| 1. |
- Anderson, Dan, 1943, et al.
(författare)
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Approximate solution of a Thomas-Fermi model equation for bulk self-gravitating stellar objects in two dimensions
- 2017
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Ingår i: European journal of physics. - : Institute of Physics (IOP). - 0143-0807 .- 1361-6404. ; 38, s. 1-8
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Tidskriftsartikel (refereegranskat)abstract
- Direct variational methods are used to find simple approximate solutions of the Thomas–Fermi equations describing the properties of self-gravitating radially symmetric stellar objects both in the non-relativistic and ultra-relativistic cases. The approximate solutions are compared and shown to be in good agreement with exact and numerically obtained solutions.
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| 2. |
- Anderson, Dan, 1943, et al.
(författare)
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Galerkin approach to approximate solutions of some nonlinear oscillator equations
- 2010
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Ingår i: American Journal of Physics. - : American Association of Physics Teachers (AAPT). - 0002-9505 .- 1943-2909. ; 78:9, s. 920-924
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Tidskriftsartikel (refereegranskat)abstract
- An analysis based on the Galerkin method is given of some nonlinear oscillator equations that have been analyzed by several other methods, including harmonic balance and direct variational methods. The present analysis is shown to provide simple yet accurate approximate solutions of these nonlinear equations and illustrates the usefulness and the power of the Galerkin method. (C) 2010 American Association of Physics Teachers. [DOI: 10.1119/1.3429974]
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| 3. |
- Anderson, Dan, 1943, et al.
(författare)
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Introduction to direct variational and moment methods and an application to the Child-Langmuir law
- 2015
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Ingår i: European Journal of Physics. - : IOP Publishing. - 1361-6404 .- 0143-0807. ; 2015:36, s. 065032-
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Tidskriftsartikel (refereegranskat)abstract
- A short introduction is given of direct variational methods and its relation to Galerkin and moment methods, all flexible and powerful approaches for finding approximate solutions of difficult physical equations. A pedagogicalapplication of moment methods is given to the physically and technically important Child–Langmuir law in electron physics. The analysis is shown to provide simple, yet accurate, approximate solutions of the two-dimensionalproblem (a problem which does not allow an exact analytical solution)and illustrates the usefulness and the power of moment methods.
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| 4. |
- Anderson, Dan, 1943, et al.
(författare)
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The least uncomfortable journey from A to B
- 2016
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Ingår i: American Journal of Physics. - : American Association of Physics Teachers (AAPT). - 0002-9505 .- 1943-2909. ; 84:9, s. 690-695
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Tidskriftsartikel (refereegranskat)abstract
- A short introduction is given about direct variational methods and their relation to Galerkin and moment methods, all flexible and powerful approaches for finding approximate solutions to difficult physicalequations. An application of these methods is given in the form of the variational problem of minimizing the discomfort experienced during different journeys, between two fixed horizontal points while keeping the travel time constant. The analysis is shown to provide simple, yet accurate, approximate solutions of the problem and illustrates the usefulness and the power of direct variational and moment methods. It also demonstrates the problem of a priori assessing the accuracy of the approximate solutions and illustrates that the variational solution does not necessarily provide a more accurate solution than that obtained by moment methods.
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| 5. |
- Anderson, D., et al.
(författare)
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The optimal journey from A to B
- 2008
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Ingår i: American Journal of Physics. - : American Association of Physics Teachers. - 0002-9505 .- 1943-2909. ; 76:9, s. 863-866
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Tidskriftsartikel (refereegranskat)abstract
- How fast can you comfortably travel between two points A and B? This question is formulated as a minimization problem of a functional where the discomfort is quantified in terms of the integral of the square of the acceleration between A and B. The problem is solved in terms of the corresponding Euler-Lagrange equation and approximately using a direct variational approach based on trial functions and Ritz optimization. The main purpose of the analysis is to introduce undergraduate students to variational calculus in an interesting and pedagogical way.
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| 6. |
- Andersson, Dan, 1966, et al.
(författare)
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Wave-breaking-free pulses in nonlinear-optical fibers
- 1993
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Ingår i: Journal of the Optical Society of America B: Optical Physics. - 1520-8540 .- 0740-3224. ; 10:7, s. 1185-1190
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Tidskriftsartikel (refereegranskat)abstract
- A qualitative as well as quantitative investigation is made of the conditions for avoiding wave breaking during pulse propagation in optical fibers. In particular, it is shown that pulses having a parabolic intensity variation are approximate wave-breaking-free solutions of the nonlinear Schrodinger equation in the high-intensity limit. A simple expression for the compression factor of a fiber-grating compressor based on parabolic pulses is also derived.
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| 7. |
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| 8. |
- Desaix, Mats, 1963
(författare)
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Contributions to the Study of Nonlinear Optics
- 1994
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Nonlinear optics, the study of how high intensity light propagates through and interacts with matter, is a subject which is rich in scientific problems and offers many technologically important applications. In this thesis, a number of physical and mathematical problems concerning nonlinear Kerr media are addressed. The papers of the thesis can be divided into four categories. (i) Nonlinear pulse propagation in optical fibers. Soliton pulses can be established in optical fibers but require high peak intensities for short pulse durations. The possibility to obtain low intensity solitons at the zero dispersion wavelength has been examined. A phenomenon which tends to deteriorate the obtainable compression factor in a fiber grating compressor is optical wave breaking. Analytical estimates of the wave breaking distance for different pulse shapes have been obtained. Approximately wave-breaking free pulses which can be used to enhance the compression factor have also been found. (ii) Phase-shift-keying coherent optical transmission systems. An approximate analytical expression has been derived for the intensity modulation index, accounting for both group-velocity dispersion and higher order dispersion. (iii) Optical beam propagation in a Kerr nonlinear medium. Beam propagation has traditionally been analyzed by means of the aberrationless paraxial-ray approximation method. This method sometimes gives poor results and it is demonstrated that a direct variational approach based on trial functions and a subsequent Ritz optimization procedure is a more reliable method for obtaining approximate solutions. (iv) Analysis. The accuracy of a direct variational approach is known to depend crucially on the choice of the trial functions. It is demonstrated that the ability to preserve invariants can be used as an a priori qualitative assessment of the accuracy. The nonlinear Schrödinger equation can, according to the inverse scattering transform method, be solved by essentially linear methods. However, the discrete eigenvalues of the Zakharov-Shabat scattering problem, at the heart of the method, can only be found in closed form for a very restricted set of initial conditions. The possibility of using direct variational methods to obtain good approximations to the discrete eigenvalues has been examined.
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| 9. |
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| 10. |
- Desaix, Mats, et al.
(författare)
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Nonlinear Schrödinger Solitons with non-zero velocities emerging from real symmetric initial conditions
- 2009
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Konferensbidrag (refereegranskat)abstract
- Solutions of the nonlinear Schrodinger equation for initial conditions in the form of two separated sech-shaped in-phase pulsed,; are analyzed. It is found that; this initial condition, with appropriate amplitude, may give rise to, not; only stationary solitons, but also to symmetrically separating solitons, if the initial distance of separation is large enough. The condition for the generation of a separating soliton pair is derived from the Zakharov-Shabat eigenvalue problem using a variational approach.
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