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Journal articles on the topic 'Time dependent solution'

Author: Grafiati

Published: 2 December 2022

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1

Feistauer, Miloslav, Jaromír Horáček, Václav Kučera, and Jaroslava Prokopová. "On numerical solution of compressible flow in time-dependent domains." Mathematica Bohemica 137, no. 1 (2012): 1–16. http://dx.doi.org/10.21136/mb.2012.142782.

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2

Los, V. F., and M. V. Los. "An Exact Solution of the Time-Dependent Schrödinger Equation with a Rectangular Potential for Real and Imaginary Times." Ukrainian Journal of Physics 61, no. 4 (April 2016): 331–41. http://dx.doi.org/10.15407/ujpe61.04.0331.

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3

López, L. A., Omar Pedraza, and V. E. Ceron. "Time-dependent solution from Myers–Perry." Canadian Journal of Physics 94, no. 2 (February 2016): 177–79. http://dx.doi.org/10.1139/cjp-2015-0354.

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We present a three-parameter time-dependent solution of the vacuum Einstein equations in five dimensions. The solution is obtained by applying the Wick rotation to the Myers–Perry solution that represents a rotating black hole in five dimensions. The new interpretation of the Myers–Perry solution can be considered among the generalized Einstein–Rosen type that can be interpreted as plane-symmetric waves, cylindrical waves or cosmological space–time in five dimensions. In some limits the solution has boost-rotational symmetry and it is asymptotically flat. In the case that the solution represents a cylindrical space–time, the E-energy is analyzed.

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4

Vidal, Thibaut, Rafael Martinelli, Tuan Anh Pham, and Minh Hoàng Hà. "Arc Routing with Time-Dependent Travel Times and Paths." Transportation Science 55, no. 3 (May 2021): 706–24. http://dx.doi.org/10.1287/trsc.2020.1035.

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Vehicle routing algorithms usually reformulate the road network into a complete graph in which each arc represents the shortest path between two locations. Studies on time-dependent routing followed this model and therefore defined the speed functions on the complete graph. We argue that this model is often inadequate, in particular for arc routing problems involving services on edges of a road network. To fill this gap, we formally define the time-dependent capacitated arc routing problem (TDCARP), with travel and service speed functions given directly at the network level. Under these assumptions, the quickest path between locations can change over time, leading to a complex problem that challenges the capabilities of current solution methods. We introduce effective algorithms for preprocessing quickest paths in a closed form, efficient data structures for travel time queries during routing optimization, and heuristic and exact solution approaches for the TDCARP. Our heuristic uses the hybrid genetic search principle with tailored solution-decoding algorithms and lower bounds for filtering moves. Our branch-and-price algorithm exploits dedicated pricing routines, heuristic dominance rules, and completion bounds to find optimal solutions for problems counting up to 75 services. From these algorithms, we measure the benefits of time-dependent routing optimization for different levels of travel-speed data accuracy.

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5

Vardy, Alan E., and James M. B. Brown. "Laminar pipe flow with time-dependent viscosity." Journal of Hydroinformatics 13, no. 4 (October 1, 2010): 729–40. http://dx.doi.org/10.2166/hydro.2010.073.

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A general solution is obtained for laminar flow in axisymmetric pipes, allowing for prescribed timedependent viscosity and time-dependent pressure gradients. In both cases, the only restriction on the prescribed time dependence is that it must vary continuously; it is not necessary for rates of change to be continuous. The general solution is obtained using the Finite Hankel Transform method. This makes it possible to allow explicitly for time-dependent viscosity, but it does not permit the spatial dependence of viscosity. This contrasts with Laplace transforms, which allow spatial, but not general, temporal variations. The general solution is used to study a selection of particular flows chosen to illustrate distinct forms of physical behaviour and to demonstrate the ease with which solutions are obtained. The methodology is also applied to the simple case of constant (Newtonian) viscosity. In this case, it yields the same solutions as previously published methods, but it does so in a much simpler manner.

