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Journal articles on the topic 'Generalized Pochhammer- Chree equation'

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Published: 3 June 2025
Last updated: 13 July 2025

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1

Zayed, E. M. E., and K. A. E. Alurrfi. "The Generalized Projective Riccati Equations Method for Solving Nonlinear Evolution Equations in Mathematical Physics." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/259190.

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We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.
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2

Liu, Chunyan. "The traveling wave solution and dynamics analysis of the fractional order generalized Pochhammer–Chree equation." AIMS Mathematics 9, no. 12 (2024): 33956–72. https://doi.org/10.3934/math.20241619.

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<p>This article studies the phase portraits, chaotic patterns, and traveling wave solutions of the fractional order generalized Pochhammer–Chree equation. First, the fractional order generalized Pochhammer–Chree equation is transformed into an ordinary differential equation. Second, the dynamic behavior is analyzed using the planar dynamical system, and some three-dimensional and two-dimensional phase portraits are drawn using Maple software to reflect its chaotic behaviors. Finally, many solutions were constructed using the polynomial complete discriminant system method, including rational, trigonometric, hyperbolic, Jacobian elliptic function, and implicit function solutions. Two-dimensional graphics, three-dimensional graphics, and contour plots of some solutions are drawn.</p>
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3

Li, Jibin, and Lijun Zhang. "Bifurcations of traveling wave solutions in generalized Pochhammer–Chree equation." Chaos, Solitons & Fractals 14, no. 4 (2002): 581–93. http://dx.doi.org/10.1016/s0960-0779(01)00248-x.

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4

Li, Biao, Yong Chen, and Hongqing Zhang. "Travelling Wave Solutions for Generalized Pochhammer-Chree Equations." Zeitschrift für Naturforschung A 57, no. 11 (2002): 874–82. http://dx.doi.org/10.1515/zna-2002-1106.

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In this paper, by means of a proper transformation and symbolic computation, we study the travelling wave reduction for the generalized Pochhammer-Chree (PC) equations (1.3) and (1.4) by use of the recently proposed extended-tanh method. As a result, rich travelling wave solutions, which include kink-shaped solitons, bell-shaped solitons, periodic solutions, rational solutions, singular solitons, are obtained. At the same time, using a direct assumption method, the more general bell-shaped solitons for the generalized PC Eq. (1.3) are obtained. The properties of the solutions are show in figures.
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5

Weiguo, Zhang, and Ma Wenxiu. "Explicit solitary-wave solutions to generalized Pochhammer-Chree equations." Applied Mathematics and Mechanics 20, no. 6 (1999): 666–74. http://dx.doi.org/10.1007/bf02464941.

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6

LI, JIBIN, and GUANRONG CHEN. "EXACT TRAVELING WAVE SOLUTIONS AND THEIR BIFURCATIONS FOR THE GENERALIZED POCHHAMMER–CHREE EQUATIONS." International Journal of Bifurcation and Chaos 22, no. 09 (2012): 1250233. http://dx.doi.org/10.1142/s0218127412502331.

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By using the method of dynamical systems to study the generalized Pochhammer–Chree equations, the dynamics of traveling wave solutions are characterized under different parameter conditions. Some exact parametric representations of the traveling wave solutions are obtained. Thus, many results reported in the literature can be completed.
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7

El-Ganaini, Shoukry Ibrahim Atia. "Travelling Wave Solutions to the Generalized Pochhammer-Chree (PC) Equations Using the First Integral Method." Mathematical Problems in Engineering 2011 (2011): 1–13. http://dx.doi.org/10.1155/2011/629760.

