Journal articles: 'B-evolution Equations'

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Banasiak, J. "B-bounded semigroups and implicit evolution equations." Abstract and Applied Analysis 5, no. 1 (2000): 13–32. http://dx.doi.org/10.1155/s1085337500000087.

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The aim of this paper is to show an application of the recently introducedB-bounded semigroups in the theory of implicit and degenerate evolution equations. The most interesting feature of this approach is its applicability to problems with noncloseable operators.

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Dalsen, Marié Grobbelaar-Van. "Semilinear evolution equations and fractional powers of a closed pair of operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 105, no. 1 (1987): 101–15. http://dx.doi.org/10.1017/s0308210500021946.

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SynopsisThe nonlinear evolution problem [Bu(t)]′ = A(t, Bu)u + f(t, Bu) with B a constant linear operator and A = A(t, Bu) a time-dependent nonlinear operator from one Banach space to another, is studied. Existence and uniqueness results are obtained by making use of the theory of B-evolutions and the fractional powers of A and B. Two examples are presented in which the theory is applied to nonlinear equations with dynamic boundary conditions.

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Qiao, Zhijun, and Walter Strampp. "L–A–B representation for nonlinear evolution equations and its applications." Physica A: Statistical Mechanics and its Applications 293, no. 1-2 (April 2001): 143–56. http://dx.doi.org/10.1016/s0378-4371(00)00593-8.

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Ahmed, N. U., and S. Kerbal. "Stochastic systems governed by B-evolutions on Hilbert spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 5 (1997): 903–20. http://dx.doi.org/10.1017/s0308210500026792.

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SynopsisIn this paper, we consider the question of existence of solutions and their regularity properties for a large class of stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions together with noisy boundary data. They cover also stochastic boundary value problems. Our results are illustrated by two practical examples.

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Gühring, Gabriele, and Frank Räbiger. "Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations." Abstract and Applied Analysis 4, no. 3 (1999): 169–94. http://dx.doi.org/10.1155/s1085337599000214.

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We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation(d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t∈ℝ, where(A,D(A))is a Hille-Yosida operator on a Banach spaceX,B(t),t∈ℝ, is a family of operators inℒ(D(A)¯,X)satisfying certain boundedness and measurability conditions andf∈L loc 1(ℝ,X). The solutions of the corresponding homogeneous equations are represented by an evolution family(UB(t,s))t≥s. For various function spacesℱwe show conditions on(UB(t,s))t≥sandfwhich ensure the existence of a unique solution contained inℱ. In particular, if(UB(t,s))t≥sisp-periodic there exists a unique bounded solutionusubject to certain spectral assumptions onUB(p,0),fandu. We apply the results to nonautonomous semilinear retarded differential equations. For certainp-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of(UB(t,s))t≥s.

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Bertsch, M., R. Dal Passo, and R. Kersner. "The evolution of turbulent bursts: the b—ε model." European Journal of Applied Mathematics 5, no. 4 (December 1994): 537–57. http://dx.doi.org/10.1017/s0956792500001601.

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We study the semi-empirical b—ε model which describes the time evolution of turbulent spots in the case of equal diffusivity of the turbulent energy density b and the energy dissipation rate ε. We prove that the system of two partial differential equations possesses a solution, and that after some time this solution exhibits self-similar behaviour, provided that the system has self-similar solutions. The existence of such self-similar solutions depends upon the value of a parameter of the model.

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Wang, Xiao Li, Jian Qin Mei, and Zhen Hua Wu. "Auto-Bäcklund Transformations Derived via the Implicit C-D Pair." Applied Mechanics and Materials 268-270 (December 2012): 1345–48. http://dx.doi.org/10.4028/www.scientific.net/amm.268-270.1345.

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By the implicit C-D pairs, a new approach to auto-B¨acklund transformations of nonlinear evolution equations is presented in this paper. The auto-B¨acklund transformations for combined KdV and MKdV equations are obtained by this approach.

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Mahmudov, N. I. "Existence and approximate controllability of Sobolev type fractional stochastic evolution equations." Bulletin of the Polish Academy of Sciences Technical Sciences 62, no. 2 (June 1, 2014): 205–15. http://dx.doi.org/10.2478/bpasts-2014-0020.

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Abstract We study the existence of mild solutions and the approximate controllability concept for Sobolev type fractional semilinear stochastic evolution equations in Hilbert spaces. We prove existence of a mild solution and give sufficient conditions for the approximate controllability. In particular, we prove that the fractional linear stochastic system is approximately controllable in [0, b] if and only if the corresponding deterministic fractional linear system is approximately controllable in every [s, b], 0 ≤ s < b. An example is provided to illustrate the application of the obtained results.

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Arlotti, Luisa. "A new characterization ofB-bounded semigroups with application to implicit evolution equations." Abstract and Applied Analysis 5, no. 4 (2000): 227–44. http://dx.doi.org/10.1155/s1085337501000331.

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We consider the one-parameter family of linear operators that A. Belleni Morante recently introduced and calledB-bounded semigroups. We first determine all the properties possessed by a couple(A,B)of operators if they generate aB-bounded semigroup(Y(t))t≥0. Then we determine the simplest further property of the couple(A,B)which can assure the existence of aC0-semigroup(T(t))t≥0such that for allt≥0,f∈D(B)we can writeY(t)f=T(t)Bf. Furthermore, we compare our result with the previous ones and finally we show how our method allows to improve the theory developed by Banasiak for solving implicit evolution equations.

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CONTE, R. "ANALYTIC PATTERNS FOR CHAOTIC EQUATIONS." International Journal of Modern Physics B 21, no. 23n24 (September 30, 2007): 3918–24. http://dx.doi.org/10.1142/s0217979207044962.

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Chaotic evolution equations sometimes display regular patterns (fronts, kinks, holes), which often correspond to closed form analytic solutions. In the dissipative-dispersive Kuramoto and Sivashinsky travelling wave reduction u(x − ct), νu‴ + bu″ + μu′ + u2/2 + A = 0, ν ≠ 0, with (ν,b,μ,A) constants, such analytic solutions are known for heteroclinic solutions, but one has also observed (Toh, 1987) homoclinic solutions without corresponding analytic solutions yet. We review the search for the most general analytic solution admissible by this chaotic differential equation. Several investigations, both analytic by the Painlevé test (Thual and Frisch, 1986) and numerical by Padé approximants (Yee, Conte, Musette, 2003) indicate its quite probable single valuedness for any (ν,b,μ,A). Moreover, Nevanlinna theory on the growth of solutions near infinity rules out (Eremenko, 2005) the possibility for the singularities of this unknown closed form single valued expression to be only poles. We propose a qualitative closed form involving a deformed elliptic function.