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6

MEYLAN, MICHAEL H. "Spectral solution of time-dependent shallow water hydroelasticity." Journal of Fluid Mechanics 454 (March 10, 2002): 387–402. http://dx.doi.org/10.1017/s0022112001007273.

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The spectral theory of a thin plate floating on shallow water is derived and used to solve the time-dependent motion. This theory is based on an energy inner product in which the evolution operator becomes unitary. Two solution methods are presented. In the first, the solution is expanded in the eigenfunctions of a self-adjoint operator, which are the incoming wave solutions for a single frequency. In the second, the scattering theory of Lax–Phillips is used. The Lax–Phillips scattering solution is suitable for calculating only the free motion of the plate. However, it determines the modes of vibration of the plate–water system. These modes, which both oscillate and decay, are found by a complex search algorithm based contour integration. As well as an application to modelling floating runways, the spectral theory for a floating thin plate on shallow water is a solvable model for more complicated hydroelastic systems.

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7

Li, Nan, and Shripad Tuljapurkar. "The solution of time‐dependent population models." Mathematical Population Studies 7, no. 4 (January 2000): 311–29. http://dx.doi.org/10.1080/08898480009525464.

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8

Evans, D. J., and A. A. Al-kharafi. "Finite element solution of time-dependent problems." International Journal of Computer Mathematics 22, no. 3-4 (January 1987): 287–302. http://dx.doi.org/10.1080/00207168708803599.

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9

Paasschens, J. C. J. "Solution of the time-dependent Boltzmann equation." Physical Review E 56, no. 1 (July 1, 1997): 1135–41. http://dx.doi.org/10.1103/physreve.56.1135.

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10

Abdou, M. A. "On the solution of time-dependent problems." Journal of Quantitative Spectroscopy and Radiative Transfer 95, no. 2 (October 2005): 271–84. http://dx.doi.org/10.1016/j.jqsrt.2004.08.044.

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11

Conte, Enrico, and Antonello Troncone. "One-dimensional consolidation under general time-dependent loading." Canadian Geotechnical Journal 43, no. 11 (November 1, 2006): 1107–16. http://dx.doi.org/10.1139/t06-064.

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This paper presents an analytical solution for the analysis of one-dimensional consolidation of saturated soil layers subjected to general time-dependent loading. A simple calculation procedure that makes use of the Fourier series is proposed for practical applications. Both single loads and cyclic loads can be considered by choosing a suitable period for the Fourier series. A number of comparisons with existing theoretical solutions are shown to assess the accuracy of the proposed procedure. Moreover, the experimental results from oedometer tests performed in the present study and from a well-documented case history concerning a large embankment constructed on compressible soils are analysed using this solution to evaluate the coefficient of consolidation of the soil.Key words: one-dimensional consolidation, time-dependent loading, excess pore-water pressure, theoretical solution, Fourier series.

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12

CARNEIRO, S., and J. A. S. LIMA. "TIME DEPENDENT COSMOLOGICAL TERM AND HOLOGRAPHY." International Journal of Modern Physics A 20, no. 11 (April 30, 2005): 2465–69. http://dx.doi.org/10.1142/s0217751x0502478x.

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The physical space is believed to contain a kind of energy whose gravitational effect resembles the cosmological Λ-term introduced by Einstein. We discuss how the holographic conjecture, relating the universe entropy with the horizon area, may fix the late time variation of this effective Λ-term and the induced time dependence of the gravitational constant. The resulting cosmological solutions at late times present a constant ratio between the matter density and the vacuum energy density. It is then argued that, among the non-decelerating solutions, the coasting expansion is the only acceptable from a thermodynamic viewpoint. This solution is characterized by a universe age given by Ht = 1, and by density parameters Ωm = 1/3 and ΩΛ = 2/3, as required by the cosmic concordance model.

