Several Solutions of the Damped Harmonic Oscillator with Time-Dependent Frictional Coefficient and Time-Dependent Frequency
|
|
- Gwen Nichols
- 5 years ago
- Views:
Transcription
1 Advanced Studies in Theoretical Physics Vol. 11, 017, no. 6, HIKARI Ltd, Several Solutions of the Damped Harmonic Oscillator with Time-Dependent Frictional Coefficient and Time-Dependent Frequency Eun Ji Jang and Won Sang Chung 1 Department of Physics and Research Institute of Natural Science College of Natural Science, Gyeongsang National University Jinju , Korea Copyright c 017 Eun Ji Jang and Won Sang Chung. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we discuss several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. We consider three types of frictional coefficient and frequency. We discuss the matrix formulation for the frictional coefficient γt) = γ 0 1+qt and frequency wt) = w 0 1+qt, q > 0). 1 Introduction The harmonic oscillator is a system playing an important role both in classical and quantum mechanics. It appears in various physical applications running from condensed matter to semiconductors see e.g.[1] for references to such problems). The harmonic oscillator equation with time-dependent parameters [-6] has been discussed for a sudden frequency change using a continuous treatment based on an invariant formalism [7]. This analytic treatment requires that the time-dependent parameter be a monotonic function whose variation is short compared with the typical period of the system. In 1931, Bateman presented [8] the so-called dual or mirror image formalism for damped oscillator. The Lagrangian for the linearly damped free 1 Corresponding author
2 64 Eun Ji Jang and Won Sang Chung particle was first obtained by Caldirola [9] and Kanai [10]. They considered the following Lagrangian L = e λt 1 mẋ, which gives the equation of motion for the the linearly damped free particle ẍ = λẋ In general the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency is governed by the following equation of motion: ẍ + γt)ẋ + w t)x = 0, 1) we set the mass of a particle to be unity. The Lagrangian L = 1 v Sx, t) does not give the eq.1). One can make use of Jacobis Last Multiplier Mt) to derive a suitable Lagrangian giving the eq.1). Consider Lagrangian [ 1 L = Mt) v 1 ] w t)x last multiplier turns out to be the integrating factor of such an equation obeying ) M = L 3) ẋ The canonical momentum p is given by ) p = Mẋ 4) Comparing the equation of motion derived from the eq.) with the eq.1), one get t Mt) = e γs)ds 0 5) By using the standard Legendre transformation it follows that the corresponding Hamiltonian is H = 1 Mt) p + 1 Mt)w t)x 6) In this paper we discuss several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. The paper is organized as follows: In section II we obtain solutions for the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency for γt) = γ 0, wt) = w 0, q > 0) and γt) = γ 1+qt 1+qt 0e ɛt, wt) = w 0 e ɛt, ɛ > 0) and γt) = γ 0 tanh t, wt) = w 0 tanh t. In section III we discuss the matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0). 1+qt 1+qt
3 Several solutions of the damped harmonic oscillator 65 Solution of the equation of motion In this section we solve the eq.1) for three cases. First we present the general solving technique. If we introduce the function yt) of the form we obtain yt) = e α t 0 γs)ds xt) 7) ÿ + 1 α)γẏ + [αα 1)γ α γ + w ]y = 0 8) Let us change the variable z = wt). If we replace some t-dependent functions with z-dependent ones through the eq.8) reduces to γt) = Λz), ẇ = fz), γ = fλ z) 9) f y + f[f + 1 α)λ]y + [αα 1)Λ αλ f + z ]y = 0 10) In the following subsections we obtain the solutions of the eq.10) for three choices of γt), wt). Here we impose the following initial condition: x0) = A, ẋ0) = 0 11).1 γt) = γ 0 1+qt, wt) = w 0 1+qt, q > 0), α = 1 In this case we have Thus, the eq.10) reduces to The solution to the eq.13) is Λz) = γ 0 w 0 z, fz) = q w 0 z 1) q z y + qγ 0 + q)zy + qγ 0 + w 0)y = 0 13) c 1 z λ + + c z λ γ 0 q > w 0 ) yz) = z λ 0 c 1 + c ln z) γ 0 q = w 0 ), 14) z λ 0 [c 1 cosν ln z) + c sinν ln z)] γ 0 q < w 0 ) λ ± = 1 q [ q + γ 0) ± γ 0 q) w 0 ) ] 15) λ 0 = 1 q q + γ 0) 16)
4 66 Eun Ji Jang and Won Sang Chung ν = 1 q w 0 ) γ 0 q) 17) For γ 0 q > w 0, using the initial condition 11), we have xt) = Aγ 0 + qλ ) qλ λ + ) 1 + qt) λ + γ 0 /q Aγ 0 + qλ + ) qλ λ + ) 1 + qt) λ γ 0 /q, 18) which is a q-deformed over damping case. For γ 0 q = w 0, using the initial condition 11), we have xt) = A1 + qt) λ 0 γ 0 /q 1 + γ 0 + qλ 0 q ln1 + qt) ), 19) which is a q-deformed critical damping case. For γ 0 q < w 0, using the initial condition 11), we have xt) = A1 + qt) λ 0 γ 0 /q [cosν ln1 + qt)) + γ 0 + qλ 0 νq which is a q-deformed under damping case.. γt) = γ 0 e ɛt, wt) = w 0 e ɛt, ɛ > 0) In this case we have Thus, the eq.10) reduces to sinν ln1 + qt))] 0) Λz) = γ 0 w 0 z, fz) = ɛz 1) [ ɛ z y + ɛ z ɛγ ] [ ) 0 γ 1 α)z y + 0 αα 1) + 1 z + αɛγ ] 0 z y = 0 w 0 w0 w 0 ) Now we choose α so that it obeys Then, we get The eq.) can be written as γ0 w0 αα 1) + 1 = 0 3) α = 1 1 ± 1 4 w 0 4) γ0 zy + [1 σz]y + µy = 0, 5) σ = γ 0 w 0 ɛ 1 α), µ = αγ 0 w 0 ɛ 6)