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Gou, Haide, and Baolin Li. "Existence of Mild Solutions for Sobolev-Type Hilfer Fractional Nonautonomous Evolution Equations with Delay." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 5 (July 26, 2018): 481–92. http://dx.doi.org/10.1515/ijnsns-2017-0183.

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AbstractThis paper treats the existence of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations with delay in Banach spaces. We first characterize the definition of mild solutions for the studied problem which was given based on an operator family generated by the operator pair (A,B) and probability density function. And then via Hilfer fractional derivative and combining the techniques of fractional calculus, measure of noncompactness and Sadovskii fixed-point theorem, we obtain new existence result of mild solutions for Sobolev-type Hilfer fractional nonautonomous evolution equations. Particularly, the existence or compactness of an operator $B^{-1} $ is not necessarily needed in our results. Furthermore, our results obtained improve and extend some related conclusions on this topic. At last, an example is given to illustrate our main results.

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Eikmeier, André, Etienne Emmrich, and Hans-Christian Kreusler. "Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability." Computational Methods in Applied Mathematics 20, no. 1 (January 1, 2020): 89–108. http://dx.doi.org/10.1515/cmam-2018-0268.

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AbstractThe initial value problem for an evolution equation of type {v^{\prime}+Av+BKv=f} is studied, where {A:V_{A}\to V_{A}^{\prime}} is a monotone, coercive operator and where {B:V_{B}\to V_{B}^{\prime}} induces an inner product. The Banach space {V_{A}} is not required to be embedded in {V_{B}} or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

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POGUTSE, IGOR O., ANDREI I. SMOLYAKOV, and AKIRA HIROSE. "Magnetohydrodynamic equations for plasmas with finite-Larmor-radius effects." Journal of Plasma Physics 60, no. 1 (August 1998): 133–49. http://dx.doi.org/10.1017/s0022377898006552.

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It is shown that the truncation of the infinite hierarchy of fluid equations obtained as moments of the Vlasov kinetic equation leads to a system of nonlinear equations that describe finite-Larmor-radius effects with good accuracy. Inertial terms in the momentum balance, viscosity and heat-flux evolution equations are crucial for a uniform description of the plasma response with an arbitrary Larmor radius. In the low-frequency ordering, the obtained equations are simplified by an expansion in the parameter 1/B, where B is the equilibrium magnetic field. The results of the second-order [Oscr ](1/B2) and the fourth-order [Oscr ](1/B4) closures are compared. It is shown that the accuracy of the description improves for higher-order closures.

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Meškauskas, Tadas, and Feliksas Ivanauskas. "Role of parabolic viscosity in numerical analysis of derivative nonlinear evolution equations." Nonlinear Analysis: Modelling and Control 2 (December 21, 1998): 75–80. http://dx.doi.org/10.15388/na.1998.2.0.15288.

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We consider the difference schemes applied to a derivative nonlinear system of evolution equations. For the boundary-value problem with initial conditions ∂u/∂t = A ∂2u/∂x2 + B ∂u/∂x + f(x,u) + g(x,u) ∂u/∂x, (t,x) ∈ (0, T] x (0, 1),u(t,0) = u(t,1) = 0, t ∈ [0, T],u(0,x) = u(0)(x), x ∈ (0, 1) we use the Crank-Nicolson discretization. A is complex and B – real diagonal matrixes; u, f and g are complex vector-functions. The analysis shows that proposed schemes are uniquely solvable, convergent and stable in a grid norm W22 if all (diagonal) elements in Re(A) are positive. This is true without any restriction on the ratio of time and space grid steps.

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Jolly, Michael S. "Exponential Attractors for Dissipative Evolution Equations (A. Eden, C. Foias, B. Nicolaenko, and R. Témam)." SIAM Review 38, no. 1 (March 1996): 171–72. http://dx.doi.org/10.1137/1038025.

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16

Mao, Jin-Jin, Shou-Fu Tian, Xing-Jie Yan, and Tian-Tian Zhang. "Lump solutions and interaction phenomena of the (3 + 1)-dimensional nonlinear evolution equations." International Journal of Numerical Methods for Heat & Fluid Flow 29, no. 9 (September 2, 2019): 3417–36. http://dx.doi.org/10.1108/hff-02-2019-0160.

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Purpose The purpose of this study is to examine the lump solutions of the (3 + 1)-dimensional nonlinear evolution equations by considering a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and a (3 + 1)-dimensional variable-coefficient generalized B-type Kadomtsev–Petviashvili (vcgBKP) equation as examples. Design/methodology/approach Based on Hirota’s bilinear theory, a direct method is used to examine the lump solutions of these two equations. Findings The complete non-elastic interaction solutions between a lump and a stripe are also discussed for the equations, which show that the lump solitons are swallowed by the stripe solitons. Originality/value The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional KP-type nonlinear wave equations.

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WEBB, G. M., A. ZAKHARIAN, M. BRIO, and G. P. ZANK. "Wave interactions in magnetohydrodynamics, and cosmic-ray-modified shocks." Journal of Plasma Physics 61, no. 2 (February 1999): 295–346. http://dx.doi.org/10.1017/s0022377898007399.

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Multiple-scales perturbation methods are used to study wave interactions in magnetohydrodynamics (MHD), in one Cartesian space dimension, with application to cosmic-ray-modified shocks. In particular, the problem of the propagation and interaction of short wavelength MHD waves, in a large-scale background flow, modified by cosmic rays is studied. The wave interaction equations consist of seven coupled evolution equations for the backward and forward Alfvén waves, the backward and forward fast and slow magnetoacoustic waves and the entropy wave. In the linear wave regime, the waves are coupled by wave mixing due to gradients in the background flow, cosmic-ray squeezing instability effects, and damping due to the diffusing cosmic rays. In the most general case, the evolution equations also contain nonlinear wave interaction terms due to Burgers self wave steepening for the magnetoacoustic modes, resonant three wave interactions, and mean wave field interaction terms. The form of the wave interaction equations in the ideal MHD case is also discussed. Numerical simulations of the fully nonlinear cosmic ray MHD model equations are compared with spectral code solutions of the linear wave interaction equations for the case of perpendicular, cosmic-ray-modified shocks. The solutions are used to illustrate how the different wave modes can be generated by wave mixing, and the modification of the cosmic ray squeezing instability due to wave interactions. It is shown that the Alfvén waves are coupled to the magnetoacoustic and entropy waves due to linear wave mixing, only in background flows with non-zero field aligned electric current and/or vorticity (i.e. if B·∇×B≠0 and/or B·∇×u≠0, where B and u are the magnetic field induction and fluid velocity respectively).