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13

Caillerie, D., and C. Dascalu. "One-dimensional Localization Solutions for Time-dependent Damage." International Journal of Damage Mechanics 20, no. 8 (November 2011): 1178–97. http://dx.doi.org/10.1177/1056789510395553.

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This article presents analytical solutions for a class of one-dimensional time-dependent elasto-damage problems. The considered damage evolution law may be seen as a one-dimensional version of the Kachanov–Rabotnov creep damage model with classical loading–unloading conditions. We construct analytical solutions for the quasistatic one-dimensional problem. The evolution consists of a first regime, in which damage and strain grow uniformly, followed by a regime in which localization occurs. In the second regime, the uniqueness of the solution is lost and the deformation of the body is represented by a sequence of arbitrary alternate loading/unloading regions. Complex evolutions with progressive enlargement of the unloading regions in a finite number of steps are also constructed. We study analytically and numerically the features of the obtained bifurcated solutions. It is shown that, at every instant of time, a lower limit exists for the size of the localization zone. This lower limit is actually realized by the solution with successive unloadings constructed in this article. These features help us to understand the behavior of numerical solutions for time-dependent damage in the quasistatic approximation.

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14

Phan, Dung T., Chao Liu, Murtadha J. AlTammar, Yanhui Han, and Younane N. Abousleiman. "Application of Artificial Intelligence To Predict Time-Dependent Mud-Weight Windows in Real Time." SPE Journal 27, no. 01 (October 26, 2021): 39–59. http://dx.doi.org/10.2118/206748-pa.

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Summary Selection of a safe mud weight is crucial in drilling operations to reduce costly wellbore-instability problems. Advanced physics models and their analytical solutions for mud-weight-window computation are available but still demanding in terms of central-processing-unit (CPU) time. This paper presents an artificial-intelligence (AI) solution for predicting time-dependent safe mud-weight windows and very refined polar charts in real time. The AI agents are trained and tested on data generated from a time-dependent coupled analytical solution (poroelastic) because numerical solutions are prohibitively slow. Different AI techniques, including linear regression, decision tree, random forest, extra trees, adaptive neuro fuzzy inference system (ANFIS), and neural networks are evaluated to select the most suitable one. The results show that neural networks have the best performances and are capable of predicting time-dependent mud-weight windows and polar charts as accurately as the analytical solution, with 1/1,000 of the computer time needed, making them very applicable to real-time drilling operations. The trained neural networks achieve a mean squared error (MSE) of 0.0352 and a coefficient of determination (R2) of 0.9984 for collapse mud weights, and an MSE of 0.0072 and an R2 of 0.9998 for fracturing mud weights on test data sets. The neural networks are statistically guaranteed to predict mud weights that are within 5% and 10% of the analytical solutions with probability up to 0.986 and 0.997, respectively, for collapse mud weights, and up to 0.9992 and 0.9998, respectively, for fracturing mud weights. Their time performances are significantly faster and less demanding in computing capacity than the analytical solution, consistently showing three-orders-of-magnitude speedups in computational speed tests. The AI solution is integrated into a deployed wellbore-stability analyzer, which is used to demonstrate the AI’s performances and advantages through three case studies.

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15

Vabishchevich, P. N., and P. E. Zakharov. "Numerical Solution of Time-Dependent Problems with Different Time Scales." Computational Mathematics and Mathematical Physics 58, no. 10 (October 2018): 1552–61. http://dx.doi.org/10.1134/s0965542518100123.

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16

Dunajski, Maciej, and Prim Plansangkate. "Topology and energy of time-dependent unitons." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2080 (January 9, 2007): 945–59. http://dx.doi.org/10.1098/rspa.2006.1799.

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We consider a class of time-dependent finite energy multi-soliton solutions of the U ( N ) integrable chiral model in (2+1) dimensions. The corresponding extended solutions of the associated linear problem have a pole with arbitrary multiplicity in the complex plane of the spectral parameter. Restrictions of these extended solutions to any space-like plane in have trivial monodromy and give rise to maps from a three-sphere to U ( N ). We demonstrate that the total energy of each multi-soliton is quantized at the classical level and given by the third homotopy class of the extended solution. This is the first example of a topological mechanism explaining the classical energy quantization of moving solitons.