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By using the first integral method, the traveling wave solutions for the generalized Pochhammer-Chree (PC) equations are constructed. The obtained results include complex exponential function solutions, complex traveling solitary wave solutions, complex periodic wave solutions, and complex rational function solutions. The power of this manageable method is confirmed.
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8

Yépez-Martínez, Huitzilin, Mir Sajjad Hashemi, Ali Saleh Alshomrani, and Mustafa Inc. "Analytical solutions for nonlinear systems using Nucci's reduction approach and generalized projective Riccati equations." AIMS Mathematics 8, no. 7 (2023): 16655–90. http://dx.doi.org/10.3934/math.2023852.

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<abstract><p>In this study, the Nucci's reduction approach and the method of generalized projective Riccati equations (GPREs) were utilized to derive novel analytical solutions for the (1+1)-dimensional classical Boussinesq equations, the generalized reaction Duffing model, and the nonlinear Pochhammer-Chree equation. The nonlinear systems mentioned earlier have been solved using analytical methods, which impose certain limitations on the interaction parameters and the coefficients of the guess solutions. However, in the case of the double sub-equation guess solution, analytic solutions were allowed. The soliton solutions that were obtained through this method display real positive values for the wave phase transformation, which is a novel result in the application of the generalized projective Riccati method. In previous applications of this method, the real positive properties of the solutions were not thoroughly investigated.</p></abstract>
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9

EL Achab, Abdelfattah. "On the integrability of the generalized Pochhammer–Chree (PC) equations." Physica A: Statistical Mechanics and its Applications 545 (May 2020): 123576. http://dx.doi.org/10.1016/j.physa.2019.123576.

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10

Jaradat, Imad, Marwan Alquran, Sania Qureshi, Tukur A. Sulaiman, and Abdullahi Yusuf. "Convex-rogue, half-kink, cusp-soliton and other bidirectional wave-solutions to the generalized Pochhammer-Chree equation." Physica Scripta 97, no. 5 (2022): 055203. http://dx.doi.org/10.1088/1402-4896/ac5f25.

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Abstract The generalized Pochhammer-Chree equation is considered and studied for different orders of its nonlinearity terms The Kudryashov-expansion method is used and bidirectional kink, singular-kink, rogue-periodic, and V-shaped wave-solutions are obtained. Moreover, we modify the sine-cosine function method to accommodate the current model and obtain symmetric half-kink, convex-rogue, and cusp bidirectional waves. On the other side, a graphical analysis is conducted to identify the physical shapes of the obtained solutions to the proposed model. Finally, the polynomial function method is implemented to validate the reported solutions.
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11

FAN, HUI-LING, and XIN LI. "The classification of the single travelling wave solutions to the generalized Pochhammer–Chree equation." Pramana 81, no. 6 (2013): 925–41. http://dx.doi.org/10.1007/s12043-013-0626-0.

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12

Runzhang, Xu, and Liu Yacheng. "Global existence and blow-up of solutions for generalized Pochhammer-Chree equations." Acta Mathematica Scientia 30, no. 5 (2010): 1793–807. http://dx.doi.org/10.1016/s0252-9602(10)60173-7.

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13

Ali, Asghar, Aly R. Seadawy, and Dumitru Baleanu. "Propagation of harmonic waves in a cylindrical rod via generalized Pochhammer-Chree dynamical wave equation." Results in Physics 17 (June 2020): 103039. http://dx.doi.org/10.1016/j.rinp.2020.103039.

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14

Zhao, Yan, and Weiguo Zhang. "Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer-Chree Equation with a Dissipation Term." Studies in Applied Mathematics 121, no. 4 (2008): 369–94. http://dx.doi.org/10.1111/j.1467-9590.2008.00420.x.

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15

Mohebbi, Akbar. "Solitary wave solutions of the nonlinear generalized Pochhammer–Chree and regularized long wave equations." Nonlinear Dynamics 70, no. 4 (2012): 2463–74. http://dx.doi.org/10.1007/s11071-012-0634-5.

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16

Ilyashenko, Alla V., and Sergey V. Kuznetsov. "Polarization of the Longitudinal Pochhammer–Chree Waves." Mechanics and Mechanical Engineering 22, no. 4 (2020): 1329–36. http://dx.doi.org/10.2478/mme-2018-0103.