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WEBB, G. M., Q. HU, B. DASGUPTA, and G. P. ZANK. "Double Alfvén waves." Journal of Plasma Physics 78, no. 1 (October 17, 2011): 71–85. http://dx.doi.org/10.1017/s0022377811000420.

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AbstractDouble Alfvén wave solutions of the magnetohydrodynamic equations in which the physical variables (the gas density ρ, fluid velocity u, gas pressure p, and magnetic field induction B) depend only on two independent wave phases ϕ1(x,t) and ϕ2(x,t) are obtained. The integrals for the double Alfvén wave are the same as for simple waves, namely, the gas pressure, magnetic pressure, and group velocity of the wave are constant. Compatibility conditions on the evolution of the magnetic field B due to changes in ϕ1 and ϕ2, as well as constraints due to Gauss's law ∇ · B = 0 are discussed. The magnetic field lines and hodographs of B in which the tip of the magnetic field B moves on the sphere |B| = B = const. are used to delineate the physical characteristics of the wave. Hamilton's equations for the simple Alfvén wave with wave normal n(ϕ), and with magnetic induction B(ϕ) in which ϕ is the wave phase, are obtained by using the Frenet–Serret equations for curves x=X(ϕ) in differential geometry. The use of differential geometry of 2D surfaces in a 3D Euclidean space to describe double Alfvén waves is briefly discussed.

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BEDNAREK, ILONA, and RYSZARD MAŃKA. "THE SOLITON STARS EVOLUTION." International Journal of Modern Physics D 07, no. 02 (April 1998): 225–36. http://dx.doi.org/10.1142/s0218271898000188.

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The evolution of a soliton star filled with fermions is studied in the framework of general relativity. Such a system could be described by the surface tension σ0, the bag constant B, and the fermion number density ρ. One of the parameters mentioned above could prevail in the system and thus affect the spacetime inside the bubble. Whether it is described by Friedman or de Sitter metric depends on the prevailing parameter. In broad outline, the whole spacetime is divided by the surface of the soliton into the false vacuum region inside the soliton and the true vacuum region outside, the latter being described by the Schwarzschild line element. The aim of this paper is to study the equations of motion of the domain wall in two cases. In both of them, the Schwarzschild metric is outside the soliton. The de Sitter metric describes the interior in the first case, and in the second case, it is replaced by the Friedman metric. Analyzing obtained equations, one can draw conclusions concerning the further evolution of the soliton star.

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Yang, Xin, Qiang Xu, Zhong Yu Lu, and Simon Barrans. "Preliminary Review of the Influence of Cavitation Behavior in Creep Damage Constitutive Equations." Advanced Materials Research 940 (June 2014): 46–51. http://dx.doi.org/10.4028/www.scientific.net/amr.940.46.

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This review paper mainly consists of from two aspects: (a) the evolution of the cavitation damage equation from Dyson to current application in high Cr steels by traditional techniques; (b) quantitation analyses of cavitation behavior in brass, copper, dual phase steel from X-ray microtomtograph. Though there is a lack of experimental data for high Cr steels by X-ray microtomography currently, but (b) has provided reference value for studying creep cavitation behavior in high Cr steels. This paper will be the fundamental of development new creep damage constitutive equation through quantitation analyses of X-ray tomography.

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Norouzi, Fatemeh, and Gaston M. N’guérékata. "Existence results to a ψ- Hilfer neutral fractional evolution equation with infinite delay." Nonautonomous Dynamical Systems 8, no. 1 (January 1, 2021): 101–24. http://dx.doi.org/10.1515/msds-2020-0128.

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Abstract In this paper, we prove the existence and uniqueness of a mild solution to the system of ψ- Hilfer neutral fractional evolution equations with infinite delay H 𝔻0 αβ;ψ [x(t) − h(t, xt )] = A x(t) + f (t, x(t), xt ), t ∈ [0, b], b > 0 and x(t) = ϕ(t), t ∈ (−∞, 0]. We first obtain the Volterra integral equivalent equation and propose the mild solution of the system. Then, we prove the existence and uniqueness of solution by using the Banach contraction mapping principle and the Leray-Schauder alternative theorem.

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Crouseilles, Nicolas, and Thomas Respaud. "A Charge Preserving Scheme for the Numerical Resolution of the Vlasov-Ampère Equations." Communications in Computational Physics 10, no. 4 (October 2011): 1001–26. http://dx.doi.org/10.4208/cicp.210410.211210a.

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AbstractIn this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson’s equation, or equivalently under charge conservation, the Ampère’s one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.

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Landrini, M., G. Grytøyr, and O. M. Faltinsen. "A B-Spline Based BEM for Unsteady Free-Surface Flows." Journal of Ship Research 43, no. 01 (March 1, 1999): 13–24. http://dx.doi.org/10.5957/jsr.1999.43.1.13.

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Fully nonlinear free-surface flows are numerically studied in the framework of the potential theory. The problem is formulated in terms of boundary integral equations which are solved by means of an arbitrary high-order boundary element method based on B-Spline representation of both the geometry and the fluid dynamic variables along the domain boundary. The solution is stepped forward in time either by following Lagrangian points attached to the free surface or by a less conventional scheme in which evolution equations for the B-Spline coefficients are integrated in time. Numerical examples for inner and outer free-surface flows are shown. The accuracy of the numerical solution is assessed either by checking mass and energy conservation or by comparing with reference solutions. Good results are generally obtained. Extended use of the developed algorithm to more applied problems in the context of naval hydrodynamics is now under development.

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Lovegrove, A. F., I. M. Moroz, and P. L. Read. "Bifurcations and instabilities in rotating, two-layer fluids: II. β-plane." Nonlinear Processes in Geophysics 9, no. 3/4 (August 31, 2002): 289–309. http://dx.doi.org/10.5194/npg-9-289-2002.