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17

Huang, Y., X. Duan, X. Lan, Z. Tan, N. Wang, X. Tang, and Y. He. "Time-dependent neutral-plasma isothermal expansions into a vacuum." Laser and Particle Beams 26, no. 4 (December 2008): 671–75. http://dx.doi.org/10.1017/s0263034608000748.

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AbstractA time-dependent solution for neutral-plasma isothermal expansions into a vacuum in a special-transformation coordinate system is obtained. In this new coordinate system, the special self-similar solutions of it were given by Huang and co-workers (Appl. Phys. Lett. 92, 031501). Combining the time-dependent solution and the quasi-linear increase of the electron density due to the hot-electron recirculation, an analytic model is proposed to reveal the influence of the hot-electron recirculation on the increase of electric field and on the acceleration of ions of different masses and charges.

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18

Keto, Eric. "Stability and solution of the time-dependent Bondi–Parker flow." Monthly Notices of the Royal Astronomical Society 493, no. 2 (February 21, 2020): 2834–40. http://dx.doi.org/10.1093/mnras/staa529.

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ABSTRACT Bondi and Parker derived a steady-state solution for Bernoulli’s equation in spherical symmetry around a point mass for two cases, respectively, an inward accretion flow and an outward wind. Left unanswered were the stability of the steady-state solution, the solution itself of time-dependent flows, whether the time-dependent flows would evolve to the steady state, and under what conditions a transonic flow would develop. In a Hamiltonian description, we find that the steady-state solution is equivalent to the Lagrangian implying that time-dependent flows evolve to the steady state. We find that the second variation is definite in sign for isothermal and adiabatic flows, implying at least linear stability. We solve the partial differential equation for the time-dependent flow as an initial-value problem and find that a transonic flow develops under a wide range of realistic initial conditions. We present some examples of time-dependent solutions.

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19

GUO HUA. "EXACT SOLUTION OF THE TIME-DEPENDENT DIRAC EQUATION." Acta Physica Sinica 48, no. 6 (1999): 983. http://dx.doi.org/10.7498/aps.48.983.

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20

Takahashi, Isao, Hideyuki Kobayashi, Jiro Ryuta, Mikio Kishimoto, and Takayuki Shingyouji. "Time-Dependent Variation of Composition of SC1 Solution." Japanese Journal of Applied Physics 32, Part 2, No. 9A (September 1, 1993): L1183—L1185. http://dx.doi.org/10.1143/jjap.32.l1183.

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21

Fityo, T. V., and V. M. Tkachuk. "Time-dependent Schrödinger equation with one known solution." Journal of Physical Studies 9, no. 4 (2005): 299–303. http://dx.doi.org/10.30970/jps.09.299.

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22

Ferracane, J. L., P. Mafiana, C. Cooper, and T. Okabe. "Time-dependent Dissolution of Amalgams into Saline Solution." Journal of Dental Research 66, no. 8 (August 1987): 1331–35. http://dx.doi.org/10.1177/00220345870660080801.

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The dissolution of mercury, silver, and copper from polished and unpolished surfaces of low- and high-copper amalgams into saline was investigated with respect to time via atomic absorption spectrophotometry. Greater amounts of mercury and silver were released from unpolished than from polished surfaces. Ion release was greatest during the first three hours after trituration of all amalgams. The rate of dissolution of ions from amalgams into saline is considered to be very low once the amalgam has set, and is probably inhibited by the formation of a surface film formed during immersion.

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23

Fujita, Masako, and Hiroyuki Hata. "Time dependent solution in cubic string field theory." Journal of High Energy Physics 2003, no. 05 (May 17, 2003): 043. http://dx.doi.org/10.1088/1126-6708/2003/05/043.