5 Several solutions of the damped harmonic oscillator 67 We set the solution to the eq.5) like yz) = a n z n+λ 7) n=0 Inserting the eq.7) into the eq.5), we have the following characteristic equation for λ: a 0 λ = 0 8) Here we set a 0 = 1 hence λ = 0. The recurrence relation is given by Solving the eq.9) we get a n+1 = σn µ n + 1) a n n 0) 9) a n = σn µ/σ) n n!), 30) a) 0 = 1, a) n = aa + 1)a + ) a + n 1). Thus, we have y 1 = 1 F 1 µ σ ; 1 : σz ) 31) Because the characteristic equation has double root, the second solution takes the following form : y = y 1 ln z + b m z m 3) Inserting the eq.3) into the eq.5) we get y 1 σy 1 + m b m z m 1 + µ σm)b m z m = 0 33) Solving the eq.33), we obtain b 1 = σa 0 a 1 34) and n + 1) b n+1 + µ σn)b n σa n + n + 1)a n+1 = 0, n 0) 35) The first few b n s are b 1 = σa 0 a 1 36) b = 1 4 [σ µ)b 1 + σa 1 4a ] 37)
6 68 Eun Ji Jang and Won Sang Chung Thus, we have b 3 = 1 9 [σ µ)b + σa 6a 3 ] 38) xt) = e αγ 0 ɛw 0 z [c 11 F 1 µ σ ; 1 : σz ) +c ln z 1 F 1 µ σ ; 1 : σz ) +c Because t = 0 gives z = w 0, from x0) = A, we get +c [µ ln w 0 µ 1 F 1 µ σ + 1; : σw 0 we used Because dx dt e αγ 0 ɛ A = c 1 µ 1 F 1 µ ) σ + 1; : σw 0 ) 1 1F 1 µ ) w0 σ ; 1 : σw 0 b m z m ] 39) mb m w m 1 0 ] 40) d dz µ 1F 1 a; 1 : σz) = aσµ 1 F 1 a + 1; : σz) 41) dx = ɛz, from v0) = 0, we obtain dz e αγ 0 ɛ A = c 11 F 1 µ ) σ ; 1 : σw 0 + c [ln w 01 F 1 µ ) σ ; 1 : σw γt) = γ 0 tanh t, wt) = w 0 tanh t In this case we have If we set z = w 0 η, we have b m w0 m ] 4) Λz) = γ 0 w 0 z, fz) = w 0 1 w 0 z 43) 1 η ) y +[1 α)γ 0 ]η1 η )y +[w 0 +αα 1)γ 0 +αγ 0 )η αγ 0 ]y = 0 44) Now we choose α so that it obeys Then, we get w 0 + αα 1)γ 0 = 0 45) α = 1 ) γ 0 ± γ0 4w0, 46) γ 0 α is real when γ0 > 4w0 over-damping case). We set the solution to the eq.44) like yη) = a n η n+λ 47) n=0
7 Several solutions of the damped harmonic oscillator 69 Inserting the eq.47) into the eq.44), we have the following characteristic equation for λ: a 0 λλ 1) = 0, a 1 λλ + 1) = 0 48) Here we set a 0 = 1, a 1 = 0 hence λ = 0, 1. The recurrence relation is given by a n+ = n + λ)n + λ α)γ 0) + αγ 0 a n n 0) 49) n + λ + )n + λ + 1) Solving the eq.49) for λ = 0, we obtain a m = m m)! n +) m n ) m, 50) n ± = 1 4 [1 1 α)γ 0 ± 1 γ 0 ) w 0] 51) Thus, we have the solution for λ = 0 like y 1 = F 1 n+, n ; 1/ : η ) 5) Solving the eq.49) for λ = 1, we obtain a m = Thus, we have the solution for λ = 1 like m m)! n + + 1/) m n + 1/) m 53) y = η F 1 n+ + 1/, n + 1/; 1/ : η ) 54) The general solution of the eq.44) is then given by and y = c 1 y 1 + c y 55) xt) = cosh t) αγ 0 [c 1 F 1 n+, n ; 1/ : η ) +c η F 1 n+ + 1/, n + 1/; 1/ : η ) ] 56) Using the initial condition we get c 1 = A, c = 0, hence xt) = Acosh t) αγ 0 F 1 n+, n ; 1/ : η ) 57)
8 70 Eun Ji Jang and Won Sang Chung 3 Matrix formulation In this section we consider the Matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0), α = 1. For the eq.1), we set 1+qt 1+qt xt) = e 1 Inserting the eq.58) into the eq.1) we get t 0 γs)ds ψt) 58) ψ + Ω t)ψ = 0, 59) Ω t) = w t) 1 4 γ t) 1 γ 60) In order to convert this second order equation to a first order system we introduce ) ψ1 t) ψt) =, 61) ψ t) ψ 1, ψ are two independent solutions of the eq.1). Now consider the following matrix equation: If we set d dt from the eq.63) we have ) ψ1 t) = ψ t) f1 t) f t) f 3 t) f 4 t) ) ) ψ1 t), 6) ψ t) f 1 f + f + f f 4 = 0 f 1 + f1 + f f 3 = Ω t) f 1 f 3 + f 3 + f 3 f 4 = 0 f 4 + f4 + f f 3 = Ω t) 63) f i t) = α i, i = 1,, 3, 4), 64) 1 + qt α 1 + α 4 = q α 1 qα 1 + α α 3 = w γ 0 1 qγ 0 65) Here we choose α 1 = α 4 = q/. Then, we have and the solution of the eq.6) is ) γ0 q α α 3 = w 0 66) ψt) = e t q) A ψ0) = e ln et q )A ψ0), 67)
9 Several solutions of the damped harmonic oscillator 71 ) α1 α A = α 3 α 4 In the eq.65) we choose α, α 3 as 68) α = α 3 = w = for q-deformed over damping and γ0 q ) w 0 69) ) α = α 3 = w 1 = iw = w0 γ0 q 70) for q-deformed under damping. For the q-deformed over damping case we have Using the relation A n = [n/] j=0 n j ) q ) n j w j I + q ) w A = σ = w [n 1)/] j=0 q ) Inserting the eq.7) into the eq.67), we obtain 71) ) ) n q n 1 j w j+1 σ, 7) j ) ) ψ1 t) = [e t ψ t) q] q/ cosh[w ln e t q] sinh[w ln e t ) ) q] ψ1 0) sinh[w ln e t q] cosh[w ln e t, 74) q] ψ 0) which gives the same result as the eq.18). Similarly, if we replace α = α 3 = w 1 = iw, we can have the same solution for the q-deformed under damping case. Acknowledgements. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government NRF- 015R1D1A1A ). 4 Conclusion In this paper we discussed several solutions of the damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. We considered the damped harmonic oscillator with time-dependent frictional