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Abstract. In this paper, we show that the behavior of weakly nonlinear waves in a 2-layer model of baroclinic instability on a b-plane with varying viscosity is determined by a single, degenerate codimension three bifurcation. In the process, we show how previous studies, using the method of multiple scales to derive evolution equations for the slowly varying amplitude of the growing wave, arise as special limits of the general evolution description.

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NASHINE, HEMANT KUMAR, HE YANG, and RAVI P. AGARWAL. "Fractional evolution equations with nonlocal conditions in partially ordered Banach space." Carpathian Journal of Mathematics 34, no. 3 (2018): 379–90. http://dx.doi.org/10.37193/cjm.2018.03.13.

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In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form where C Dσ t denotes the Caputo fractional derivative of order σ ∈ (0, 1), −A : D(A) ⊂ X → X generates a positive C0-semigroup T(t)(t ≥ 0) of uniformly bounded linear operator in X, b > 0 is a constant, f is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone K is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.

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BEUF, GUILLAUME. "HIGH-ENERGY FACTORIZATION AND EVOLUTION WITH IMPROVED KINEMATICS." International Journal of Modern Physics: Conference Series 04 (January 2011): 56–66. http://dx.doi.org/10.1142/s2010194511001565.

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The high-energy factorization and the associated B-JIMWLK or BK evolution equations are presented, using the example of DIS structure functions. The necessity of taking gluon saturation into account is discussed, and also the various approximations underlying high-energy factorization. The appearance of large NLL corrections in such a framework with or without gluon saturation is recalled, and their physical origin is explained. Finally, old and new results are presented about the resummation of some of those large corrections, related to kinematical approximations.

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Carvalho, Ricardo Nolasco, Marcelo A. C. Ferreira, Dagoberto Brandão Santos, and Ronaldo Barbosa. "Mathematical Modeling of Microstructure Evolution of V Steels during Hot Rolling of Seamless Tubes." Materials Science Forum 638-642 (January 2010): 2537–42. http://dx.doi.org/10.4028/www.scientific.net/msf.638-642.2537.

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Torsion and compression testing have been used to simulate microstructure evolution of industry processes. Additionally, mathematical modeling of the industry hot rolling processes has been carried out by several researchers. These models employed equations published in the literature describing kinetics of softening, grain size evolution and grain growth. Validation of the models was carried, in some cases, by comparing the microstructure or the average stress per pass, the latter as calculated from industry rolling mill loads. In the present work, torsion simulation and industry trial results were used to validate the mathematical model presented. Equations used in the model were mostly taken from literature and appropriate modifications were implemented concerning basically two points: a) the transfer time between CMM and SRM, a step in the production line typical for seamless rolling and rather unusual for other industry rolling processes and b) the chemical composition used in tube rolling industry where C equivalent values are usually higher than those used in the rolling of flats.

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Wodarz, Dominik, and David N. Levy. "Correction for Wodarz and Levy, Human immunodeficiency virus evolution towards reduced replicative fitness in vivo and the development of AIDS." Proceedings of the Royal Society B: Biological Sciences 275, no. 1653 (December 22, 2008): 2897. http://dx.doi.org/10.1098/rspb.2008.3001.

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Correction for ‘Human immunodeficiency virus evolution towards reduced replicative fitness in vivo and the development of AIDS’ by Dominik Wodarz and David N. Levy (Proc. R. Soc. B 274 , 2481–2490. (doi: 10.1098/rspb.2007.0413 )). On p. 2482, equations (2.1), (2.2) and the equation in the penultimate sentence of the paragraph following equation (2.1) as well as references to a x in sections 2a, 2b and 3b were presented incorrectly.

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Álvarez-Estrada, Ramon. "Non-Equilibrium Liouville and Wigner Equations: Classical Statistical Mechanics and Chemical Reactions for Long Times." Entropy 21, no. 2 (February 14, 2019): 179. http://dx.doi.org/10.3390/e21020179.

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We review and improve previous work on non-equilibrium classical and quantum statistical systems, subject to potentials, without ab initio dissipation. We treat classical closed three-dimensional many-particle interacting systems without any “heat bath” ( h b ), evolving through the Liouville equation for the non-equilibrium classical distribution W c , with initial states describing thermal equilibrium at large distances but non-equilibrium at finite distances. We use Boltzmann’s Gaussian classical equilibrium distribution W c , e q , as weight function to generate orthogonal polynomials ( H n ’s) in momenta. The moments of W c , implied by the H n ’s, fulfill a non-equilibrium hierarchy. Under long-term approximations, the lowest moment dominates the evolution towards thermal equilibrium. A non-increasing Liapunov function characterizes the long-term evolution towards equilibrium. Non-equilibrium chemical reactions involving two and three particles in a h b are studied classically and quantum-mechanically (by using Wigner functions W). Difficulties related to the non-positivity of W are bypassed. Equilibrium Wigner functions W e q generate orthogonal polynomials, which yield non-equilibrium moments of W and hierarchies. In regimes typical of chemical reactions (short thermal wavelength and long times), non-equilibrium hierarchies yield approximate Smoluchowski-like equations displaying dissipation and quantum effects. The study of three-particle chemical reactions is new.

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Morel, P. "The Evolutionary Code Cesam: Numerical Techniques." International Astronomical Union Colloquium 137 (1993): 445–47. http://dx.doi.org/10.1017/s0252921100018200.

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AbstractCESAM is a consistent set of programs and routines designed for the calculations of stellar evolution. Untill now it allows the computation of the evolution from PMS or ZAMS to helium flash for stellar masses of some solar mass. It is constructed in such a way that all the physics works as external routines. The numerical techniques are based on the B-spline formalism. This formalism used both for the integration of the differential equations and for 1D and 2D interpolation schemes of various tables of physical data.

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Huy, Nguyen, and Pham Bang. "Dichotomy and positivity of neutral equations with nonautonomous past." Applicable Analysis and Discrete Mathematics 8, no. 2 (2014): 224–42. http://dx.doi.org/10.2298/aadm140816015h.

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Consider the linear partial neutral functional differential equations with nonautonomous past of the form (?/?t) F(u(t, ?)) = BFu(t, ?) + ?u(t, ?), t ? 0; (? / ?t) u(t, s) = (? / ?s) u(t, s) + A(s)u(t, s), t ? 0 ? s, where the function u(?, ?) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator ? we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U(t, s))t?s?0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (etB)t?0 on X. Furthermore, under the positivity conditions on (etB)t?0, U, F and ? we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup.