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AbstractThe exact solutions of the linear Pochhammer – Chree equation for propagating harmonic waves in a cylindrical rod, are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of wave polarization on the free surface due to variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear wave speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer – Chree wave can propagate at this phase speed.
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17

Kuznetsov, Sergey V. "Abnormality of the longitudinal Pochhammer–Chree waves in the vicinity of C2 phase speed." Journal of Vibration and Control 24, no. 23 (2018): 5642–49. http://dx.doi.org/10.1177/1077546318763205.

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The exact solutions of the linear Pochhammer–Chree equation for propagating harmonic waves in a cylindrical rod are analyzed. Spectral analysis of the matrix dispersion equation for longitudinal axially symmetric modes is performed and analytical expressions for displacement fields are obtained. The variation of wave polarization on the free surface due to the variation of Poisson’s ratio and circular frequency is analyzed. It is observed that at the phase speed coinciding with the bulk shear speed all the components of the displacement field vanish, meaning that no longitudinal axisymmetric Pochhammer–Chree waves can propagate at this phase speed.
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18

Tao, Zhao-Ling. "Variational Principles for Some Nonlinear Wave Equations." Zeitschrift für Naturforschung A 63, no. 5-6 (2008): 237–40. http://dx.doi.org/10.1515/zna-2008-5-601.

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Using the semi-inverse method proposed by Ji-Huan He, variational principles are established for some nonlinear wave equations arising in physics, including the Pochhammer-Chree equation, Zakharov-Kuznetsov equation, Korteweg-de Vries equation, Zhiber-Shabat equation, Kawahara equation, and Boussinesq equation.
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19

Ilyashenko, Alla, and Sergey Kuznetsov. "On degeneracy of dispersive waves at the bulk wave velocities." E3S Web of Conferences 97 (2019): 03004. http://dx.doi.org/10.1051/e3sconf/20199703004.

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Degeneracy of the linear dispersion wave equation at the phase velocities coinciding with the bulk wave velocities is observed and analysed. Spectral analysis of Pochhammer – Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.
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20

Yue, Liu. "Existence and blow up of a nonlinear Pochhammer-Chree equation." Indiana University Mathematics Journal 45, no. 3 (1996): 0. http://dx.doi.org/10.1512/iumj.1996.45.1121.

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21

Ilyashenko, A. V., and S. V. Kuznetsov. "Longitudinal Pochhammer — Chree Waves in Mild Auxetics and Non-Auxetics." Journal of Mechanics 35, no. 3 (2018): 327–34. http://dx.doi.org/10.1017/jmech.2018.13.

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ABSTRACTThe exact solutions of Pochhammer — Chree equation for propagating harmonic waves in isotropic elastic cylindrical rods, are analyzed. Spectral analysis of the matrix dispersion equation for the longitudinal axially symmetric modes is performed. Analytical expressions for displacement fields are obtained. Variation of the wave polarization due to variation of Poisson’s ratio for mild auxetics (Poisson’s ratio is greater than -0.5) is analyzed and compared with the non-auxetics. It is observed that polarization of the waves for both considered cases (auxetics and non-auxetics) exhibits abnormal behavior in the vicinity of the bulk shear wave speed.
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22

Novotny, S. V. "Modes of the Lamb's Type in the Generalized Pochhammer-Chree Problem." International Journal of Fluid Mechanics Research 29, no. 2 (2002): 13. http://dx.doi.org/10.1615/interjfluidmechres.v29.i2.90.

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23

Yusuf, Abdullahi, Mustafa Inc, and Aliyu Isa Aliyu. "Fractional solitons for the nonlinear Pochhammer-Chree equation with conformable derivative." Journal of Coupled Systems and Multiscale Dynamics 6, no. 2 (2018): 158–62. http://dx.doi.org/10.1166/jcsmd.2018.1149.