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24

Guang, Hou, Xu Xiu-Wei, and Zhang Yong-De. "Analytical Solution for Single-Mode Time-Dependent Oscillator." Communications in Theoretical Physics 37, no. 5 (May 15, 2002): 531–34. http://dx.doi.org/10.1088/0253-6102/37/5/531.

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25

Cveticanin, Livija. "Approximate solution of a time-dependent differential equation." Meccanica 30, no. 6 (December 1995): 665–71. http://dx.doi.org/10.1007/bf00986572.

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26

Fu, Jin, Sheng Wu, and Linda R. Petzold. "Time dependent solution for acceleration of tau-leaping." Journal of Computational Physics 235 (February 2013): 446–57. http://dx.doi.org/10.1016/j.jcp.2012.10.036.

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27

Storchak, S. N. "Path integral solution for some time-dependent potential." Physics Letters A 161, no. 5 (January 1992): 397–402. http://dx.doi.org/10.1016/0375-9601(92)90676-d.

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28

de Freitas, J. A. Teixeira. "Mixed finite element solution of time-dependent problems." Computer Methods in Applied Mechanics and Engineering 197, no. 45-48 (August 2008): 3657–78. http://dx.doi.org/10.1016/j.cma.2008.02.014.

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29

Bertolini, D., M. Cassettari, G. Salvetti, E. Tombari, S. Veronesi, and G. Squadrito. "Time-dependent heat capacity of water-lysozyme solution." Il Nuovo Cimento D 14, no. 2 (February 1992): 199–205. http://dx.doi.org/10.1007/bf02457352.

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30

Schnell, S., and C. Mendoza. "Closed Form Solution for Time-dependent Enzyme Kinetics." Journal of Theoretical Biology 187, no. 2 (July 1997): 207–12. http://dx.doi.org/10.1006/jtbi.1997.0425.

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31

Durkee, J. W., P. P. Antich, and C. E. Lee. "Exact solutions to the multiregion time-dependent bioheat equation. I: Solution development." Physics in Medicine and Biology 35, no. 7 (July 1, 1990): 847–67. http://dx.doi.org/10.1088/0031-9155/35/7/004.

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32

Saadati, Reza, Andrea Giusti, Valerio Faraoni, and Fatimah Shojai. "New time-dependent solutions of viable Horndeski gravity." Journal of Cosmology and Astroparticle Physics 2022, no. 09 (September 1, 2022): 067. http://dx.doi.org/10.1088/1475-7516/2022/09/067.

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Abstract We generate new spherical and time-dependent solutions of viable Horndeski gravity by disforming a solution of the Einstein equations with scalar field source and positive cosmological constant. They describe dynamical objects embedded in asymptotically FLRW spacetimes and contain apparent horizons and a finite radius singularity that evolve in time in peculiar ways apparently not encountered before in Einstein and “old” scalar-tensor gravity.

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33

Ludu, Andrei. "Nonlocal Symmetries for Time-Dependent Order Differential Equations." Symmetry 10, no. 12 (December 19, 2018): 771. http://dx.doi.org/10.3390/sym10120771.

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A new type of ordinary differential equation is introduced and discussed: time-dependent order ordinary differential equations. These equations are solved via fractional calculus by transforming them into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equation represent deformations of the solutions of the classical (integer order) differential equations, mapping them into one-another as limiting cases. This equation can also move, remove or generate singularities without involving variable coefficients. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers is observed.

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34

Wang, Yang, Jin Xin Cao, Xia Xi Li, and Ri Dong Wang. "Time-Dependent Discrete Transportation Network Design." Applied Mechanics and Materials 505-506 (January 2014): 533–36. http://dx.doi.org/10.4028/www.scientific.net/amm.505-506.533.

