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Journal articles on the topic 'Spin-wave separation of variables'

Author: Grafiati

Published: 7 September 2024

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1

Kalnins, E. G., and G. C. Williams. "Symmetry operators and separation of variables for spin‐wave equations in oblate spheroidal coordinates." Journal of Mathematical Physics 31, no. 7 (July 1990): 1739–44. http://dx.doi.org/10.1063/1.528670.

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2

Amico, Luigi, Holger Frahm, Andreas Osterloh, and Tobias Wirth. "Separation of variables for integrable spin–boson models." Nuclear Physics B 839, no. 3 (November 2010): 604–26. http://dx.doi.org/10.1016/j.nuclphysb.2010.07.005.

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3

Zhdanov, R. Z. "Separation of variables in the nonlinear wave equation." Journal of Physics A: Mathematical and General 27, no. 9 (May 7, 1994): L291—L297. http://dx.doi.org/10.1088/0305-4470/27/9/009.

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4

BEREST, YURI, and PAVEL WINTERNITZ. "HUYGENS' PRINCIPLE AND SEPARATION OF VARIABLES." Reviews in Mathematical Physics 12, no. 02 (February 2000): 159–80. http://dx.doi.org/10.1142/s0129055x00000071.

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We demonstrate a close relation between the algebraic structure of the (local) group of conformal transformations on a smooth Lorentzian manifold [Formula: see text] and the existence of nontrivial hierarchies of wave-type hyperbolic operators satisfying Huygens' principle on [Formula: see text]. The mechanism of such a relation is provided through a local separation of variables for linear second order partial differential operators with a metric principal symbol. The case of flat (Minkowski) spaces is studied in detail. As a result, some new nontrivial classes of Huygens operators are constructed. Their relation to the classical Hadamard conjecture and its modifications is discussed.

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5

Zhdanov, R. Z., I. V. Revenko, and V. I. Fushchich. "Separation of variables in two-dimensional wave equations with potential." Ukrainian Mathematical Journal 46, no. 10 (October 1994): 1480–503. http://dx.doi.org/10.1007/bf01066092.

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6

Smirnov, Yu G., V. Yu Martynova, M. A. Moskaleva, and A. V. Tikhonravov. "MODIFIED METHOD OF SEPARATION OF VARIABLES FOR SOLVING DIFFRACTION PROBLEMS ON MULTILAYER DIELECTRIC GRATINGS." Eurasian Journal of Mathematical and Computer Applications 9, no. 4 (December 2021): 76–88. http://dx.doi.org/10.32523/2306-6172-2021-9-4-76-88.

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A modified method of separation of variables is proposed for solving the direct problem of diffraction of electromagnetic wave by multilayer dielectric gratings (MDG). To apply this method, it is necessary to solve a one-dimensional eigenvalue problem for a 2nd- order differential equation on a segment with piecewise constant coefficients. The accuracy of the method is verified by comparison with the results obtained by the commercially available RCWA method. It is demonstrated that the method can be applied not only to commonly used MDG elements with one line in a grating period but also to potentially promising MDG elements with several different lines in a grating period.

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7

Sergeev, S. M. "Functional Equations and Quantum Separation of Variables for 3d Spin Models." Theoretical and Mathematical Physics 138, no. 2 (February 2004): 226–37. http://dx.doi.org/10.1023/b:tamp.0000015070.88403.f9.

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8

Osetrin, Konstantin, and Evgeny Osetrin. "Shapovalov Wave-Like Spacetimes." Symmetry 12, no. 8 (August 18, 2020): 1372. http://dx.doi.org/10.3390/sym12081372.

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A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

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9

Casals, Marc, Adrian C. Ottewill, and Niels Warburton. "High-order asymptotics for the spin-weighted spheroidal equation at large real frequency." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2222 (February 2019): 20180701. http://dx.doi.org/10.1098/rspa.2018.0701.

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The spin-weighted spheroidal eigenvalues and eigenfunctions arise in the separation by variables of spin-field perturbations of Kerr black holes. We derive a large, real-frequency asymptotic expansion of the spin-weighted spheroidal eigenvalues and eigenfunctions to high order. This expansion corrects and extends existing results in the literature and we validate it via a high-precision numerical calculation.