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24

Kadkhoda, Nematollah, Michal Feckan та Yasser Khalili. "Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation". Filomat 32, № 9 (2018): 3347–54. http://dx.doi.org/10.2298/fil1809347k.

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In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.
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25

ZHANG, WEIGUO, YAN ZHAO, GANG LIU, and TONGKE NING. "PERIODIC WAVE SOLUTIONS FOR POCHHAMMER–CHREE EQUATION WITH FIVE ORDER NONLINEAR TERM AND THEIR RELATIONSHIP WITH SOLITARY WAVE SOLUTIONS." International Journal of Modern Physics B 24, no. 19 (2010): 3769–83. http://dx.doi.org/10.1142/s0217979210056268.

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In this paper, periodic wave solutions for Pochhammer–Chree equation (PC-equation) with fifth order nonlinear term and their relationship with solitary wave solutions are studied. By designing innovative structure of solution, sixteen bounded periodic wave solutions in fractional form of Jacobi elliptic function (JEF) for PC-equation are given. Furthermore, global phase figure in the plane of the traveling solution for the PC-equation are obtained through dynamic systematic method, we indicate the region in the phase where the given sixteen solutions for PC-equation belong to. We find that two couples of these solutions change into two bell profile solitary wave solutions as k → 1 and four solutions change into four periodic wave solutions in fractional form of cosine function as k → 0. Finally, four figures are shown to describe the evolvement from periodic wave solutions to bell profile solitary wave solutions as k → 1.
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26

Rani, Attia, Nawab Khan, Kamran Ayub, et al. "Solitary Wave Solution of Nonlinear PDEs Arising in Mathematical Physics." Open Physics 17, no. 1 (2019): 381–89. http://dx.doi.org/10.1515/phys-2019-0043.

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Abstract The solution of nonlinear mathematical models has much importance and in soliton theory its worth has increased. In the present article, we have investigated the Caudrey-Dodd-Gibbon and Pochhammer-Chree equations, to discuss the physics of these equations and to attain soliton solutions. The exp(−ϕ(ζ ))-expansion technique is used to construct solitary wave solutions. A wave transformation is applied to convert the problem into the form of an ordinary differential equation. The drawn-out novel type outcomes play an essential role in the transportation of energy. It is noted that in the study, the approach is extremely reliable and it may be extended to further mathematical models signified mostly in nonlinear differential equations.
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27

EME, Zayed, and Shohib RMA. "The unified sub-equation method and its applications to conformable space-time fractional fourth-order pochhammer-chree equation." Physics & Astronomy International Journal 2, no. 5 (2018): 452–63. http://dx.doi.org/10.15406/paij.2018.02.00124.

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28

Seadawy, Aly R., S. U. Rehman, M. Younis, S. T. R. Rizvi, Saad Althobaiti, and M. M. Makhlouf. "Modulation instability analysis and longitudinal wave propagation in an elastic cylindrical rod modelled with Pochhammer-Chree equation." Physica Scripta 96, no. 4 (2021): 045202. http://dx.doi.org/10.1088/1402-4896/abdcf7.

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29

Jawernah, Yousef, Qasem M. Tawhari, Musaad S. Aldhabani, Ali H. Hakami, and Hussain Gissy. "An analytical examination of bright and dark kink solitons in Conformable Pochhammer-Chree equation arising in elastic medium." Ain Shams Engineering Journal 16, no. 9 (2025): 103539. https://doi.org/10.1016/j.asej.2025.103539.

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30

Doungmo Goufo, Emile Franc, and Stella Mugisha. "Mathematical Solvability of a Caputo Fractional Polymer Degradation Model Using Further Generalized Functions." Mathematical Problems in Engineering 2014 (2014): 1–5. http://dx.doi.org/10.1155/2014/392792.