10 7 Eun Ji Jang and Won Sang Chung coefficient and time-dependent frequency governed by the equation of motion ẍ + γt)ẋ + w t)x = 0. We introduced the function yt) of the form yt) = e α t γs)ds 0 xt) and solved the second order differential equations for three types of time-dependent frictional coefficient and time-dependent frequency. First, we considered the case of γt) = γ 0, wt) = w 0, q > 0), α = 1 1+qt 1+qt for q-deformed over damping case, q-deformed critical damping case and q- deformed under damping case. Second, we considered the case of γt) = γ 0 e ɛt, wt) = w 0 e ɛt, ɛ > 0). In this case we obtained the solutions expressed in terms of the hypergeometric functions. Third, we considered the case of γt) = γ 0 tanh t, wt) = w 0 tanh t. In this case we also obtained the solutions expressed in terms of the hypergeometric functions. Finally, we discussed the Matrix formulation for the case of γt) = γ 0, wt) = w 0, q > 0), α = 1. 1+qt 1+qt We solved the matrix differential equation to obtain the same solution for the q-deformed under damping case. References [1] J. Yu and S. Dong, Exactly solvable potentials for the Schrodinger equation with spatially dependent mass, Phys. Lett. A, ), [] H.R. Lewis, Classical and Quantum Systems with Time-Dependent Harmonic-Oscillator-Type Hamiltonians, Phys. Rev. Lett., ), [3] V.V. Dodonov and V.I. Manko, Coherent states and the resonance of a quantum damped oscillator, Phys. Rev. A, ), [4] J. Janszky and Y.Y. Yushin, Squeezing via frequency jump, Opt. Commun., ), [5] F. Haas and J. Goedert, Dynamical symmetries and the Ermakov invariant, Phys. Lett. A, ), [6] S. Bouquet and H.R. Lewis, A second invariant for one-degree-of-freedom, time-dependent Hamiltonians given a first invariant, J. Math. Phys., ), [7] H. Moya-Cessa and M. Fernandez Guasti, Coherent states for the time
11 Several solutions of the damped harmonic oscillator 73 dependent harmonic oscillator: the step function, Phys. Lett. A, ), [8] H. Bateman, On Dissipative Systems and Related Variational Principles, Phys. Rev., ), [9] P. Caldirola, Forze non conservative nella meccanica quantistica, Il Nuovo Cimento, ), [10] E. Kanai, On the Quantization of the Dissipative Systems, Progress of Theoretical Physics, ), Received: August 1, 016; Published: May 15, 017
Damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency. Abstract
Damped harmonic oscillator with time-dependent frictional coefficient and time-dependent frequency Eun Ji Jang, Jihun Cha, Young Kyu Lee, and Won Sang Chung Department of Physics and Research Institute
More informationTwo Constants of Motion in the Generalized Damped Oscillator
Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 2, 57-65 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2016.511107 Two Constants o Motion in the Generalized Damped Oscillator
More informationOn the Deformed Theory of Special Relativity
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 6, 275-282 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.61140 On the Deformed Theory of Special Relativity Won Sang Chung 1
More informationHall Effect on Non-commutative Plane with Space-Space Non-commutativity and Momentum-Momentum Non-commutativity
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 8, 357-364 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.614 Hall Effect on Non-commutative Plane with Space-Space Non-commutativity
More informationA Symmetric Treatment of Damped Harmonic Oscillator in Extended Phase Space
Proceedings of Institute of Mathematics of NAS of Ukraine 00, Vol. 43, Part, 645 65 A Symmetric Treatment of Damped Harmonic Oscillator in Extended Phase Space S. NASIRI and H. SAFARI Institute for Advanced
More informationOn the f-deformed Boson Algebra and its Application to Thermodynamics
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 4, 143-162 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.61135 On the f-deformed Boson Algebra and its Application to Thermodynamics
More informationOn the q-deformed Thermodynamics and q-deformed Fermi Level in Intrinsic Semiconductor
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 5, 213-223 HIKARI Ltd, www.m-hikari.com htts://doi.org/10.12988/ast.2017.61138 On the q-deformed Thermodynamics and q-deformed Fermi Level in
More informationTime dependent quantum harmonic oscillator subject to a sudden change of mass: continuous solution
INVESTIGACIÓN REVISTA MEXICANA DE FÍSICA 53 1 4 46 FEBRERO 7 Time dependent quantum harmonic oscillator subject to a sudden change of mass: continuous solution H. Moya-Cessa INAOE, Coordinación de Óptica,
