Articles de revues : « Fractional obstacle »

1

Focardi, Matteo. « Aperiodic fractional obstacle problems ». Advances in Mathematics 225, n^o 6 (décembre 2010) : 3502–44. http://dx.doi.org/10.1016/j.aim.2010.06.014.

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Allen, Mark, et Mariana Smit Vega Garcia. « The fractional unstable obstacle problem ». Nonlinear Analysis 193 (avril 2020) : 111459. http://dx.doi.org/10.1016/j.na.2019.02.012.

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Caffarelli, Luis, et Antoine Mellet. « Random homogenization of fractional obstacle problems ». Networks & ; Heterogeneous Media 3, n^o 3 (2008) : 523–54. http://dx.doi.org/10.3934/nhm.2008.3.523.

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Allen, Mark, Erik Lindgren et Arshak Petrosyan. « The Two-Phase Fractional Obstacle Problem ». SIAM Journal on Mathematical Analysis 47, n^o 3 (janvier 2015) : 1879–905. http://dx.doi.org/10.1137/140974195.

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Bonder, Julián Fernández, Zhiwei Cheng et Hayk Mikayelyan. « Fractional optimal maximization problem and the unstable fractional obstacle problem ». Journal of Mathematical Analysis and Applications 495, n^o 1 (mars 2021) : 124686. http://dx.doi.org/10.1016/j.jmaa.2020.124686.

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Wen, Shuhuan, Xueheng Hu, Zhen Li, Hak Keung Lam, Fuchun Sun et Bin Fang. « NAO robot obstacle avoidance based on fuzzy Q-learning ». Industrial Robot : the international journal of robotics research and application 47, n^o 6 (16 octobre 2019) : 801–11. http://dx.doi.org/10.1108/ir-01-2019-0002.

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Purpose This paper aims to propose a novel active SLAM framework to realize avoid obstacles and finish the autonomous navigation in indoor environment. Design/methodology/approach The improved fuzzy optimized Q-Learning (FOQL) algorithm is used to solve the avoidance obstacles problem of the robot in the environment. To reduce the motion deviation of the robot, fractional controller is designed. The localization of the robot is based on FastSLAM algorithm. Findings Simulation results of avoiding obstacles using traditional Q-learning algorithm, optimized Q-learning algorithm and FOQL algorithm are compared. The simulation results show that the improved FOQL algorithm has a faster learning speed than other two algorithms. To verify the simulation result, the FOQL algorithm is implemented on a NAO robot and the experimental results demonstrate that the improved fuzzy optimized Q-Learning obstacle avoidance algorithm is feasible and effective. Originality/value The improved fuzzy optimized Q-Learning (FOQL) algorithm is used to solve the avoidance obstacles problem of the robot in the environment. To reduce the motion deviation of the robot, fractional controller is designed. To verify the simulation result, the FOQL algorithm is implemented on a NAO robot and the experimental results demonstrate that the improved fuzzy optimized Q-Learning obstacle avoidance algorithm is feasible and effective.

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Jeon, S., et A. Petrosyan. « Almost minimizers for certain fractional variational problems ». St. Petersburg Mathematical Journal 32, n^o 4 (9 juillet 2021) : 729–51. http://dx.doi.org/10.1090/spmj/1667.

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A notion of almost minimizers is introduced for certain variational problems governed by the fractional Laplacian, with the help of the Caffarelli–Silvestre extension. In particular, almost fractional harmonic functions and almost minimizers for the fractional obstacle problem with zero obstacle are treated. It is shown that for a certain range of parameters, almost minimizers are almost Lipschitz or C 1 , β C^{1,\beta } -regular.

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Moreno Mérida, Lourdes, et Raúl Emilio Vidal. « The obstacle problem for the infinity fractional laplacian ». Rendiconti del Circolo Matematico di Palermo Series 2 67, n^o 1 (8 novembre 2016) : 7–15. http://dx.doi.org/10.1007/s12215-016-0286-2.

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Duhé, Jean-François, Stéphane Victor, Kendric Ruiz et Pierre Melchior. « Study on obstacle avoidance for fractional artificial potential fields ». IFAC-PapersOnLine 53, n^o 2 (2020) : 3725–30. http://dx.doi.org/10.1016/j.ifacol.2020.12.2059.