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The continuous fission equation with derivative of fractional orderα, describing the polymer chain degradation, is solved explicitly. We prove that, whether the breakup rate depends on the size of the chain breaking up or not, the evolution of the polymer sizes distribution is governed by a combination of higher transcendental functions, namely, Mittag-Leffler function, the further generalizedG-function, and the Pochhammer polynomial. In particular, this shows the existence of an eigenproperty; that is, the system describing fractional polymer chain degradation contains replicated and partially replicated fractional poles, whose effects are given by these functions.
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31

Savalia, Rajesh V., and B. I. Dave. "p-Deformation of a General Class of Polynomials and its Properties." Journal of the Indian Mathematical Society 85, no. 1-2 (2018): 226. http://dx.doi.org/10.18311/jims/2018/17945.

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The work incorporates the extension of the Srivastava-Pathan’s generalized polynomial by means of p-generalized gamma function: Γ<sub>p</sub> and Pochhammer p-symbol (x)<sub>n,p</sub> due to Rafael Dıaz and Eddy Pariguan [Divulgaciones Mathematicas Vol.15, No. 2(2007), pp. 179-192]. We establish the inverse series relation of this extended polynomial with the aid of general inversion theorem. We also obtain the generating function relations and the differential equation. Certain <em>p</em>-deformed combinatorial identities are illustrated in the last section.
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32

Kukla, Stanisław, and Urszula Siedlecka. "On Solutions of the Initial Value Problem for the Three-Term Fractional Differential Equation with Caputo Derivatives." Symmetry 12, no. 8 (2020): 1355. http://dx.doi.org/10.3390/sym12081355.

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In this paper, two forms of an exact solution and an analytical–numerical solution of the three-term fractional differential equation with the Caputo derivatives are presented. The Prabhakar function and an asymptotic expansion are utilized to present the double series solution. Using properties of the Pochhammer symbol, a solution is obtained in the form of an infinite series of generalized hypergeometric functions. As an alternative for the series solutions of the fractional commensurate equation, a solution received by an analytical–numerical method based on the Laplace transform technique is proposed. This solution is obtained in the form of a finite sum of the Mittag-Leffler type functions. Numerical examples and a discussion are presented.
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33

Chang, Jian Mei, and Huai Ping Feng. "Wave Propagation in a Magnetoelectroelastic Rod." Applied Mechanics and Materials 79 (July 2011): 237–41. http://dx.doi.org/10.4028/www.scientific.net/amm.79.237.

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In this paper the propagation of axially symmetric compression wave in an infinite transversely isotropic magnetoelectroelastic cylinder is presented. Following Pan(2002), the general boundary conditions on the surface of the magnetoelectroelastic cylinder are written by a simple vector equation that is similar to the purely elastic counterpart. By assuming the possible form of the solution, the generalized Pochhammer frequency equations are presented and the corresponding phase velocity curves in some range are given. Numerical examples are also presented to show phase velocity curves under the first 4 kinds of boundary conditions. Other boundary conditions can be studied under farther research.
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34

Yang, Jun, Junhua He, Dezhi Zhang, et al. "Local Phase-Amplitude Joint Correction for Free Surface Velocity of Hopkinson Pressure Bar." Applied Sciences 10, no. 15 (2020): 5390. http://dx.doi.org/10.3390/app10155390.