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10

Osetrin, Konstantin, Ilya Kirnos, Evgeny Osetrin, and Altair Filippov. "Wave-Like Exact Models with Symmetry of Spatial Homogeneity in the Quadratic Theory of Gravity with a Scalar Field." Symmetry 13, no. 7 (June 29, 2021): 1173. http://dx.doi.org/10.3390/sym13071173.

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Exact solutions are obtained in the quadratic theory of gravity with a scalar field for wave-like models of space–time with spatial homogeneity symmetry and allowing the integration of the equations of motion of test particles in the Hamilton–Jacobi formalism by the method of separation of variables with separation of wave variables (Shapovalov spaces of type II). The form of the scalar field and the scalar field functions included in the Lagrangian of the theory is found. The obtained exact solutions can describe the primary gravitational wave disturbances in the Universe (primary gravitational waves).

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11

Shermenev, Alexander. "Separation of variables for the nonlinear wave equation in cylindrical coordinates." Physica D: Nonlinear Phenomena 212, no. 3-4 (December 2005): 205–15. http://dx.doi.org/10.1016/j.physd.2005.09.009.

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12

Shermenev, Alexander. "Separation of variables for the nonlinear wave equation in polar coordinates." Journal of Physics A: Mathematical and General 37, no. 45 (October 29, 2004): 10983–91. http://dx.doi.org/10.1088/0305-4470/37/45/016.

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13

Derkachov, Sergey É., Gregory P. Korchemsky, and Alexander N. Manashov. "Separation of variables for the quantum SL(2,Bbb R) spin chain." Journal of High Energy Physics 2003, no. 07 (July 21, 2003): 047. http://dx.doi.org/10.1088/1126-6708/2003/07/047.

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14

von Gehlen, G., N. Iorgov, S. Pakuliak, and V. Shadura. "Factorized finite-size Ising model spin matrix elements from separation of variables." Journal of Physics A: Mathematical and Theoretical 42, no. 30 (July 14, 2009): 304026. http://dx.doi.org/10.1088/1751-8113/42/30/304026.

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15

Noula Tefouet, Joseph D., and David Yemélé. "Soliton Domain Wall Concept: Analytical and Numerical Investigation in Digital Magnetic Recording System." European Journal of Applied Physics 3, no. 2 (April 30, 2021): 56–66. http://dx.doi.org/10.24018/ejphysics.2021.3.2.64.

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To introduce Soliton theory in magnetic recording systems, we begin with the profile of Domain Wall, which is key elements of recording systems, knowing that many different Domain Walls shape exist. For this, we consider the recording media as chain of atoms (spin) and, we use the Hamiltonian to describe the global state of the system; by taking into consideration interaction between the neighboring spin and anisotropic interaction. Spins are considered as classical vector; for that, we defined the cosine and sine of angles that specify the position of the spins. They are developed in Taylor’s series until second order then using the approximation of continuous medium we obtained the Lagragian relation. This Lagragian enables us to describe the dynamics of spin through the wave velocity. As we are fine just the profile of domain wall it is beneficial for us to consider the wall at rest (static) and by the aid of Euler equation we obtain two simple equations; using the equilibrium conditions, the differential equation is obtained and solved by the quadratic method and separation variables method. The profile of domain wall that we obtain is at a particular position, then analytical and numerical simulation give us the opportunity to see that profile of that domain wall is a Kink, anti-Kink Soliton and also Soliton Train. Using this magnetic Soliton wave (Domain Wall), we also evaluate the playback voltage V (x), the peak voltage and the half pulse width PW50 to confirm the uses of this DW profile in magnetic recording systems and insure validity of this work.

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16

Alvarez Alvarado, Manuel, and Fernando Vaca Urbano. "EURISTIC INSTRUCTION FOR WAVE EQUATION PROBLEM-SOLVING USING VARIABLE SEPARATION METHOD." Revista Bases de la Ciencia. e-ISSN 2588-0764 2, no. 3 (December 31, 2017): 74. http://dx.doi.org/10.33936/rev_bas_de_la_ciencia.v2i3.1028.