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Xiong, Na, Wen-Tao Li, Biao Li, and Zine El Abiddine Fellah. "Weakly Coupled B-Type Kadomtsev-Petviashvili Equation: Lump and Rational Solutions." Advances in Mathematical Physics 2020 (September 28, 2020): 1–8. http://dx.doi.org/10.1155/2020/6185391.

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Through the method of Z N -KP hierarchy, we propose a new ( 3 + 1 )-dimensional weakly coupled B-KP equation. Based on the bilinear form, we obtain the lump and rational solutions to the dimensionally reduced cases by constructing a symmetric positive semidefinite matrix. Then, we do numerical analysis on the rational solutions and fit the trajectory equation of the crest. Furthermore, we verify the accuracy of the trajectory equation by numerical analysis. This method of solving the lump and rational solutions can also be applied to other nonlinear evolution equations.

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KLEIDIS, KOSTAS, APOSTOLOS KUIROUKIDIS, DEMETRIOS PAPADOPOULOS, and LOUKAS VLAHOS. "DYNAMO EFFECTS IN MAGNETIZED IDEAL PLASMA COSMOLOGIES." International Journal of Modern Physics A 23, no. 11 (April 30, 2008): 1697–710. http://dx.doi.org/10.1142/s0217751x08039542.

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The excitation of cosmological perturbations in an anisotropic cosmological model and in the presence of a homogeneous magnetic field has been studied, using the ideal magnetohydrodynamic (MHD) equations. In this case, the system of partial differential equations which governs the evolution of the magnetized cosmological perturbations can be solved analytically. Our results verify that fast-magnetosonic modes propagating normal to the magnetic field, are excited. But, what is most important, is that, at late times, the magnetic-induction contrast(δB/B) grows, resulting in the enhancement of the ambient magnetic field. This process can be particularly favored by condensations, formed within the plasma fluid due to gravitational instabilities.

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Jovanović, D., H. L. Pécseli, J. Juul Rasmussen, and K. Thomsen. "Modified convective cells in plasmas." Journal of Plasma Physics 37, no. 1 (February 1987): 81–95. http://dx.doi.org/10.1017/s0022377800012009.

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Two coupled equations describing the nonlinear evolution of modified convective cells with a finite but small wavenumber parallel to the external magnetic field B are derived. It is found that, for oblique propagation, the electrostatic and magnetostatic convective mode do not decouple as in the strictly perpen-dicular case. Various aspects of the governing equations are investigated. In the weakly nonlinear lirait, a three-wave interaction study shows the possibility of dual energy cascading, or of cascading to larger wavenumbers only, depending on the direction of the parallel wavenumber component of the pump wave. Furthermore, solitary wave solutions in the form of double vortices are considered.

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Jendoubi, Ch. "On the theory of integral manifolds for some delayed partial differential equations with nondense domain}." Ukrains’kyi Matematychnyi Zhurnal 72, no. 6 (June 17, 2020): 776–89. http://dx.doi.org/10.37863/umzh.v72i6.6020.

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UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation ⅆ u ⅆ t = ( A + B ( t ) ) u ( t ) + f ( t , u t ) , t ∈ R , ( 1 ) where ( A , D ( A ) ) satisfies the Hille – Yosida condition, ( B ( t ) ) t ∈ R is a family of operators in ℒ ( D ( A ) ¯ , X ) satisfying some measurability and boundedness conditions, and the nonlinear forcing term f satisfies ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 , here, φ belongs to some admissible spaces and ϕ , ψ ∈ 𝒞 : = C ( [ - r ,0 ] , X ) . We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.

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Chen, Wei, Hanlin Chen, and Zhengde Dai. "Breather-type kink wave of the (2+1)-dimensional BKP equation." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 1 (January 5, 2015): 146–52. http://dx.doi.org/10.1108/hff-02-2013-0046.

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Purpose – The purpose of this paper is to find solutions for the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation and to research the quality of B-type Kadomtsev-Petviashvili equation. Design/methodology/approach – The authors apply the extended three-wave approach and the homoclinic test technique to solve the B-type Kadomtsev-Petviashvili equation. Findings – The authors obtain breather type of cross-kink solutions, doubly breather type of kink solitary solutions and the breather type of kink wave solutions for B-type Kadomtsev-Petviashvili equation. Research limitations/implications – As nonlinear evolution equations are characterized by rich dynamical behaviors, the authors have just found some of them and others are still to be found. Originality/value – These results may help us to investigate the local structure and the interaction of waves in high-dimensional models.

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Rouhani, Behzad Djafari, and Mohsen Rahimi Piranfar. "Nonhomogeneous nonlinear oscillator with damping: asymptotic analysis in continuous and discrete time." Demonstratio Mathematica 52, no. 1 (September 13, 2019): 274–82. http://dx.doi.org/10.1515/dema-2019-0028.

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AbstractWe consider the following second order evolution equation modelling a nonlinear oscillator with damping$$\ddot{u} (t) + \gamma \dot u(t) + Au\left( t \right) = f\left( t \right),\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\rm{SEE}}} \right)$$where A is a maximal monotone and α-inverse strongly monotone operator in a real Hilbert space H. With suitable assumptions on γ and f(t) we show that A−1(0) ≠ ∅, if and only if (SEE) has a bounded solution and in this case we provide approximation results for elements of A−1(0) by proving weak and strong convergence theorems for solutions to (SEE) showing that the limit belongs to A−1(0). As a discrete version of (SEE), we consider the following second order difference equation$${u_{n + 1}} - {u_n} - {\alpha _n}\left( {{u_n} - {u_{n - 1}}} \right) + {\lambda _n}A{u_{n + 1}\ni} f\left( t \right),$$where A is assumed to be only maximal monotone (possibly multivalued). By using the results in [Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417], we prove ergodic, weak and strong convergence theorems for the sequence un, and show that the limit is the asymptotic center of un and belongs to A−1(0). This again shows that A−1(0) ≠ ∅ if and only if un is bounded. Also these results solve an open problem raised in [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11], namely the study of the convergence results for the inexact inertial proximal algorithm. Our paper is motivated by the previous results in [Djafari Rouhani B., Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl., 1990, 147, 465–476; Djafari Rouhani B., Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl., 1990, 151, 226–235; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to some second order evolution systems, Rocky Mountain J. Math., 2010, 40, 1289–1311; Djafari Rouhani B., Khatibzadeh H., A strong convergence theorem for solutions to a nonhomogeneous second order evolution equation, J. Math. Anal. Appl., 2010, 363, 648–654; Djafari Rouhani B., Khatibzadeh H., Asymptotic behavior of bounded solutions to a class of second order nonhomogeneous evolution equations, Nonlinear Anal., 2009, 70, 4369–4376; Djafari Rouhani B., Khatibzadeh H., On the proximal point algorithm, J. Optim. Theory Appl., 2008, 137, 411–417] and significantly improves upon the results of [Attouch H., Maingé P. E., Asymptotic behavior of second-order dissipative evolution equations combining potential with non-potential effects, ESAIM Control Optim. Calc. Var., 2011, 17(3), 836–857], and [Alvarez F., Attouch H., An inertial proximal method for maximal monotone operators via dicretization of a nonlinear oscillator with damping, Set Valued Anal., 2001, 9, 3–11].