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The Hopkinson pressure bar is widely used to measure the reflected pressure of blast waves over a short distance. However, dispersion effects will occur when the elastic stress waves propagate in the pressure bar due to lateral inertia, and there will be errors between the signals obtained from the sensors and the actual loading. For the free surface velocity measured in our system, we developed a local phase-amplitude joint correction method to convert the measured velocity into the average reflected pressure of a shock wave at the impact end of the bar, considering factors such as propagation modes of the elastic wave, the frequency components’ time of arrival, velocity variation over the bar axis, and the stress–velocity relationship. Firstly, the Pochhammer–Chree frequency equation is calculated numerically, and the first to fourth orders of phase velocity, group velocity, normalized frequency, and propagation time curves of elastic wave propagation in 35CrMnSiA steel are obtained. Secondly, the phase and amplitude correction formulas for calculating average reflected pressure from center velocity are derived based on the propagation mode of the axial elastic wave in the pressure bar by analyzing the time-frequency combined spectrum obtained by short-time Fourier transform. Thirdly, a local phase-amplitude joint correction algorithm based on propagation mode is proposed in detail. The experimental tests and data analyses are carried out for eight sets of pressure bar. The results show that this method can identify the propagation mode of elastic waves in the bar intuitively and clearly. The first three orders of propagation modes are stimulated in the bar 04, while only the first order of propagation is stimulated in the other eight bars. The local phase-amplitude joint correction algorithm can avoid correcting the component of the non-axial elastic wave. The rising edge of the average stress curve on the impact surface of bar 01 and bar 04 is corrected from 4.13 μs and 4.09 μs to 2.70 μs, respectively.
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35

Kourosh, Parand, and Amani Rad Jamal. "Some Solitary Wave Solutions of Generalized Pochhammer-Chree Equation via Exp-function Method." July 25, 2010. https://doi.org/10.5281/zenodo.1079408.

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In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.
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36

AL Nuwairan, Muneerah. "The exact solutions of the conformable time fractional version of the generalized Pochhammer–Chree equation." Mathematical Sciences, May 13, 2022. http://dx.doi.org/10.1007/s40096-022-00471-3.

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AbstractThe time-fractional version of the generalized Pochhammer–Chree equation is analyzed. In this paper, the equation is converted into an ordinary differential equation by applying certain real transformation, then the discrimination of polynomials system is used to find exact solutions depending on the fractional order derivative. The obtained solutions are graphically illustrated for different values of the fractional order derivative keeping the other parameters fixed.
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37

Hussain, A., M. Usman, F. D. Zaman, and S. M. Eldin. "Double reductions and traveling wave structures of the generalized Pochhammer–Chree equation." Partial Differential Equations in Applied Mathematics, May 2023, 100521. http://dx.doi.org/10.1016/j.padiff.2023.100521.

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38

Abbas, Naseem, Amjad Hussain, Aziz Khan, and Thabet Abdeljawad. "Bifurcation analysis, quasi-periodic and chaotic behavior of generalized Pochhammer-Chree equation." Ain Shams Engineering Journal, May 2024, 102827. http://dx.doi.org/10.1016/j.asej.2024.102827.

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39

Cong, Hongzi, Siming Li, Yingte Sun, and Xiaoqing Wu. "Birkhoff Normal Form and Long Time Existence for d-Dimensional Generalized Pochhammer–Chree Equation." Journal of Statistical Physics 192, no. 2 (2025). https://doi.org/10.1007/s10955-025-03409-w.

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40

Zulfiqar, Aniqa, Jamshad Ahmad, and Qazi Mahmood Ul-Hassan. "Analysis of some new wave solutions of fractional order generalized Pochhammer-chree equation using exp-function method." Optical and Quantum Electronics 54, no. 11 (2022). http://dx.doi.org/10.1007/s11082-022-04141-5.

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41

Kumar, Ajay, and Prachi Fartyal. "Dynamical behavior for the approximate solutions and different wave profiles nonlinear fractional generalised pochhammer-chree equation in mathematical physics." Optical and Quantum Electronics 55, no. 13 (2023). http://dx.doi.org/10.1007/s11082-023-05416-1.

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42

Marques, A., and M. S. Rodrigues. "Frequency dependence of the speed of sound in metallic rods." Physica Scripta, October 24, 2023. http://dx.doi.org/10.1088/1402-4896/ad0693.