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This study focuses on heuristic instruction as a method of logical thought processes. The objective is to raise and establish a framework for problem-solving in the application of separation of variables for the wave equation. This leads to a simple and easy pathway to make the topic at a manageable level for the students. This paper proposes a base problem, over which the heuristic instruction is applied. Eventually, the result is the determination of a system that allows a simple approach to avoid difficulties in solving problems in which the separation of variables in the equation of the wave is employed. In order to determine the impact of heuristic in students learning gain, two groups of 20 students each (experimental and control) were subject to a pre- and post-test. To the control group, the scheme proposed was not employed. In contrast to the experimental the proposed approach was employed. The results reveal that the heuristic scheme for wave equation problem-solving has a significant impact on students’ learning process. Key words: Physics, Heuristic Instruction, Differential Equations, Wave Equation.

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17

Lai, Xian-Jing, and Jie-Fang Zhang. "Novel Excitations of the Nonlinear Schrödinger Equation by Separation of Variables." Zeitschrift für Naturforschung A 64, no. 1-2 (February 1, 2009): 21–29. http://dx.doi.org/10.1515/zna-2009-1-204.

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By means of an extended tanh method, a new type of variable separation solutions with two arbitrary lower-dimensional functions of the (2+1)-dimensional nonlinear Schrödinger (NLS) equation is derived. Based on the derived variable separation excitation, some special types of localized solutions such as a curved soliton, a straight-line soliton and a periodic soliton are constructed by choosing appropriate functions. In addition, one dromion changes its shape during the collision with a folded solitary wave.

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18

Yaro, Rasmane, Youssouf Pare, and Bakari Abbo. "Comparison of SBA Numerical Method and Method of Separation of Variables (Fourier) on Wawe Equations." European Journal of Pure and Applied Mathematics 12, no. 3 (July 25, 2019): 1260–76. http://dx.doi.org/10.29020/nybg.ejpam.v12i3.3471.

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In this paper, our aim is to use the SBA numerical method (combination of Adomian method and Picard successive approximations) and Fourier method or method of separation of variables to construct the solution of some wave equations. We compare the two methods and apply them to some wave equations.

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19

Pei, Hao, and Véronique Terras. "On scalar products and form factors by separation of variables: the antiperiodic XXZ model." Journal of Physics A: Mathematical and Theoretical 55, no. 1 (December 13, 2021): 015205. http://dx.doi.org/10.1088/1751-8121/ac3b85.

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Abstract We consider the XXZ spin-1/2 Heisenberg chain with antiperiodic boundary conditions. The inhomogeneous version of this model can be solved by separation of variables, and the eigenstates can be constructed in terms of Q-functions, solution of a Baxter TQ-equation, which have double periodicity compared to the periodic case. We compute in this framework the scalar products of a particular class of separate states which notably includes the eigenstates of the transfer matrix. We also compute the form factors of local spin operators, i.e. their matrix elements between two eigenstates of the transfer matrix. We show that these quantities admit determinant representations with rows and columns labelled by the roots of the Q-functions of the corresponding separate states, as in the periodic case, although the form of the determinant are here slightly different. We also propose alternative types of determinant representations written directly in terms of the transfer matrix eigenvalues.

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20

Osetrin, Konstantin, Altair Filippov, and Evgeny Osetrin. "Wave-like spatially homogeneous models of Stäckel spacetimes (2.1) type in the scalar-tensor theory of gravity." Modern Physics Letters A 35, no. 33 (August 26, 2020): 2050275. http://dx.doi.org/10.1142/s0217732320502752.

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Six exact solutions are obtained in the general scalar-tensor theory of gravity related to spatially homogeneous wave-like models of the Universe. Wave-like spacetime models allow the existence of privileged coordinate systems where the eikonal equation and the Hamilton–Jacobi equation of test particles can be integrated by the method of complete separation of variables with the separation of isotropic (wave) variables on which the space metric depends (non-ignored variables). An explicit form of the scalar field and two functions of the scalar field that are part of the general scalar-tensor theory of gravity are found. The explicit form of the eikonal function and the action function for test particles in the considered models is given. The obtained solutions are of type III according to the Bianchi classification and type N according to the Petrov classification. Wave-like spatially homogeneous spacetime models can describe primordial gravitational waves of the Universe.

