1142/s0217984921502547.

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The conformable derivative and adequate fractional complex transform are implemented to discuss the fractional higher-dimensional Ito equation analytically. The Jacobi elliptic function method and Riccati equation mapping method are successfully used for this purpose. New exact solutions in terms of linear, rational, periodic and hyperbolic functions for the wave amplitude are derived. The obtained solutions are entirely new and can be considered as a generalization of the existing results in the ordinary derivative case. Numerical simulations of some obtained solutions with special choices of free constants and various fractional orders are displayed.

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44

Sadek, I. S., J. M. Sloss, S. Adali, and J. C. Bruch. "Maximum Principle for the Optimal Control of a Hyperbolic Equation in Two Space Dimensions." Journal of Vibration and Control 2, no. 1 (January 1996): 3–15. http://dx.doi.org/10.1177/107754639600200101.

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A maximum principle is developed for a class of problems involving the optimal control of a damped parameter system governed by a not-necessarily separable linear hyperbolic equation in two space dimensions. An index of performance is formulated, which consists of functions of the state variable, its first and second order space derivatives and first order time derivative, and a penalty function involving the open-loop control force. The solution of the optimal control problem is shown to be unique using convexity arguments. The maximum principle given involves a Hamiltonian, which contains an adjoint variable as well as an admissible control function. The state and adjoint variables are linked by terminal conditions leading to a boundary/initial/terminal value problem. The maximum principle can be used to compute the optimal control function and is particularly suitable for problems involving the active control of two-dimensional structural elements for vibration suppression.

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Bradley, David. "Ramanujan's formula for the logarithmic derivative of the gamma function." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 3 (October 1996): 391–401. http://dx.doi.org/10.1017/s030500410007496x.

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AbstractWe prove a remarkable formula of Ramanujan for the logarithmic derivative of the gamma function, which converges more rapidly than classical expansions, and which is stated without proof in the notebooks [5]. The formula has a number of very interesting consequences which we derive, including an elegant hyperbolic summation, Ramanujan's formula for the Riemann zeta function evaluated at the odd positive integers, and new formulae for Euler's constant γ.

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46

MORANDO, ALESSANDRO, and PAOLO SECCHI. "REGULARITY OF WEAKLY WELL POSED HYPERBOLIC MIXED PROBLEMS WITH CHARACTERISTIC BOUNDARY." Journal of Hyperbolic Differential Equations 08, no. 01 (March 2011): 37–99. http://dx.doi.org/10.1142/s021989161100238x.

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We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be "weakly" well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.

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47

Al-Hawasy, Jamil A. Ali. "The Continuous Classical Boundary Optimal Control of Couple Nonlinear Hyperbolic Boundary Value Problem with Equality and Inequality Constraints." Baghdad Science Journal 16, no. 4(Suppl.) (December 18, 2019): 1064. http://dx.doi.org/10.21123/bsj.2019.16.4(suppl.).1064.

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The paper is concerned with the state and proof of the existence theorem of a unique solution (state vector) of couple nonlinear hyperbolic equations (CNLHEQS) via the Galerkin method (GM) with the Aubin theorem. When the continuous classical boundary control vector (CCBCV) is known, the theorem of existence a CCBOCV with equality and inequality state vector constraints (EIESVC) is stated and proved, the existence theorem of a unique solution of the adjoint couple equations (ADCEQS) associated with the state equations is studied. The Frcéhet derivative derivation of the "Hamiltonian" is obtained. Finally the necessary theorem (necessary conditions "NCs") and the sufficient theorem (sufficient conditions" SCs") for optimality of the state constrained problem are stated and proved.

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Lipcius, R. N., D. B. Eggleston, D. L. Miller, and T. C. Luhrs. "The habitat-survival function for Caribbean spiny lobster: an inverted size effect and non-linearity in mixed algal and seagrass habitats." Marine and Freshwater Research 49, no. 8 (1998): 807. http://dx.doi.org/10.1071/mf97094.

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The habitat-survival function (HSF) defines changes in survival relative to habitat structure; forms include linear, hyperbolic and sigmoid (threshold) curves, whose consequences on predator–prey dynamics are illustrated by their first derivatives. Survival of two juvenile size classes of Caribbean spiny lobster was evaluated as a function of plant biomass in tethering experiments in mixed algal and seagrass patches adjacent to Bahía de la Ascensión, Mexico, which serves as nursery habitat. The HSF was hyperbolic for algal biomass; even modest increases of algal biomass significantly enhanced lobster survival. The rate of change in survival as a function of algal biomass (i.e. an approximation of the first derivative) was greatest at low-to-moderate levels of habitat structure. Hence, survival in these microhabitats is either low or rapidly changing with alterations in habitat structure, and they should be avoided by juveniles. Seagrass biomass did not significantly influence survival, although its levels were relatively low. Smaller juveniles had significantly higher survival rates than larger juveniles, probably because of the limited availability of appropriately scaled refugia for larger juveniles; large juveniles may display an ontogenetic shift from these habitats to coral reefs because of elevated predation risk as they grow.

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Dattoli, Giuseppe, and Paolo E. Ricci. "Laguerre-Type Exponentials, and the Relevant 𝐿-Circular and 𝐿-Hyperbolic Functions." gmj 10, no. 3 (September 2003): 481–94. http://dx.doi.org/10.1515/gmj.2003.481.

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Abstract Some classes of entire functions which are eigenfunctions of generalizations of the Laguerre derivative operator are considered. Since this property is an analog of the one characterizing the exponential function, we refer to such functions as Laguerre-type exponentials, or shortly 𝐿-exponentials. The definition of 𝐿-circular and 𝐿-hyperbolic functions easily follows. Applications in the framework of generalized evolution problems are also mentioned.

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LIANG, JIANFENG. "HYPERBOLIC SMOOTHING EFFECT FOR SEMILINEAR WAVE EQUATIONS AT A FOCAL POINT." Journal of Hyperbolic Differential Equations 06, no. 01 (March 2009): 1–23. http://dx.doi.org/10.1142/s0219891609001745.

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For semi-linear dissipative wave equation □u + |ut|p - 1ut = 0, we consider finite energy solutions with singularities propagating along a focusing light cone. At the tip of cone, the singularities are focused and partially smoothed out under strong nonlinear dissipation, i.e. the solution gets up to 1/2 more L2 derivative after the focus. The smoothing phenomenon is in fact the result of simultaneous action of focusing and nonlinear dissipation.

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