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Wirbel, Anna, and Alexander Helmut Jarosch. "Inequality-constrained free-surface evolution in a full Stokes ice flow model (<i>evolve_glacier v1.1</i>)." Geoscientific Model Development 13, no. 12 (December 21, 2020): 6425–45. http://dx.doi.org/10.5194/gmd-13-6425-2020.

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Abstract. Like any gravitationally driven flow that is not constrained at the upper surface, glaciers and ice sheets feature a free surface, which becomes a free-boundary problem within simulations. A kinematic boundary condition is often used to describe the evolution of this free surface. However, in the case of glaciers and ice sheets, the naturally occurring constraint that the ice surface elevation (S) cannot fall below the bed topography (B) (S-B≥0), in combination with a non-zero mass balance rate complicates the matter substantially. We present an open-source numerical simulation framework to simulate the free-surface evolution of glaciers that directly incorporates this natural constraint. It is based on the finite-element software package FEniCS solving the Stokes equations for ice flow and a suitable transport equation, i.e. “kinematic boundary condition”, for the free-surface evolution. The evolution of the free surface is treated as a variational inequality, constrained by the bedrock underlying the glacier or the topography of the surrounding ground. This problem is solved using a “reduced space” method, where a Newton line search is performed on a subset of the problem (Benson and Munson, 2006). Therefore, the “constrained” non-linear problem-solving capabilities of PETSc's (Portable, Extensible Toolkit for Scientific Computation, Balay et al., 2019) SNES (Scalable Non-linear Equations Solver) interface are used. As the constraint is considered in the solving process, this approach does not require any ad hoc post-processing steps to enforce non-negativity of ice thickness and corresponding mass conservation. The simulation framework provides the possibility to divide the computational domain into different subdomains so that individual forms of the relevant equations can be solved for different subdomains all at once. In the presented setup, this is used to distinguish between glacierised and ice-free regions. The option to chose different time discretisations, spatial stabilisation schemes and adaptive mesh refinement make it a versatile tool for glaciological applications. We present a set of benchmark tests that highlight that the simulation framework is able to reproduce the free-surface evolution of complex geometries under different conditions for which it is mass-conserving and numerically stable. Real-world glacier examples demonstrate high-resolution change in glacier geometry due to fully resolved 3D velocities and spatially variable mass balance rate, whereby realistic glacier recession and advance states can be simulated. Additionally, we provide a thorough analysis of different spatial stabilisation techniques as well as time discretisation methods. We discuss their applicability and suitability for different glaciological applications.

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Ayala, Diego, Charles R. Doering, and Thilo M. Simon. "Maximum palinstrophy amplification in the two-dimensional Navier–Stokes equations." Journal of Fluid Mechanics 837 (January 5, 2018): 839–57. http://dx.doi.org/10.1017/jfm.2017.874.

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We derive and assess the sharpness of analytic upper bounds for the instantaneous growth rate and finite-time amplification of palinstrophy in solutions of the two-dimensional incompressible Navier–Stokes equations. A family of optimal solenoidal fields parametrized by initial values for the Reynolds number $Re$ and palinstrophy ${\mathcal{P}}$ which maximize $\text{d}{\mathcal{P}}/\text{d}t$ is constructed by numerically solving suitable optimization problems for a wide range of $Re$ and ${\mathcal{P}}$, providing numerical evidence for the sharpness of the analytic estimate $\text{d}{\mathcal{P}}/\text{d}t\leqslant (a+b\sqrt{\ln Re+c}){\mathcal{P}}^{3/2}$ with respect to both $Re$ and ${\mathcal{P}}$. This family of instantaneously optimal fields is then used as initial data in fully resolved direct numerical simulations, and the time evolution of different relevant norms is carefully monitored as the palinstrophy is transiently amplified before decaying. The peak values of the palinstrophy produced by these initial data, i.e. $\sup _{t>0}{\mathcal{P}}(t)$, are observed to scale with the magnitude of the initial palinstrophy ${\mathcal{P}}(0)$ in accord with the corresponding a priori estimate. Implications of these findings for the question of finite-time singularity formation in the three-dimensional incompressible Navier–Stokes equation are discussed.

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Shekh, Salim Harun, Pedro H. R. S. Moraes, and Pradyumn Kumar Sahoo. "Physical Acceptability of the Renyi, Tsallis and Sharma-Mittal Holographic Dark Energy Models in the f(T,B) Gravity under Hubble’s Cutoff." Universe 7, no. 3 (March 12, 2021): 67. http://dx.doi.org/10.3390/universe7030067.

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In the present article, we investigate the physical acceptability of the spatially homogeneous and isotropic Friedmann–Lemâitre–Robertson–Walker line element filled with two fluids, with the first being pressureless matter and the second being different types of holographic dark energy. This geometric and material content is considered within the gravitational field equations of the f(T,B) (where T is the torsion scalar and the B is the boundary term) gravity in Hubble’s cut-off. The cosmological parameters, such as the Equation of State (EoS) parameter, during the cosmic evolution, are calculated. The models are stable throughout the universe expansion. The region in which the model is presented is dependent on the real parameter δ of holographic dark energies. For all δ≥4.5, the models vary from ΛCDM era to the quintessence era.