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21

Chen, Jing, Ling Liu, and Li Liu. "Separation Transformation and a Class of Exact Solutions to the Higher-Dimensional Klein-Gordon-Zakharov Equation." Advances in Mathematical Physics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/974050.

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The separation transformation method is extended to then+1-dimensional Klein-Gordon-Zakharov equation describing the interaction of the Langmuir wave and the ion acoustic wave in plasma. We first reduce then+1-dimensional Klein-Gordon-Zakharov equation to a set of partial differential equations and two nonlinear ordinary differential equations of the separation variables. Then the general solutions of the set of partial differential equations are given and the two nonlinear ordinary differential equations are solved by extendedF-expansion method. Finally, some new exact solutions of then+1-dimensional Klein-Gordon-Zakharov equation are proposed explicitly by combining the separation transformation with the exact solutions of the separation variables. It is shown that, for the case ofn≥2, there is an arbitrary function in every exact solution, which may reveal more nontrivial nonlinear structures in the high-dimensional Klein-Gordon-Zakharov equation.

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22

Druzhinin, A. "Separation of Variables Applied to the Wave Equation in Laterally Inhomogeneous Media." Pure and Applied Geophysics 160, no. 7 (July 1, 2003): 1225–44. http://dx.doi.org/10.1007/s000240300003.

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23

Belov, V. V., S. Yu Dobrokhotov, and T. Ya Tudorovskiy. "Operator separation of variables for adiabatic problems in quantum and wave mechanics." Journal of Engineering Mathematics 55, no. 1-4 (July 25, 2006): 183–237. http://dx.doi.org/10.1007/s10665-006-9044-3.

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24

Barannyk, A. F., T. A. Barannyk, and I. I. Yuryk. "Generalized procedure of separation of variables and reduction of nonlinear wave equations." Ukrainian Mathematical Journal 61, no. 7 (July 2009): 1055–74. http://dx.doi.org/10.1007/s11253-009-0270-5.

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25

Su, Ting, and Hui Hui Dai. "A New Integrable Variable-Coefficient 2+1-Dimensional Long Wave-Short Wave Equation and the Generalized Dressing Method." Advances in Mathematical Physics 2018 (July 9, 2018): 1–7. http://dx.doi.org/10.1155/2018/7286574.

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Based on the generalized dressing method, we propose a new integrable variable-coefficient 2+1-dimensional long wave-short wave equation and derive its Lax pair. Using separation of variables, we have derived the explicit solutions of the equation. With the aid of Matlab, the curves of the solutions are drawn.

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26

Ovsiyuk, E., A. Safronov, A. Ivashkevich, and O. Semenyuk. "St¨uckelberg particle in external magnetic field. The method of projective operators." Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences, no. 5 (December 20, 2022): 69–78. http://dx.doi.org/10.19110/1994-5655-2022-5-69-78.

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We study the St¨uckelberg equation for a relativistic particle with two spin states S = 1 and S = 0 in the presence of an external uniform magnetic field. The particle is described by an 11-component wave function consisting of a scalar, a vector, and an antisymmetric tensor. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, a system for 11 radial functions is obtained. Its solution is based on the use of the Fedorov-Gronsky method, in which all 11 radial functions are expressed in terms of three main functions. Exact solutions with cylindrical symmetry are constructed. Three series of energy levels are found.

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27

Ovsiyuk, E., A. Safronov, A. Ivashkevich, and O. Semenyuk. "St¨uckelberg particle in external magnetic field. Nonrelativistic approximation. Exact solutions." Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences, no. 5 (December 20, 2022): 79–88. http://dx.doi.org/10.19110/1994-5655-2022-5-79-88.

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The St¨uckelberg equation for a particle with two spin states, S = 1 and S = 0, is studied in the presence of an external uniform magnetic field. In relativistic case, the particle is described by an 11-component wave function. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, we derive a system for 11 functions depending on one variable. We perform the nonrelativistic approximation in this system. For this we apply the known method of deriving nonrelativistic equations from relativistic ones, which is based on projective operators related to the matrix Γ0 of the relativistic equation. The nonrelativistic wave function turns out to be 4-dimensional. We derive the system for 4 functions. It is solved in terms of confluent hypergeometric functions. There arise three series of energy levels with corresponding solutions. This result agrees with that obtained for the relativistic St¨uckelberg equation.