More informationMadelung Representation and Exactly Solvable Schrödinger-Burgers Equations with Variable Parameters
arxiv:005.5059v [nlin.si] 7 May 00 Madelung Representation and Exactly Solvable Schrödinger-Burgers Equations with Variable Parameters May 30, 08 Şirin A. Büyükaşık, Oktay K. Pashaev Dept. of Mathematics,
More informationSolution for a non-homogeneous Klein-Gordon Equation with 5th Degree Polynomial Forcing Function
Advanced Studies in Theoretical Physics Vol., 207, no. 2, 679-685 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/astp.207.7052 Solution for a non-homogeneous Klein-Gordon Equation with 5th Degree
More informationKKM-Type Theorems for Best Proximal Points in Normed Linear Space
International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed
More informationPoincaré`s Map in a Van der Pol Equation
International Journal of Mathematical Analysis Vol. 8, 014, no. 59, 939-943 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.411338 Poincaré`s Map in a Van der Pol Equation Eduardo-Luis
More informationA Damped Oscillator as a Hamiltonian System
A Damped Oscillator as a Hamiltonian System 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (June 9, 015; updated July 1, 015 It is generally considered
More informationOn a Certain Representation in the Pairs of Normed Spaces
Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida
More informationIdentities of Symmetry for Generalized Higher-Order q-euler Polynomials under S 3
Applied Mathematical Sciences, Vol. 8, 204, no. 3, 559-5597 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/ams.204.4755 Identities of Symmetry for Generalized Higher-Order q-euler Polynomials under
More informationElectric and Magnetic Forces in Lagrangian and Hamiltonian Formalism
Electric and Magnetic Forces in Lagrangian and Hamiltonian Formalism Benjamin Hornberger 1/26/1 Phy 55, Classical Electrodynamics, Prof. Goldhaber Lecture notes from Oct. 26, 21 Lecture held by Prof. Weisberger
More informationOn the Three-Phase-Lag Heat Equation with Spatial Dependent Lags
Nonlinear Analysis and Differential Equations, Vol. 5, 07, no., 53-66 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/nade.07.694 On the Three-Phase-Lag Heat Equation with Spatial Dependent Lags Yang
More informationOn Symmetric Bi-Multipliers of Lattice Implication Algebras
International Mathematical Forum, Vol. 13, 2018, no. 7, 343-350 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2018.8423 On Symmetric Bi-Multipliers of Lattice Implication Algebras Kyung Ho
More informationAdvanced Studies in Theoretical Physics Vol. 8, 2014, no. 22, HIKARI Ltd,
Advanced Studies in Theoretical Physics Vol. 8, 204, no. 22, 977-982 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/astp.204.499 Some Identities of Symmetry for the Higher-order Carlitz Bernoulli
More informationPath Integral Quantization of the Electromagnetic Field Coupled to A Spinor
EJTP 6, No. 22 (2009) 189 196 Electronic Journal of Theoretical Physics Path Integral Quantization of the Electromagnetic Field Coupled to A Spinor Walaa. I. Eshraim and Nasser. I. Farahat Department of
More informationQuantization of the LTB Cosmological Equation
Adv. Studies Theor. Phys., Vol. 7, 2013, no. 15, 723-730 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2013.3660 Quantization of the LTB Cosmological Equation Antonio Zecca 1 2 Dipartimento
More informationLinearization of Two Dimensional Complex-Linearizable Systems of Second Order Ordinary Differential Equations
Applied Mathematical Sciences, Vol. 9, 2015, no. 58, 2889-2900 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121002 Linearization of Two Dimensional Complex-Linearizable Systems of
More informationAn Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh
International Mathematical Forum, Vol. 8, 2013, no. 9, 427-432 HIKARI Ltd, www.m-hikari.com An Abundancy Result for the Two Prime Power Case and Results for an Equations of Goormaghtigh Richard F. Ryan
More informationHeisenberg-picture approach to the exact quantum motion of a. time-dependent forced harmonic oscillator. Abstract
KAIST-CHEP-96/01 Heisenberg-picture approach to the exact quantum motion of a time-dependent forced harmonic oscillator Hyeong-Chan Kim,Min-HoLee, Jeong-Young Ji,andJaeKwanKim Department of Physics, Korea
More informationA path integral approach to the Langevin equation
A path integral approach to the Langevin equation - Ashok Das Reference: A path integral approach to the Langevin equation, A. Das, S. Panda and J. R. L. Santos, arxiv:1411.0256 (to be published in Int.
More informationSymmetric Properties for the (h, q)-tangent Polynomials
Adv. Studies Theor. Phys., Vol. 8, 04, no. 6, 59-65 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/astp.04.43 Symmetric Properties for the h, q-tangent Polynomials C. S. Ryoo Department of Mathematics