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Simon, Laurent, Christophe Lécuyer, and Thibaut Putelat. "The calculation of water-rock ratios using trace element (Li, B) stable isotopes." Annales Universitatis Mariae Curie-Sklodowska, sectio AAA – Physica 71 (February 23, 2017): 79. http://dx.doi.org/10.17951/aaa.2016.71.79.

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The amount of aqueous fluids circulating into the oceanic crust can be estimated using mass balance equations based on stable isotope exchange between rock and water. Unlike oxygen and strontium, isotopic exchange of trace elements (such as B or Li) between fluids and rocks, operates along with a chemical evolution of the rocks (e.g. a large enrichment of B or Li) that must be integrated into any model of water-rock interaction. We propose a general dimensionless mass balance equation for single-pass open systems that describes the equilibrium elemental distribution and the isotopic composition of reacting rocks as a function of the amount of circulating water. Water-rock ratios calculated from B compositions of hydrothermally-altered basalts range from 8 to 100. They are lower than those previously published (most W/R > 300) but comparable to those inferred from Sr isotope ratios measured in the same samples (3 < W/R < 30). Similar low water-rock ratios from 2 to 20 are calculated from Li isotope compositions of altered basalts and serpentinized peridotites.

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Flores, María, Luis Payá, David Valiente, Julio Gallego, and Oscar Reinoso. "Deployment of a Software to Simulate Control Systems in the State-Space." Electronics 8, no. 11 (October 23, 2019): 1205. http://dx.doi.org/10.3390/electronics8111205.

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In this work, we present a simulation software that permits designing and testing several types of controllers based on both classical and modern control theory. It has been created using Easy JavaScript Simulations, since this software permits implementing interactive simulations of physical systems in a quick and intuitive way. This laboratory contains a SISO (Single-Input and Single-Output) and a MIMO (Multiple-Input and Multiple-Output) plant, which are hydraulic and nonlinear, thus the linear model (linearized equations) and the original model (nonlinerized equations) have been implemented. The user can choose any of these physical systems and they have the options to control them using either continuous-time or discrete-time controllers. All parameters of the plant are fully configurable by the user. After that, the controller can be designed and tested. This simulation software offers several configurations: (a) PID (Proportional, Integral and Derivative controller); (b) state feedback; (c) observer and state feedback; and (d) integral controller, observer and state feedback control. The evolution of the controlled system is visualized using an animation of the virtual plant and a graphical representation of the evolution of the most important variables. In this paper, the steps for the implementation of this simulation software are detailed.

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Clark, R. C., R. D. Santer, and J. S. Brebner. "A generalized equation for the calculation of receptor noise limited colour distances in n -chromatic visual systems." Royal Society Open Science 4, no. 9 (September 2017): 170712. http://dx.doi.org/10.1098/rsos.170712.

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Researchers must assess similarities and differences in colour from an animal's eye view when investigating hypotheses in ecology, evolution and behaviour. Nervous systems generate colour perceptions by comparing the responses of different spectral classes of photoreceptor through colour opponent mechanisms, and the performance of these mechanisms is limited by photoreceptor noise. Accordingly, the receptor noise limited (RNL) colour distance model of Vorobyev and Osorio (Vorobyev & Osorio 1998 Proc. R. Soc. Lond. B 265 , 351–358 ( doi:10.1098/rspb.1998.0302 )) generates predictions about the discriminability of colours that agree with behavioural data, and consequently it has found wide application in studies of animal colour vision. Vorobyev and Osorio (1998) provide equations to calculate RNL colour distances for animals with di-, tri- and tetrachromatic vision, which is adequate for many species. However, researchers may sometimes wish to compute RNL colour distances for potentially more complex colour visual systems. Thus, we derive a simple, single formula for the computation of RNL distance between two measurements of colour, equivalent to the published di-, tri- and tetrachromatic equations of Vorobyev and Osorio (1998), and valid for colour visual systems with any number of types of noisy photoreceptors. This formula will allow the easy application of this important colour visual model across the fields of ecology, evolution and behaviour.

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AREF'EVA, I. YA, and I. V. VOLOVICH. "ON THE LARGE TIME BEHAVIOR OF QUANTUM SYSTEMS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 03, no. 04 (December 2000): 453–82. http://dx.doi.org/10.1142/s0219025700000297.

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There exists the well-known approximate expression describing the large time behavior of matrix elements of the evolution operator in quantum theory: < U(t)>≃ exp (at). This expression plays the crucial role in considerations of problems of quantum decoherence, radiation, decay, scattering theory, stochastic limit, derivation of master and kinetic equations etc. This expression was obtained in the Weisskopf–Wigner approximation and in the van Hove (stochastic) limit. We derive the exact general formula which includes the higher order corrections to the above approximate expression: <U(t)>= exp (At+B+C(t)). The constants A and B and the oscillating function C(t) are computed in perturbation theory. The method of perturbation of spectra and renormalized wave operators is used. The formula is valid for a general class of Hamiltonians used in statistical physics and quantum field theory.

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Hassan, Hany N. "Numerical Solution of a Boussinesq Type Equation Using Fourier Spectral Methods." Zeitschrift für Naturforschung A 65, no. 4 (April 1, 2010): 305–14. http://dx.doi.org/10.1515/zna-2010-0407.

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Efficient numerical methods for solving nonlinear wave equations and studying the propagation and stability properties of their solitary waves (solitons) are applied to a Boussinesq type equation in one space dimension. These methods use a pseudospectral (Fourier transform) treatment of the space dependence, together with (a) finite differences, or (b) a fourth-order Runge-Kutta scheme (RK4), for the time evolution. Our schemes follow very accurately single solitons, which are given by simple closed formulas and are known to be stable for all allowed velocities. However, as a parameter of the problem tends to the critical value b = 0.5, where the velocity of the exact soliton vanishes, our solutions destabilize due to numerical errors, producing two small solitons in the place of the exact one. On the other hand, when we study the interaction of two such solitons, starting far apart from each other, we find in the b1, b2 parameter plane a curve beyond which the solution becomes unstable by exponential blow-up of the amplitudes, independently of our space and time discretization. We claim that this is due to a dynamical resonance rather than the accumulation of numerical errors, in agreement with theoretical predictions. Our implementation relies on the fast Fourier transform (FFT) algorithm and no major differences are observed, when either scheme (a) or (b) is used for the evolution of time.