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28

Qu, Chang-Zheng, and Shun-Li Zhang. "Extended Group Foliation Method and Functional Separation of Variables to Nonlinear Wave Equations." Communications in Theoretical Physics 44, no. 4 (October 2005): 577–82. http://dx.doi.org/10.1088/6102/44/4/577.

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29

Buryy, A. V., A. V. Ivashkevich, and O. A. Semenyuk. "A spin 1 particle in a cylindric basis: the projective operator method." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 58, no. 4 (January 1, 2023): 398–411. http://dx.doi.org/10.29235/1561-2430-2022-58-4-398-411.

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In this paper, the system of equations describing a spin 1 particle is studied in cylindric coordinates with the use of tetrad formalism and the matrix 10-dimension formalism of Duffin – Kemmer – Petieau. After separating the variables, we apply the method proposed by Fedorov – Gronskiy and based on the use of projective operators to resolve the system of 10 equations in the r variable. In the presence of an external uniform magnetic field, we construct in an explicit form three independent classes of wave functions with corresponding energy spectra. Separately the massless field with spin 1 is studied; there are found four linearly independent solutions, two of which are gauge ones, and other two do not contain gauge degrees of freedom. Meanwhile, the method of Fedorov – Gronskiy is also used.

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30

SHAN, YUEH. "GENERALIZED SPHEROIDAL WAVE EQUATIONS FOR IMPURITY STATES IN A HETEROSTRUCTURE." Modern Physics Letters B 04, no. 17 (September 20, 1990): 1099–102. http://dx.doi.org/10.1142/s0217984990001380.

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Within the framework of effective mass theory, a set of generalized spheroidal wave equations for the exact treatment of a shallow donor impurity in a semiconductor-semiconductor heterostructure is obtained from the relevant Schrödinger equation by the method of separation of variables in prolate spheroidal coordinates. The way of calculating the eigensolutions of these wave equations is briefly discussed.

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31

Seheda, M. S., Ye V. Cheremnykh, P. F. Gogolyuk, T. A. Mazur, and Y. V. Blyznak. "MATHEMATICAL MODEL OF WAVE PROCESSES IN TWO-WINDING TRANSFORMERS." Tekhnichna Elektrodynamika 2020, no. 6 (October 21, 2020): 5–14. http://dx.doi.org/10.15407/techned2020.06.005.

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A mathematical model has been developed to study the wave processes of two-winding transformers, taking into account the electromagnetic connections between the turns of the winding and between the windings. To solve differential-integral equations in partial derivatives, a method of separation of variables is proposed. References 12, figures 2.

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32

Benkemache, Ibtisam, Mohammad Al-horani, and Roshdi Rashid Khalil. "Tensor Product and Certain Solutions of Fractional Wave Type Equation." European Journal of Pure and Applied Mathematics 14, no. 3 (August 5, 2021): 942–48. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.4012.

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In this paper we Ã–nd certain solutions of some fractional partial diÂ§erential equations. Tensor product of Banach spaces is used to Ã–nd some solutions where separation of variables does not work. We solve the fractional wave type equation using fractional Fourier Series

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33

Wei, Chun-Fu. "Local fractional heat and wave equations with Laguerre type derivatives." Thermal Science 24, no. 4 (2020): 2575–80. http://dx.doi.org/10.2298/tsci2004575w.

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In this paper, we investigate a local fractional PDE with Laguerre type derivative. The considered equation represents a general extension of the classical heat and wave equations. The method of separation of variables is used to solve the differential equation defined in a bounded domain.

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34

Ivashkevich, A. V., E. M. Ovsiyuk, V. V. Kisel, and V. M. Red’kov. "Spherical solutions of the wave equation for a spin 3/2 particle." Doklady of the National Academy of Sciences of Belarus 63, no. 3 (June 28, 2019): 282–90. http://dx.doi.org/10.29235/1561-8323-2019-63-3-282-290.