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Abstract The speed of sound waves in rods depends on the relationship between wavelength and rod dimensions. 
It differs from the speeds readily available in tables, and from what is often learned during most introductory courses on solid-state physics. Metallic rods with diameters in the centimetre range excited with sound waves of tens of kHz will behave as dispersive media.
Here, the speed of sound in metallic titanium rods of different lengths is measured, by using two different methodologies: 1) from the time of flight and 2) from the wavelength and frequency of standing waves that form in the rod. The latter allows analyzing the results in light of Pochhammer-Cree dispersion. The reflection coefficient is also determined both from time and from frequency response. 
Two off-the-shelf piezoelectric transducers, a function generator, an oscilloscope, and a lock-in amplifier were used.
We have used a low-frequency square wave (of tens of Hz) in the first case and a sine wave with frequencies that range from audible to ultrasound in the second case.
Experimental results show that the speed of sound decreases as the wavelength decreases. The Pochhammer-Chree dispersion equation was numerically solved to fit the experimental data that can be used to estimate both the Young modulus and the Poisson ratio. A practical empirical formula that allows the analysis of data without explicitly solving the Pochhammer-Chree equation is suggested.
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43

Khater, Mostafa M. A., and Suleman H. Alfalqi. "High accuracy solutions for the Pochhammer–Chree equation in elastic media." Scientific Reports 14, no. 1 (2024). http://dx.doi.org/10.1038/s41598-024-68051-0.

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44

Khater, Mostafa M. A. "Nonlinear effects in quantum field theory: Applications of the Pochhammer–Chree equation." Modern Physics Letters B, November 25, 2024. http://dx.doi.org/10.1142/s0217984925500708.

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This study aims to solve the nonlinear Pochhammer–Chree ([Formula: see text]) equation to understand its physical implications and establish connections with other nonlinear evolution equations, particularly in plasma dynamics. By using the Khater II ([Formula: see text]hat. II) method for analytical solutions and validating these solutions numerically with the variational iteration method, this study offers a detailed understanding of the equation’s behavior. The results demonstrate the effectiveness of these methods in accurately modeling the system, highlighting the importance of combining analytical and numerical approaches for reliable solutions. This research significantly advances the field of nonlinear dynamics, especially in plasma physics, by employing multidisciplinary methods to tackle complex physical processes. Moreover, the [Formula: see text] equation is relevant in various physical contexts beyond plasma dynamics such as optical and quantum fields. In optics, it models the propagation of nonlinear waves in fiber optics, where similar nonlinear evolution equations describe wave interactions. In quantum field theory, the [Formula: see text] equation helps in understanding the behavior of quantum particles and fields under nonlinear effects, making it a versatile tool for studying complex phenomena across different domains of physics.
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45

Shin, Hyunho. "Sound Speed and Poisson's Ratio Calibration of (Split) Hopkinson Bar via Iterative Dispersion Correction of Elastic Wave." Journal of Applied Mechanics, March 17, 2022, 1–43. http://dx.doi.org/10.1115/1.4054107.

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Abstract A process of calibrating a one-dimensional sound speed (c_o) and Poisson's ratio (v) of a (split) Hopkinson bar is presented. This process consists of Fourier synthesis and iterative dispersion correction (time-domain phase shift) of the elastic pulse generated by the striker impact on a circular bar. At each iteration, a set of co and v is assumed, and the sound speed vs. frequency (c vs. f) relationship under the assumed set is obtained using the Pochhammer–Chree equation solver developed herein for ground state excitation. Subsequently, each constituting wave of the elastic pulse was phase-shifted (dispersion-corrected) using the c–f relationship. The co and v values of the bar were determined in the iteration process when the dispersion-corrected overall pulse profiles were reasonably consistent with the measured profiles at two travel distances in the bar. The observed consistency of the predicted (dispersion-corrected) wave profiles with the measured profiles is a mutually self-consistent verification of (i) the calibrated values of co and v, and (ii) the combined theories of Fourier and Pochhammer–Chree. The contributions of the calibrated values of co and v to contemporary bar technology are discussed, together with the physical significance of the tail part of a traveling wave according to the combined theories. A preprocessing template (in Excel®) and calibration platform (in MATLAB ®) for the presented calibration process are available online in a public repository.
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46

Tarla, Sibel, Karmina K. Ali, and Hatıra Günerhan. "Optical soliton solutions of generalized Pochammer Chree equation." Optical and Quantum Electronics 56, no. 5 (2024). http://dx.doi.org/10.1007/s11082-024-06711-1.