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Berrahmoune, Larbi. "Constrained null controllability for distributed systems and applications to hyperbolic-like equations." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 32. http://dx.doi.org/10.1051/cocv/2018018.

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We consider linear control systems of the form y′(t) = Ay(t) + Bu(t) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y . The aim is to study the null controllability in the case where the control u is constrained to lie in a bounded subset Γ ⊂ U. We obtain local constrained controllability properties. When (etA)t∈ℝ is a group of isometries, we establish necessary conditions and sufficient ones for global constrained controllability. Moreover, when the constraint set Γ contains the origin in its interior, the local constrained property turns out to be equivalent to a dual observability inequality of L1 type with respect to the time variable. In this setting, the study is focused on hyperbolic-like systems which can be reduced to a second order evolution equation. Furthermore, we treat the problem of determining a steering control for general constraint set Γ in nonsmooth convex analysis context. In the case where Γ contains the origin in its interior, a steering control can be obtained by minimizing a convenient smooth convex functional. Applications to the wave equation and Euler-Bernoulli beams are presented.

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Page, Jacob, and Tamer A. Zaki. "Streak evolution in viscoelastic Couette flow." Journal of Fluid Mechanics 742 (February 21, 2014): 520–51. http://dx.doi.org/10.1017/jfm.2013.686.

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AbstractThe combined effect of inertia and elasticity on streak amplification in planar Couette flow of an Oldroyd-B fluid is examined. The linear perturbation equations are solved in the form of a forced-response problem to obtain the wall-normal vorticity response to a decaying streamwise vortex. With significant disparity between the solvent diffusion and polymer relaxation time scales, two distinct responses are possible. The first is termed ‘quasi-Newtonian’ because the streak evolution collapses onto the Newtonian behaviour at the same total and solvent Reynolds numbers when relaxation is very fast or slow, respectively. The second response is labelled ‘elastic’: with a long relaxation time, the streaks can reach significant amplitudes even with very weak inertia. If the diffusion and relaxation time scales are commensurate, the streaks are able to re-energize in a periodic cycle within an envelope of overall decay. This behaviour is enhanced in the instantaneously elastic limit, where the governing equation reduces to a forced wave equation. The streak re-energization is demonstrated to be a superposition of trapped vorticity waves.

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Bittencourt, J. A., V. G. Pillat, P. R. Fagundes, Y. Sahai, and A. A. Pimenta. "LION: A dynamic computer model for the low-latitude ionosphere." Annales Geophysicae 25, no. 11 (November 29, 2007): 2371–92. http://dx.doi.org/10.5194/angeo-25-2371-2007.

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Abstract. A realistic fully time-dependent computer model, denominated LION (Low-latitude Ionospheric) model, that simulates the dynamic behavior of the low-latitude ionosphere is presented. The time evolution and spatial distribution of the ionospheric particle densities and velocities are computed by numerically solving the time-dependent, coupled, nonlinear system of continuity and momentum equations for the ions O+, O2+, NO+, N2+ and N+, taking into account photoionization of the atmospheric species by the solar extreme ultraviolet radiation, chemical and ionic production and loss reactions, and plasma transport processes, including the ionospheric effects of thermospheric neutral winds, plasma diffusion and electromagnetic E×B plasma drifts. The Earth's magnetic field is represented by a tilted centered magnetic dipole. This set of coupled nonlinear equations is solved along a given magnetic field line in a Lagrangian frame of reference moving vertically, in the magnetic meridian plane, with the electromagnetic E×B plasma drift velocity. The spatial and time distribution of the thermospheric neutral wind velocities and the pattern of the electromagnetic drifts are taken as known quantities, given through specified analytical or empirical models. The model simulation results are presented in the form of computer-generated color maps and reproduce the typical ionization distribution and time evolution normally observed in the low-latitude ionosphere, including details of the equatorial Appleton anomaly dynamics. The specific effects on the ionosphere due to changes in the thermospheric neutral winds and the electromagnetic plasma drifts can be investigated using different wind and drift models, including the important longitudinal effects associated with magnetic declination dependence and latitudinal separation between geographic and geomagnetic equators. The model runs in a normal personal computer (PC) and generates color maps illustrating the typical behavior of the low-latitude ionosphere for a given longitudinal region, for different seasons, geophysical conditions and solar activity, at each instant of time, showing the time evolution of the low-latitude ionosphere, between about 20° north and south of the magnetic equator. This paper presents a detailed description of the mathematical model and illustrative computer results.

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Mücket, J. P., H. J. Haubold, S. Gottlöber, and V. Müller. "The Coupling of Modes and the Formation of Population III Objects." Symposium - International Astronomical Union 130 (1988): 591. http://dx.doi.org/10.1017/s0074180900137179.

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We consider two density perturbation modes with significantly different length scales λ and L (λ ≪ L) in a homogeneous Universe within Newtonian approximation. For the two modes the coupling terms in the corresponding Euler-Lagrange and Poisson equations are taken into account within lowest order of approximation. We assume that the λ - nods (high-frequency mode) is superimposed on the large-scale mode in such way that by an appropriate averaging procedure, the global behaviour is determined only by the single L-mode. Locally (Δ x ≈ λ ≪ L) the space dependence of the L-mode can be neglected in comparison with the λ -mode, but its time evolution remains important for the evolution of the λ -mode perturbation δ. We obtain for δ the equation: where , b is the sound velocity with respect to the undisturbed homogeneous natter distribution, H is the Hubble parameter, .

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Patel, Apoorva, and Anjani Priyadarsini. "Efficient quantum algorithms for state measurement and linear algebra applications." International Journal of Quantum Information 16, no. 06 (September 2018): 1850048. http://dx.doi.org/10.1142/s021974991850048x.

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We present an algorithm for measurement of [Formula: see text]-local operators in a quantum state, which scales logarithmically both in the system size and the output accuracy. The key ingredients of the algorithm are a digital representation of the quantum state, and a decomposition of the measurement operator in a basis of operators with known discrete spectra. We then show how this algorithm can be combined with (a) Hamiltonian evolution to make quantum simulations efficient, (b) the Newton–Raphson method based solution of matrix inverse to efficiently solve linear simultaneous equations, and (c) Chebyshev expansion of matrix exponentials to efficiently evaluate thermal expectation values. The general strategy may be useful in solving many other linear algebra problems efficiently.

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Journal articles on the topic 'B-evolution Equations'

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