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The transportation network construction takes place over a quite long time span and need enough budget. The budget is from the allocation of funds in phases and the construction cost change in the process of the construction. The general static transportation network design problems ignores the problems above. So the optimal solution obtained by the static model is best in short time, and it is may be unfeasible in the actual situation. Based on the actual situation and the shortage of the static model, the time-dependent transport network design is proposed in this study. The plan horizon is divided into N intervals and a bi-level model is built to describe the problem. The objective of the upper-level is to minimize the total cost of the whole stages. the lower-level model is a user equilibrium model. Then the branch and bound (B-B) algorithm is designed to solve the model. It is obvious that the solution of the time-dependent simulation model is more feasible than the solution of the static sequential design.

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35

Wang, Q., H. Zhan, and Z. Tang. "Forchheimer flow to a well considering time-dependent critical radius." Hydrology and Earth System Sciences Discussions 10, no. 11 (November 19, 2013): 14095–129. http://dx.doi.org/10.5194/hessd-10-14095-2013.

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Abstract. Previous studies on the non-Darcian flow into a pumping well assumed that critical radius (RCD) was a constant or infinity, where RCD represents the location of the interface between the non-Darcian flow region and Darcian flow region. In this study, a two-region model considering time-dependent RCD was established, where the non-Darcian flow was described by the Forchheimer equation. A new iteration method was proposed to estimate RCD based on the finite-difference method. The results showed that RCD increased with time until reaching the quasi-steady state flow, and the asymptotic value of RCD only depended on the critical specific discharge beyond which flow became non-Darcian. A larger inertial force would reduce the change rate of RCD with time, and resulted in a smaller RCD at a specific time during the transient flow. The difference between the new solution and previous solutions were obvious in the early pumping stage. The new solution agreed very well with the solution of previous two-region model with a constant RCD under quasi-steady flow. It agreed with the solution of the fully Darcian flow model in the Darcian flow region, and with the solution of the fully non-Darcian flow model in the non-Darcian flow region near the well.

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36

Joshi, Raj Kishor, Indranil Chattopadhyay, and Lallan Yadav. "Radiatively driven, time dependent bipolar outflows." Monthly Notices of the Royal Astronomical Society 509, no. 1 (October 1, 2021): 85–99. http://dx.doi.org/10.1093/mnras/stab2841.

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ABSTRACT We study the radiatively driven fluid jets around a non-rotating black hole. The radiation arising from the inner compact corona and outer sub-Keplerian part of the disc accelerates the jets. We obtain the steady state, semi-analytical, radiatively driven outflow solutions. The thermodynamics of the outflow is described by a variable adiabatic index equation of state. We develop a total variation diminishing (TVD) routine to investigate the time dependent behaviour of the radiatively driven bipolar outflow. We inject with flow variables from the steady state outflow solutions in the TVD code and allow the code to settle into steady state and match the numerical results with the steady state solution. The radiation arising out of the accretion disc can provide a wide range of jet solutions, depending upon parameters like the intensity of disc, location of the inner corona etc. We induce the time dependence of the radiation field by inducing oscillation of the inner corona of the accretion disc. The radiation field then makes the bipolar outflow time dependent. We show that a non-steady radiation field arising out of disc oscillations can generate the internal shocks closer to the jet base. Depending on the disc geometry, there might be transient shocks in the jet and there might be multiple non-stationary shocks in the jet, which are of much interest in jet physics.

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37

Zhang, Hua-Xin, and Chun-Miao Zhang. "Multiobjective Green Time-Dependent Location-Routing Problem and Algorithms." Advances in Operations Research 2022 (May 21, 2022): 1–16. http://dx.doi.org/10.1155/2022/1811689.

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To reduce the logistic cost and carbon emission and improve customer satisfaction, this study proposes a multiobjective green time-dependent location routing problem (MOGTDLRP) model in which the objectives are to minimize the distribution total cost, delivery time, and fuel consumption. This model will be solved by several hyperheuristic algorithms which include the high-level heuristics and the low-level heuristics. There are three acceptance criterions for the solution: improving and equal, all moves and accept all solutions, and dynamic acceptance criteria. Through the case, the performance of the algorithm and the influence of various factors on the solution are analyzed in this study. The experimental results show that the proposed model can effectively reduce logistic costs, carbon emissions, and vehicle travel time.