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AbstractThis research investigates the utilization of a modified version of the Sardar sub-equation method to discover novel exact solutions for the generalized Pochammer Chree equation. The equation itself represents the propagation of longitudinal deformation waves in an elastic rod. By employing this modified method, we aim to identify previously unknown solutions for the equation under consideration, which can contribute to a deeper understanding of the behavior of deformation waves in elastic rods. The solutions obtained are represented by hyperbolic, trigonometric, exponential functions, dark, dark-bright, periodic, singular, and bright solutions. By selecting suitable values for the physical parameters, the dynamic behaviors of these solutions can be demonstrated. This allows for a comprehensive understanding of how the solutions evolve and behave over time. The effectiveness of these methods in capturing the dynamics of the solutions contributes to our understanding of complex physical phenomena. The study’s findings show how effective the selected approaches are in explaining nonlinear dynamic processes. The findings reveal that the chosen techniques are not only effective but also easily implementable, making them applicable to nonlinear model across various fields, particularly in studying the propagation of longitudinal deformation waves in an elastic rod. Furthermore, the results demonstrate that the given model possesses solutions with potentially diverse structures.
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47

Shin, Hyunho. "Pochhammer–Chree equation solver for dispersion correction of elastic waves in a (split) Hopkinson bar." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, May 10, 2021, 095440622098050. http://dx.doi.org/10.1177/0954406220980509.

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A robust algorithm for solving the Bancroft version of the Pochhammer–Chree (PC) equation is developed based on the iterative root-finding process. The formulated solver not only obtains the conventional n-series solutions but also derives a new series of solutions, named m-series solutions. The n-series solutions are located on the PC function surface that relatively gradually varies in the vicinity of the roots, whereas the m-series solutions are located between two PC function surfaces with (nearly) positive and negative infinity values. The proposed solver obtains a series of sound speeds at exactly the frequencies necessary for dispersion correction, and the derived solutions are accurate to the ninth decimal place. The solver is capable of solving the PC equation up to n = 20 and m = 20 in the ranges of Poisson’s ratio ( ν) of 0.02 [Formula: see text] ν [Formula: see text] 0.48, normalised frequency ( F) of F [Formula: see text] 30, and normalised sound speed ( C) of C [Formula: see text] 300. The developed algorithm was implemented in MATLAB®, which is available in the Supplemental Material (accessible online).
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48

Akinyemi, Lanre, P. Veeresha, Mehmet Şenol, and Hadi Rezazadeh. "An efficient technique for generalized conformable Pochhammer–Chree models of longitudinal wave propagation of elastic rod." Indian Journal of Physics, April 26, 2022. http://dx.doi.org/10.1007/s12648-022-02324-0.

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49

Akinyemi, Lanre, P. Veeresha, Mehmet Şenol, and Hadi Rezazadeh. "An efficient technique for generalized conformable Pochhammer–Chree models of longitudinal wave propagation of elastic rod." Indian Journal of Physics, April 26, 2022. http://dx.doi.org/10.1007/s12648-022-02324-0.

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50

Akinyemi, Lanre, P. Veeresha, Mehmet Şenol, and Hadi Rezazadeh. "An efficient technique for generalized conformable Pochhammer–Chree models of longitudinal wave propagation of elastic rod." Indian Journal of Physics, April 26, 2022. http://dx.doi.org/10.1007/s12648-022-02324-0.

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