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The wave equation for a spin 3/2 particle, described by 16-component vector-bispinor, is investigated in spherical coordinates. In the frame of the Pauli–Fierz approach, the complete equation is split into the main equation and two additional constraints, algebraic and differential. The solutions are constructed, on which 4 operators are diagonalized: energy, square and third projection of the total angular momentum, and spatial reflection, these correspond to quantum numbers {ε, j, m, P}. After separating the variables, we have derived the radial system of 8 first-order equations and 4 additional constraints. Solutions of the radial equations are constructed as linear combinations of the Bessel functions. With the use of the known properties of the Bessel functions, the system of differential equations is transformed to the form of purely algebraic equations with respect to three quantities a1, a2, a3. Its solutions may be chosen in various ways by solving the simple linear equation A1a1 + A2a2 + A3a3 = 0 where the coefficients Ai are expressed trough the quantum numbers ε, j. Two most simple and symmetric solutions have been chosen. Thus, at fixed quantum numbers {ε, j, m, P} there exists double-degeneration of the quantum states.

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35

Cong, Lin, and Xu Li. "Reduced-Order Modeling and Control of Heat-Integrated Air Separation Column Based on Nonlinear Wave Theory." Processes 11, no. 10 (October 5, 2023): 2918. http://dx.doi.org/10.3390/pr11102918.

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The process of low-temperature air separation consumes a significant amount of energy. Internal heat-integrated distillation technology has considerable energy-saving potential. Therefore, the combination of low-temperature air separation and heat-integrated distillation technology has led to the development of a heat-integrated air separation column (HIASC). Due to the heat integration and the inherent complexity of air separation, the modeling and control of this process poses significant challenges. This paper first introduces the nonlinear wave theory into the HIASC, derives the expression for the velocity of the concentration distribution curve movement and the curve describing function, and then establishes a nonlinear wave model. Compared to the traditional mechanical models, this approach greatly reduces the number of differential equations and variables while ensuring an accurate description of the system characteristics. Subsequently, based on the wave model, a model predictive control scheme is designed for the HIASC. This scheme is compared with two conventional control schemes: PID and a general model control. The simulation results demonstrate that MPC outperforms the other control schemes from the response curves and performance metrics.

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36

Osetrin, Konstantin, Altair Filippov, Ilya Kirnos, and Evgeny Osetrin. "Type I Shapovalov Wave Spacetimes in the Brans–Dicke Scalar-Tensor Theory of Gravity." Symmetry 14, no. 12 (December 13, 2022): 2636. http://dx.doi.org/10.3390/sym14122636.

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Exact solutions for Shapovalov wave spacetimes of type I in Brans–Dicke’s scalar-tensor theory of gravity are constructed. Shapovalov wave spacetimes describe gravitational wave models that allow for the the separation of wave variables in privileged coordinate systems. In contrast to general relativity, the vacuum field equations of the Brans–Dicke scalar-tensor theory of gravity lead to exact solutions for type I Shapovalov spaces, allowing for the the construction of observational tests to detect such wave disturbances. Furthermore, the equations for the trajectories of the test particles are obtained for the models considered.

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37

Pakuliak, Stanislav, and Sergei Sergeev. "Quantum relativistic Toda chain at root of unity: isospectrality, modifiedQ-operator, and functional Bethe ansatz." International Journal of Mathematics and Mathematical Sciences 31, no. 9 (2002): 513–53. http://dx.doi.org/10.1155/s0161171202105059.

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We investigate anN-state spin model called quantum relativistic Toda chain and based on the unitary finite-dimensional representations of the Weyl algebra withqbeingNth primitive root of unity. Parameters of the finite-dimensional representation of the local Weyl algebra form the classical discrete integrable system. Nontrivial dynamics of the classical counterpart corresponds to isospectral transformations of the spin system. Similarity operators are constructed with the help of modified Baxter'sQ-operators. The classical counterpart of the modifiedQ-operator for the initial homogeneous spin chain is a Bäcklund transformation. This transformation creates an extra Hirota-type soliton in a parameterization of the chain structure. Special choice of values of solitonic amplitudes yields a degeneration of spin eigenstates, leading to the quantum separation of variables, or the functional Bethe ansatz. A projector to the separated eigenstates is constructed explicitly as a product of modifiedQ-operators.