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38

Ton, Bui An. "Time-dependent Stokes equations with measure data." Abstract and Applied Analysis 2003, no. 17 (2003): 953–73. http://dx.doi.org/10.1155/s1085337503308012.

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We establish the existence of a unique solution of an initial boundary value problem for the nonstationary Stokes equations in a bounded fixed cylindrical domain with measure data. Feedback laws yield the source and its intensity from the partial measurements of the solution in a subdomain.

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39

Ajayi, O. O., and A. Falade. "Time-Dependent Stokes Flow Induced by Instantaneous Sources." Journal of Applied Mechanics 52, no. 3 (September 1, 1985): 718–24. http://dx.doi.org/10.1115/1.3169128.

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Analytical solutions are provided for Stokes flow fields induced by an instantaneous thermal source located (i) in an infinite fluid and (ii) at the center of a solid spherical surface. In both cases it is assumed that the flow is driven by a body force that arises from buoyancy effects in the fluid due to the gradient of the source’s temperature field. In accordance with Stokes approximation, convective acceleration terms are left out of the momentum equations but the local acceleration term is retained. The retention of the local acceleration term is crucial if all the salient features of the flow, especially in its early stages, are to be included in the solution for low Prandtl number. The evolution of the flow field in each of the two cases is demonstrated by plotting streamline patterns at various times. The influence of Prandtl number on flow field development is also discussed. Problem (i) is the fundamental one for this class of problems. From its solution, solutions to problems where the heat-generating rate is a general but known function of position and time to bounded and unbounded domains may be synthesised using well-known techniques. These results may be fruitfully applied to such practical problems as the motion of vessels in which nuclear wastes are stored, the fluid dynamics of weak point explosions, and the cooling of electrical and electronic equipments.

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40

Wang, Q., H. Zhan, and Z. Tang. "Forchheimer flow to a well-considering time-dependent critical radius." Hydrology and Earth System Sciences 18, no. 6 (June 27, 2014): 2437–48. http://dx.doi.org/10.5194/hess-18-2437-2014.

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Abstract. Previous studies on the non-Darcian flow into a pumping well assumed that critical radius (RCD) was a constant or infinity, where RCD represents the location of the interface between the non-Darcian flow region and Darcian flow region. In this study, a two-region model considering time-dependent RCD was established, where the non-Darcian flow was described by the Forchheimer equation. A new iteration method was proposed to estimate RCD based on the finite-difference method. The results showed that RCD increased with time until reaching the quasi steady-state flow, and the asymptotic value of RCD only depended on the critical specific discharge beyond which flow became non-Darcian. A larger inertial force would reduce the change rate of RCD with time, and resulted in a smaller RCD at a specific time during the transient flow. The difference between the new solution and previous solutions were obvious in the early pumping stage. The new solution agreed very well with the solution of the previous two-region model with a constant RCD under quasi steady flow. It agreed with the solution of the fully Darcian flow model in the Darcian flow region.

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41

Tu, Te-Wen, and Sen-Yung Lee. "Analytical Solution of Heat Conduction for Hollow Cylinders with Time-Dependent Boundary Condition and Time-Dependent Heat Transfer Coefficient." Journal of Applied Mathematics 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/203404.

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An analytical solution for the heat transfer in hollow cylinders with time-dependent boundary condition and time-dependent heat transfer coefficient at different surfaces is developed for the first time. The methodology is an extension of the shifting function method. By dividing the Biot function into a constant plus a function and introducing two specially chosen shifting functions, the system is transformed into a partial differential equation with homogenous boundary conditions only. The transformed system is thus solved by series expansion theorem. Limiting cases of the solution are studied and numerical results are compared with those in the literature. The convergence rate of the present solution is fast and the analytical solution is simple and accurate. Also, the influence of physical parameters on the temperature distribution of a hollow cylinder along the radial direction is investigated.