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Bonafini, M., V. P. C. Le, M. Novaga et G. Orlandi. « On the obstacle problem for fractional semilinear wave equations ». Nonlinear Analysis 210 (septembre 2021) : 112368. http://dx.doi.org/10.1016/j.na.2021.112368.

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Otárola, Enrique, et Abner J. Salgado. « Finite Element Approximation of the Parabolic Fractional Obstacle Problem ». SIAM Journal on Numerical Analysis 54, n^o 4 (janvier 2016) : 2619–39. http://dx.doi.org/10.1137/15m1029801.

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Focardi, Matteo. « Homogenization of Random Fractional Obstacle Problems via Γ-Convergence ». Communications in Partial Differential Equations 34, n^o 12 (22 décembre 2009) : 1607–31. http://dx.doi.org/10.1080/03605300903300728.

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Jhaveri, Yash, et Pablo Raúl Stinga. « The obstacle problem for a fractional Monge–Ampère equation ». Communications in Partial Differential Equations 45, n^o 6 (4 décembre 2019) : 457–82. http://dx.doi.org/10.1080/03605302.2019.1697885.

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Barrios, Begoña, Alessio Figalli et Xavier Ros-Oton. « Free Boundary Regularity in the Parabolic Fractional Obstacle Problem ». Communications on Pure and Applied Mathematics 71, n^o 10 (7 mars 2018) : 2129–59. http://dx.doi.org/10.1002/cpa.21745.

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Nochetto, Ricardo H., Enrique Otárola et Abner J. Salgado. « Convergence rates for the classical, thin and fractional elliptic obstacle problems ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 373, n^o 2050 (13 septembre 2015) : 20140449. http://dx.doi.org/10.1098/rsta.2014.0449.

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We review the finite-element approximation of the classical obstacle problem in energy and max-norms and derive error estimates for both the solution and the free boundary. On the basis of recent regularity results, we present an optimal error analysis for the thin obstacle problem. Finally, we discuss the localization of the obstacle problem for the fractional Laplacian and prove quasi-optimal convergence rates.

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Receveur, Jean-Baptiste, Stéphane Victor et Pierre Melchior. « New interpretation of fractional potential fields for robust path planning ». Fractional Calculus and Applied Analysis 22, n^o 1 (25 février 2019) : 113–27. http://dx.doi.org/10.1515/fca-2019-0007.

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Abstract Trajectory planning for autonomous vehicles is a research topical subject. In previous studies, optimal intermediate targets have been used in the Potential Fields (PFs). PFs are only a path planning method, or a reactive obstacle avoidance method and not a trajectory tracking method. In this article, the PFs are interpreted as an on-line control method to follow an optimal trajectory. An analysis and methodological approach to design the attractive potential as a robust controller are proposed, and a new definition of a fractional repulsive potential to characterize the dangerousness of obstacles is developed. Simulation results on autonomous vehicles are given.

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Duhé, Jean-François, Stéphane Victor et Pierre Melchior. « Contributions on artificial potential field method for effective obstacle avoidance ». Fractional Calculus and Applied Analysis 24, n^o 2 (1 avril 2021) : 421–46. http://dx.doi.org/10.1515/fca-2021-0019.

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Abstract Obstacle avoidance is one of the main interests regarding path planning. In many situations (mostly those regarding applications in urban environments), the obstacles to be avoided are dynamical and unpredictable. This lack of certainty regarding the environment introduces the need to use local path planning techniques rather than global ones. A well-known method uses artificial potential fields introduced by Khatib. The Weyl potential definition have enabled to distinguish the dangerousness of an obstacle, however acceleration oscillations appear when the considered robot enters a danger zone close to an obstacle, thus leading to high energy consumption. In order to reduce these oscillations regarding this method, four alternative formulations for the repulsive field are proposed: corrective polynomials, tangential and radial components, Poisson potential and pseudo fractional potential. Their limitations will be explored and their performances will be compared by using criteria such as length and energy in a simulation scenario.

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Bonder, Julián Fernández, Zhiwei Cheng et Hayk Mikayelyan. « Optimal rearrangement problem and normalized obstacle problem in the fractional setting ». Advances in Nonlinear Analysis 9, n^o 1 (31 mars 2020) : 1592–606. http://dx.doi.org/10.1515/anona-2020-0067.