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationBOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM. Q-Heung Choi and Tacksun Jung
Korean J. Math. 2 (23), No., pp. 8 9 http://dx.doi.org/.568/kjm.23.2..8 BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM Q-Heung Choi and Tacksun Jung Abstract. We investigate the bounded weak solutions
More informationThe Modified Adomian Decomposition Method for. Solving Nonlinear Coupled Burger s Equations
Nonlinear Analysis and Differential Equations, Vol. 3, 015, no. 3, 111-1 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/nade.015.416 The Modified Adomian Decomposition Method for Solving Nonlinear
More informationAlternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations
International Mathematical Forum, Vol. 9, 2014, no. 35, 1725-1739 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.410170 Alternate Locations of Equilibrium Points and Poles in Complex
More informationSolution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the
More informationChaos Control for the Lorenz System
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 181-188 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8413 Chaos Control for the Lorenz System Pedro Pablo Cárdenas Alzate
More informationThe Solution of the Truncated Harmonic Oscillator Using Lie Groups
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 327-335 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7521 The Solution of the Truncated Harmonic Oscillator Using Lie Groups
More information06. Lagrangian Mechanics II
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 06. Lagrangian Mechanics II Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationA Solution of the Spherical Poisson-Boltzmann Equation
International Journal of Mathematical Analysis Vol. 1, 018, no. 1, 1-7 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ijma.018.71155 A Solution of the Spherical Poisson-Boltzmann quation. onseca
More informationQualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions
Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 1, 45-59 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.3819 Qualitative Theory of Differential Equations and Dynamics of
More informationPólya-Szegö s Principle for Nonlocal Functionals
International Journal of Mathematical Analysis Vol. 12, 218, no. 5, 245-25 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ijma.218.8327 Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci
More informationA Generalization of Generalized Triangular Fuzzy Sets
International Journal of Mathematical Analysis Vol, 207, no 9, 433-443 HIKARI Ltd, wwwm-hikaricom https://doiorg/02988/ijma2077350 A Generalization of Generalized Triangular Fuzzy Sets Chang Il Kim Department
More informationA Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials
Applied Mathematical Sciences, Vol. 8, 2014, no. 35, 1723-1730 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.4127 A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationFactorized Parametric Solutions and Separation of Equations in ΛLTB Cosmological Models
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 7, 315-321 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5231 Factorized Parametric Solutions and Separation of Equations in
More informationApproximations to the t Distribution
Applied Mathematical Sciences, Vol. 9, 2015, no. 49, 2445-2449 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.52148 Approximations to the t Distribution Bashar Zogheib 1 and Ali Elsaheli
More informationSymmetric Identities for the Generalized Higher-order q-bernoulli Polynomials
Adv. Studies Theor. Phys., Vol. 8, 204, no. 6, 285-292 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.2988/astp.204.428 Symmetric Identities for the Generalized Higher-order -Bernoulli Polynomials Dae
More informationDynamic Model of Space Robot Manipulator
Applied Mathematical Sciences, Vol. 9, 215, no. 94, 465-4659 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.56429 Dynamic Model of Space Robot Manipulator Polina Efimova Saint-Petersburg
More informationSolitary Wave Solution of the Plasma Equations
Applied Mathematical Sciences, Vol. 11, 017, no. 39, 1933-1941 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.7609 Solitary Wave Solution of the Plasma Equations F. Fonseca Universidad Nacional
More informationShear Thinning Near the Rough Boundary in a Viscoelastic Flow
Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 8, 351-359 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2016.6624 Shear Thinning Near the Rough Boundary in a Viscoelastic Flow
More informationAntibound State for Klein-Gordon Equation