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42

Barbagallo, A. "Degenerate Time-dependent Variational Inequalities with Applications to Traffic Equilibrium Problems." Computational Methods in Applied Mathematics 6, no. 2 (2006): 117–33. http://dx.doi.org/10.2478/cmam-2006-0006.

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Abstract The aim of this paper is to study the continuity of the solutions to degenerate time-dependent variational inequalities. In order to obtain the continuity of the solution, a previous continuity result for strongly monotone variational inequalities and an appropriate use of the convergence set in Mosco’s sense play an important role. The continuity result allows us to provide a discretization procedure for the calculation of the solution to the variational inequality which expresses the time-dependent traffic network equilibrium problem.

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43

Szidarovszky, Ferenc, and Ioannis K. Argyros. "On time dependent multistep dynamic processes." Bulletin of the Australian Mathematical Society 43, no. 1 (February 1991): 51–61. http://dx.doi.org/10.1017/s0004972700028768.

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The discrete time scale Liapunov theory is extended to time dependent, higher order, nonlinear difference equations in a partially ordered topological space. The monotone convergence of the solution is examined and the speed of convergence is estimated.

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44

Abbaspour, R. A., and F. Samadzadeg. "A Solution for Time-Dependent Multimodal Shortest Path Problem." Journal of Applied Sciences 9, no. 21 (October 15, 2009): 3804–12. http://dx.doi.org/10.3923/jas.2009.3804.3812.

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Majdalani, J., and W. K. Van Moorhen. "Improved Time-Dependent Flowfield Solution for Solid Rocket Motors." AIAA Journal 36, no. 2 (February 1998): 241–48. http://dx.doi.org/10.2514/2.7507.

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Tohyama, M. "Stationary Solution of a Time Dependent Density Matrix Formalism." Progress of Theoretical Physics 92, no. 4 (October 1, 1994): 905–8. http://dx.doi.org/10.1143/ptp/92.4.905.

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47

Buske, Daniela, Marco T. Vilhena, Everson J. G. Silva, and Tiziano Tirabassi. "A solution of the time-dependent advection-diffusion equation." International Journal of Environment and Pollution 65, no. 1/2/3 (2019): 211. http://dx.doi.org/10.1504/ijep.2019.10023408.

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48

Tirabassi, Tiziano, Everson J. G. Silva, Daniela Buske, and Marco T. Vilhena. "A solution of the time-dependent advection-diffusion equation." International Journal of Environment and Pollution 65, no. 1/2/3 (2019): 211. http://dx.doi.org/10.1504/ijep.2019.101842.

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49

Margolius, B. H. "Transient solution to the time-dependent multiserver Poisson queue." Journal of Applied Probability 42, no. 3 (September 2005): 766–77. http://dx.doi.org/10.1239/jap/1127322026.

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We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express pĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in pĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

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50

Margolius, B. H. "Transient solution to the time-dependent multiserver Poisson queue." Journal of Applied Probability 42, no. 03 (September 2005): 766–77. http://dx.doi.org/10.1017/s0021900200000760.

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We derive an integral equation for the transient probabilities and expected number in the queue for the multiserver queue with Poisson arrivals, exponential service for time-varying arrival and departure rates, and a time-varying number of servers. The method is a straightforward application of generating functions. We can express p ĉ−1(t), the probability that ĉ − 1 customers are in the queue or being served, in terms of a Volterra equation of the second kind, where ĉ is the maximum number of servers working during the day. Each of the other transient probabilities is expressed in terms of integral equations in p ĉ−1(t) and the transition probabilities of a certain time-dependent random walk. In this random walk, the rate of steps to the right equals the arrival rate of the queue and the rate of steps to the left equals the departure rate of the queue when all servers are busy.

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