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38

Hussein, Shilan Othman. "Splitting the One-Dimensional Wave Equation. Part I: Solving by Finite-Difference Method and Separation Variables." Baghdad Science Journal 17, no. 2(SI) (June 22, 2020): 0675. http://dx.doi.org/10.21123/bsj.2020.17.2(si).0675.

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In this study, an unknown force function dependent on the space in the wave equation is investigated. Numerically wave equation splitting in two parts, part one using the finite-difference method (FDM). Part two using separating variables method. This is the continuation and changing technique for solving inverse problem part in (1,2). Instead, the boundary element method (BEM) in (1,2), the finite-difference method (FDM) has applied. Boundary data are in the role of overdetermination data. The second part of the problem is inverse and ill-posed, since small errors in the extra boundary data cause errors in the force solution. Zeroth order of Tikhonov regularization, and several parameters of regularization are employed to decrease errors for output force solution. It is obvious from figures how error affects the results and zeroth order stables the solution.

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39

Otsuki, Sora, Pauline N. Kawamoto, and Hiroshi Yamazaki. "A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables." Formalized Mathematics 27, no. 1 (April 1, 2019): 25–34. http://dx.doi.org/10.2478/forma-2019-0003.

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Summary In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.

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40

Fatibene, Lorenzo, Raymond G. McLenaghan, and Giovanni Rastelli. "Symmetry operators and separation of variables for Dirac's equation on two-dimensional spin manifolds with external fields." International Journal of Geometric Methods in Modern Physics 12, no. 01 (December 28, 2014): 1550012. http://dx.doi.org/10.1142/s0219887815500127.

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The second-order symmetry operators that commute with the Dirac operator with external vector, scalar and pseudo-scalar potentials are computed on a general two-dimensional spin manifold. It is shown that the operator is defined in terms of Killing vectors, valence two Killing tensors and scalar fields defined on the background manifold. The commuting operator that arises from a non-trivial Killing tensor is determined with respect to the associated system of Liouville coordinates and compared to the second-order operator that obtained from the unique separation scheme associated with such operators. It is shown by the study of several examples that the operators arising from these two approaches coincide.

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41

Kiselev, Aleksei P. "Generalization of Bateman–Hillion progressive wave and Bessel–Gauss pulse solutions of the wave equation via a separation of variables." Journal of Physics A: Mathematical and General 36, no. 23 (May 28, 2003): L345—L349. http://dx.doi.org/10.1088/0305-4470/36/23/103.

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42

Niccoli, G. "Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors." Nuclear Physics B 870, no. 2 (May 2013): 397–420. http://dx.doi.org/10.1016/j.nuclphysb.2013.01.017.

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43

Niccoli, G. "Form factors and complete spectrum of XXX antiperiodic higher spin chains by quantum separation of variables." Journal of Mathematical Physics 54, no. 5 (May 2013): 053516. http://dx.doi.org/10.1063/1.4807078.

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44

Geng, Xue, Liang Guan, and Dianlou Du. "Action-angle variables for the Lie-Poisson Hamiltonian systems associated with the three-wave resonant interaction system." AIMS Mathematics 7, no. 6 (2022): 9989–10008. http://dx.doi.org/10.3934/math.2022557.

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<abstract><p>The $ \mathfrak{gl}_3(\mathbb{C}) $ rational Gaudin model governed by $ 3\times 3 $ Lax matrix is applied to study the three-wave resonant interaction system (TWRI) under a constraint between the potentials and the eigenfunctions. And the TWRI system is decomposed so as to be two finite-dimensional Lie-Poisson Hamiltonian systems. Based on the generating functions of conserved integrals, it is shown that the two finite-dimensional Lie-Poisson Hamiltonian systems are completely integrable in the Liouville sense. The action-angle variables associated with non-hyperelliptic spectral curves are computed by Sklyanin's method of separation of variables, and the Jacobi inversion problems related to the resulting finite-dimensional integrable Lie-Poisson Hamiltonian systems and three-wave resonant interaction system are analyzed.</p></abstract>

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45

Khaneja, Navin. "Cone separation, quadratic control systems and control of spin dynamics in the presence of decoherence." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, no. 2088 (March 6, 2017): 20160214. http://dx.doi.org/10.1098/rsta.2016.0214.