International Journal of Mathematical Analysis Vol. 8, 2014, no. 59, 2945-2949 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.411374 Antibound State for Klein-Gordon Equation Ana-Magnolia
More informationSymmetric Identities of Generalized (h, q)-euler Polynomials under Third Dihedral Group
Applied Mathematical Sciences, vol. 8, 2014, no. 145, 7207-7212 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49701 Symmetric Identities of Generalized (h, )-Euler Polynomials under
More informationNo-Go of Quantized General Relativity
Advanced Studies in Theoretical Physics Vol. 10, 2016, no. 8, 415-420 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2016.6928 No-Go of Quantized General Relativity Johan Hansson Division of
More informationThe graded generalized Fibonacci sequence and Binet formula
The graded generaized Fibonacci sequence and Binet formua Won Sang Chung,, Minji Han and Jae Yoon Kim Department of Physics and Research Institute of Natura Science, Coege of Natura Science, Gyeongsang
More informationThe Representation of Energy Equation by Laplace Transform
Int. Journal of Math. Analysis, Vol. 8, 24, no. 22, 93-97 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ijma.24.442 The Representation of Energy Equation by Laplace Transform Taehee Lee and Hwajoon
More informationREVIEW. Hamilton s principle. based on FW-18. Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws!
Hamilton s principle Variational statement of mechanics: (for conservative forces) action Equivalent to Newton s laws! based on FW-18 REVIEW the particle takes the path that minimizes the integrated difference
More informationApproximation to the Dissipative Klein-Gordon Equation
International Journal of Mathematical Analysis Vol. 9, 215, no. 22, 159-163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.5236 Approximation to the Dissipative Klein-Gordon Equation Edilber
More informationWeyl s Theorem and Property (Saw)
International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, 433-437 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.8754 Weyl s Theorem and Property (Saw) N. Jayanthi Government
More informationOn Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions
Applied Mathematical Sciences, Vol. 9, 015, no. 5, 595-607 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.5163 On Linear Recursive Sequences with Coefficients in Arithmetic-Geometric Progressions
More informationVanishing Dimensions in Four Dimensional Cosmology with Nonminimal Derivative Coupling of Scalar Field
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 9, 423-431 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.5234 Vanishing Dimensions in Four Dimensional Cosmology with Nonminimal
More informationLogarithmic Extension of Real Numbers and Hyperbolic Representation of Generalized Lorentz Transforms
International Journal of Algebra, Vol. 11, 017, no. 4, 159-170 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ija.017.7315 Logarithmic Extension of Real Numbers and Hyperbolic Representation of Generalized
More informationDirac Equation with Self Interaction Induced by Torsion: Minkowski Space-Time
Advanced Studies in Theoretical Physics Vol. 9, 15, no. 15, 71-78 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/astp.15.5986 Dirac Equation with Self Interaction Induced by Torsion: Minkowski Space-Time
More informationSymmetric Properties for Carlitz s Type (h, q)-twisted Tangent Polynomials Using Twisted (h, q)-tangent Zeta Function
International Journal of Algebra, Vol 11, 2017, no 6, 255-263 HIKARI Ltd, wwwm-hiaricom https://doiorg/1012988/ija20177728 Symmetric Properties for Carlitz s Type h, -Twisted Tangent Polynomials Using
More information22.2. Applications of Eigenvalues and Eigenvectors. Introduction. Prerequisites. Learning Outcomes
Applications of Eigenvalues and Eigenvectors 22.2 Introduction Many applications of matrices in both engineering and science utilize eigenvalues and, sometimes, eigenvectors. Control theory, vibration
More informationSimple Harmonic Oscillator
Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in
More informationNumerical Solution of Heat Equation by Spectral Method
Applied Mathematical Sciences, Vol 8, 2014, no 8, 397-404 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ams201439502 Numerical Solution of Heat Equation by Spectral Method Narayan Thapa Department
More informationQuantum algebraic structures compatible with the harmonic oscillator Newton equation
J. Phys. A: Math. Gen. 32 (1999) L371 L376. Printed in the UK PII: S0305-4470(99)04123-2 LETTER TO THE EDITOR Quantum algebraic structures compatible with the harmonic oscillator Newton equation Metin
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationAn Exactly Solvable 3 Body Problem
An Exactly Solvable 3 Body Problem The most famous n-body problem is one where particles interact by an inverse square-law force. However, there is a class of exactly solvable n-body problems in which
More informationSeries solutions of second order linear differential equations
Series solutions of second order linear differential equations We start with Definition 1. A function f of a complex variable z is called analytic at z = z 0 if there exists a convergent Taylor series
More informationHyperbolic Functions and. the Heat Balance Integral Method
Nonl. Analysis and Differential Equations, Vol. 1, 2013, no. 1, 23-27 HIKARI Ltd, www.m-hikari.com Hyperbolic Functions and the Heat Balance Integral Method G. Nhawu and G. Tapedzesa Department of Mathematics,
More informationProx-Diagonal Method: Caracterization of the Limit
International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, 403-412 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.8639 Prox-Diagonal Method: Caracterization of the Limit M. Amin
More informationMassachusetts Institute of Technology Physics Department
Massachusetts Institute of Technology Physics Department Physics 8.32 Fall 2006 Quantum Theory I October 9, 2006 Assignment 6 Due October 20, 2006 Announcements There will be a makeup lecture on Friday,
More information15. Hamiltonian Mechanics
University of Rhode Island DigitalCommons@URI Classical Dynamics Physics Course Materials 2015 15. Hamiltonian Mechanics Gerhard Müller University of Rhode Island, gmuller@uri.edu Creative Commons License
More informationExact Solution of an Ekpyrotic Fluid and a Primordial Magnetic Field in an Anisotropic Cosmological Space-Time of Petrov D
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 12, 601-608 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7835 Exact Solution of an Ekpyrotic Fluid and a Primordial Magnetic
More informationImprovements in Newton-Rapshon Method for Nonlinear Equations Using Modified Adomian Decomposition Method
International Journal of Mathematical Analysis Vol. 9, 2015, no. 39, 1919-1928 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.54124 Improvements in Newton-Rapshon Method for Nonlinear
More informationSolutions for the Combined sinh-cosh-gordon Equation
International Journal of Mathematical Analysis Vol. 9, 015, no. 4, 1159-1163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.556 Solutions for the Combined sinh-cosh-gordon Equation Ana-Magnolia
More informationThe Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces
Applied Mathematical Sciences, Vol. 11, 2017, no. 12, 549-560 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.718 The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive
More informationDouble Contraction in S-Metric Spaces
International Journal of Mathematical Analysis Vol. 9, 2015, no. 3, 117-125 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2015.1135 Double Contraction in S-Metric Spaces J. Mojaradi Afra
More informationA Note on the Carlitz s Type Twisted q-tangent. Numbers and Polynomials
Applied Mathematical Sciences, Vol. 12, 2018, no. 15, 731-738 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ams.2018.8585 A Note on the Carlitz s Type Twisted q-tangent Numbers and Polynomials Cheon
More information1 The Quantum Anharmonic Oscillator
1 The Quantum Anharmonic Oscillator Perturbation theory based on Feynman diagrams can be used to calculate observables in Quantum Electrodynamics, like the anomalous magnetic moment of the electron, and
More informationLorentz-squeezed Hadrons and Hadronic Temperature
Lorentz-squeezed Hadrons and Hadronic Temperature D. Han, National Aeronautics and Space Administration, Code 636 Greenbelt, Maryland 20771 Y. S. Kim, Department of Physics and Astronomy, University of
More informationA Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths
International Journal of Mathematical Analysis Vol. 9, 05, no. 48, 387-406 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.589 A Present Position-Dependent Conditional Fourier-Feynman Transform
More informationThe Expansion of the Confluent Hypergeometric Function on the Positive Real Axis
Applied Mathematical Sciences, Vol. 12, 2018, no. 1, 19-26 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712351 The Expansion of the Confluent Hypergeometric Function on the Positive Real
More informationOn the Boundary Layer Flow of a Shear Thinning Liquid over a 2-Dimensional Stretching Surface
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 1, 25-36 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.71259 On the Boundary Layer Flow of a Shear Thinning Liquid over a 2-Dimensional
More informationNonexistence of Limit Cycles in Rayleigh System
International Journal of Mathematical Analysis Vol. 8, 014, no. 49, 47-431 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.4883 Nonexistence of Limit Cycles in Rayleigh System Sandro-Jose