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In this paper, we study some control problems related to the control of coupled spin dynamics in the presence of relaxation and decoherence in nuclear magnetic resonance spectroscopy. The decoherence is modelled through a master equation. We study some model problems, whereby, through an appropriate choice of state variables, the system is reduced to a control system, where the state enters linearly and controls quadratically. We study this quadratic control system. Study of this system gives us explicit bounds on how close a coupled spin system can be driven to its target state and how much coherence and polarization can be transferred between coupled spins. Optimal control for the quadratic control system can be understood as the separation of closed cones, and we show how the derived results on optimal efficiency can be interpreted in this formulation. Finally, we study some finite-time optimal control problems for the quadratic control system. This article is part of the themed issue ‘Horizons of cybernetical physics’.

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CAMMAROTA, CAMILLO. "POSITIVE AND NEGATIVE CORRELATIONS FOR CONDITIONAL ISING DISTRIBUTIONS." Reviews in Mathematical Physics 14, no. 10 (October 2002): 1099–113. http://dx.doi.org/10.1142/s0129055x02001521.

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In the Ising model at zero external field with ferromagnetic first neighbors interaction the Gibbs measure is investigated using the group properties of the contours configurations. Correlation inequalities expressing positive dependence among groups and comparison among groups and cosets are used. An improved version of the Griffiths' inequalities is proved for the Gibbs measure conditioned to a subgroup. Examples of positive and negative correlations among the spin variables are proved under conditioning to a contour or to a separation line.

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Fidelibus*, Matthew, Steven Vasquez, and Donald Katayama. "Canopy Separation and Defoliation for Dry-on-the-vine (DOV) Raisins on Traditional Trellises." HortScience 39, no. 4 (July 2004): 880B—880. http://dx.doi.org/10.21273/hortsci.39.4.880b.

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Pruning efficiency, fruitfulness, and yield and quality of raisins of `Thompson Seedless' (Vitis vinifera L.) grapevines subjected to several canopy separation and defoliation treatments for DOV raisin production were evaluated. Canopy separation treatments, tested in vineyards at Easton, and at the Kearney Agricultural Center (KAC), Parlier, Calif., were as follows; horizontal canopy separation with vine sections of fruiting or renewal zones (Peacock), horizontal canopy separation with vine sections of fruiting zones of one vine adjacent to renewal shoots of the next vine (wave), or non-separated (control). Defoliation treatments included burning or blowing leaves (Easton), application of concentrated solutions of calcium ammonium nitrate or Etherel to leaves (KAC), or no defoliation (both vineyards). Canopy separation treatments did not affect berry size, soluble solids, or raisin yield. Vines subjected to Peacock training had more cluster layers than vines subjected to wave training, at Easton, and more cluster layers than vines subjected to control training at KAC. Canopy separation reduced harvest pruning time by 20% at Easton, but not at KAC. No treatments affected raisin moisture at Easton but, at KAC, raisins of vines trained in the Peacock style had 10% higher moisture contents at harvest than raisins of wave or control vines. Vines subjected to conventional training and leaf blowing had about 40% higher “B and better” raisin grades than vines with separated canopies that were not defoliated, and about 30% higher grades than vines with conventional training and leaf burning. However, raisins of vines subjected to blowing had about 60% more mold than raisins of non-defoliated vines. Defoliation treatments at KAC did not affect any variables measured.

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Bagrov, V. G., B. F. Samsonov, and A. V. Shapovalov. "Separation of variables in the wave equation. Sets of the type (1.1) and Schr�dinger equation." Soviet Physics Journal 34, no. 2 (February 1991): 122–26. http://dx.doi.org/10.1007/bf00940949.

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Horwood, Joshua T., and Raymond G. McLenaghan. "Orthogonal separation of variables for the Hamilton-Jacobi and wave equations in three-dimensional Minkowski space." Journal of Mathematical Physics 49, no. 2 (February 2008): 023501. http://dx.doi.org/10.1063/1.2823971.

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Bagrov, V. G., B. F. Samsonov, A. V. Shapovalov, and I. V. Shirokov. "Commutative subalgebras of three first-order symmetry operators and separation of variables in the wave equation." Soviet Physics Journal 33, no. 5 (May 1990): 448–52. http://dx.doi.org/10.1007/bf00896088.

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