More informationEntropy and Lorentz Transformations
published in Phys. Lett. A, 47 343 (990). Entropy and Lorentz Transformations Y. S. Kim Department of Physics, University of Maryland, College Par, Maryland 074 E. P. Wigner Department of Physics, Princeton
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.323: Relativistic Quantum Field Theory I PROBLEM SET 2 REFERENCES: Peskin and Schroeder, Chapter 2 Problem 1: Complex scalar fields Peskin and
More information2.1 Calculation of the ground state energy via path integral
Chapter 2 Instantons in Quantum Mechanics Before describing the instantons and their effects in non-abelian gauge theories, it is instructive to become familiar with the relevant ideas in the context of
More informationPartial factorization of wave functions for a quantum dissipative system
PHYSICAL REVIEW E VOLUME 57, NUMBER 4 APRIL 1998 Partial factorization of wave functions for a quantum dissipative system C. P. Sun Institute of Theoretical Physics, Academia Sinica, Beiing 100080, China
More informationA Numerical Solution of Classical Van der Pol-Duffing Oscillator by He s Parameter-Expansion Method
Int. J. Contemp. Math. Sciences, Vol. 8, 2013, no. 15, 709-71 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2013.355 A Numerical Solution of Classical Van der Pol-Duffing Oscillator by
More informationOn Regular Prime Graphs of Solvable Groups
International Journal of Algebra, Vol. 10, 2016, no. 10, 491-495 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2016.6858 On Regular Prime Graphs of Solvable Groups Donnie Munyao Kasyoki Department
More informationOn Positive Stable Realization for Continuous Linear Singular Systems
Int. Journal of Math. Analysis, Vol. 8, 2014, no. 8, 395-400 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.4246 On Positive Stable Realization for Continuous Linear Singular Systems
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationp-adic Feynman s path integrals
p-adic Feynman s path integrals G.S. Djordjević, B. Dragovich and Lj. Nešić Abstract The Feynman path integral method plays even more important role in p-adic and adelic quantum mechanics than in ordinary
More informationStatic Hydrodynamic Equation in 4d BSBM Theory
Advanced Studies in Theoretical Physics Vol. 8, 2014, no. 23, 1015-1020 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2014.49120 Static Hydrodynamic Equation in 4d BSBM Theory Azrul S. K.
More informationPID Controller Design for DC Motor
Contemporary Engineering Sciences, Vol. 11, 2018, no. 99, 4913-4920 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.810539 PID Controller Design for DC Motor Juan Pablo Trujillo Lemus Department
More informationEnergy spectrum inverse problem of q-deformed harmonic oscillator and WBK approximation
Journal of Physics: Conference Series PAPER OPEN ACCESS Energy spectrum inverse problem of q-deformed harmonic oscillator and WBK approximation To cite this article: Nguyen Anh Sang et al 06 J. Phys.:
More informationProblem 1: Lagrangians and Conserved Quantities. Consider the following action for a particle of mass m moving in one dimension
105A Practice Final Solutions March 13, 01 William Kelly Problem 1: Lagrangians and Conserved Quantities Consider the following action for a particle of mass m moving in one dimension S = dtl = mc dt 1
More informationRemarks on Fuglede-Putnam Theorem for Normal Operators Modulo the Hilbert-Schmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 1389-1396 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on Fuglede-Putnam Theorem for Normal Operators Modulo the
More informationHamilton-Jacobi Formulation of Supermembrane
EJTP 1, No. 33 (015) 149 154 Electronic Journal of Theoretical Physics Hamilton-Jacobi Formulation of Supermembrane M. Kh. Srour 1,M.Alwer and N. I.Farahat 1 Physics Department, Al Aqsa University, P.O.
More informationRestrained Weakly Connected Independent Domination in the Corona and Composition of Graphs
Applied Mathematical Sciences, Vol. 9, 2015, no. 20, 973-978 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.4121046 Restrained Weakly Connected Independent Domination in the Corona and
More informationInternational Mathematical Forum, Vol. 9, 2014, no. 36, HIKARI Ltd,
International Mathematical Forum, Vol. 9, 2014, no. 36, 1751-1756 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.411187 Generalized Filters S. Palaniammal Department of Mathematics Thiruvalluvar
More informationClifford Algebras and Their Decomposition into Conjugate Fermionic Heisenberg Algebras
Journal of Physics: Conference Series PAPER OPEN ACCESS Clifford Algebras and Their Decomposition into Conugate Fermionic Heisenberg Algebras To cite this article: Sultan Catto et al 016 J. Phys.: Conf.
More information