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Abstract We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (–Δ)s, 0 < s < 1, and the Gagliardo seminorm |u|s. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies $$\begin{array}{} \displaystyle -(-{\it\Delta})^s U-\chi_{\{U\leq 0\}}\min\{-(-{\it\Delta})^s U^+;1\}=\chi_{\{U \gt 0\}}, \end{array}$$ which happens to be the fractional analogue of the normalized obstacle problem Δu = χ{u>0}.

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Fernández-Real, Xavier, et Xavier Ros-Oton. « The obstacle problem for the fractional Laplacian with critical drift ». Mathematische Annalen 371, n^o 3-4 (30 septembre 2017) : 1683–735. http://dx.doi.org/10.1007/s00208-017-1600-9.

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Geraci, Francesco. « The classical obstacle problem with coefficients in fractional Sobolev spaces ». Annali di Matematica Pura ed Applicata (1923 -) 197, n^o 2 (14 septembre 2017) : 549–81. http://dx.doi.org/10.1007/s10231-017-0692-x.

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Eberle, Simon, Xavier Ros-Oton et Georg S. Weiss. « Characterizing compact coincidence sets in the thin obstacle problem and the obstacle problem for the fractional Laplacian ». Nonlinear Analysis 211 (octobre 2021) : 112473. http://dx.doi.org/10.1016/j.na.2021.112473.

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22

Motreanu, Dumitru, Van Thien Nguyen et Shengda Zeng. « Existence of Solutions for Implicit Obstacle Problems of Fractional Laplacian Type Involving Set-Valued Operators ». Journal of Optimization Theory and Applications 187, n^o 2 (25 septembre 2020) : 391–407. http://dx.doi.org/10.1007/s10957-020-01752-4.

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Abstract The paper is devoted to a new kind of implicit obstacle problem given by a fractional Laplacian-type operator and a set-valued term, which is described by a generalized gradient. An existence theorem for the considered implicit obstacle problem is established, using a surjectivity theorem for set-valued mappings, Kluge’s fixed point principle and nonsmooth analysis.

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Kukuljan, Teo. « The fractional obstacle problem with drift : Higher regularity of free boundaries ». Journal of Functional Analysis 281, n^o 8 (octobre 2021) : 109114. http://dx.doi.org/10.1016/j.jfa.2021.109114.

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Athanasopoulos, Ioannis, Luis Caffarelli et Emmanouil Milakis. « On the regularity of the non-dynamic parabolic fractional obstacle problem ». Journal of Differential Equations 265, n^o 6 (septembre 2018) : 2614–47. http://dx.doi.org/10.1016/j.jde.2018.04.043.

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Lévi, Laurent, et Fabrice Peyroutet. « A Time-Fractional Step Method for Conservation Law Related Obstacle Problems ». Advances in Applied Mathematics 27, n^o 4 (novembre 2001) : 768–89. http://dx.doi.org/10.1006/aama.2001.0760.

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Noor, Muhammad, Muhammad Rafiq, Salah-Ud-Din Khan, Muhammad Qureshi, Muhammad Kamran, Shahab-Ud-Din Khan, Faisal Saeed et Hijaz Ahmad. « Analytical solutions to contact problem with fractional derivatives in the sense of Caputo ». Thermal Science 24, Suppl. 1 (2020) : 313–23. http://dx.doi.org/10.2298/tsci20313n.

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The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.

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Noor, Muhammad, Muhammad Rafiq, Salah-Ud-Din Khan, Muhammad Qureshi, Muhammad Kamran, Shahab-Ud-Din Khan, Faisal Saeed et Hijaz Ahmad. « Analytical solutions to contact problem with fractional derivatives in the sense of Caputo ». Thermal Science 24, Suppl. 1 (2020) : 313–23. http://dx.doi.org/10.2298/tsci20s1313n.

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The current study extends the applications of the variational iteration method for the analytical solution of fractional contact problems. The problem involves Caputo sense while calculating the derivative of fractional order, we apply the Penalty function technique to transform it into a system of fractional boundary value problems coupled with a known obstacle. The variational iteration method is employed to find the series solution of fractional boundary value problem. For different values of fractional parameters, residual errors of solutions are plotted to make sure the convergence and accuracy of the solution. The reasonably accurate results show that one of the highly effective and stable methods for the solution of fractional boundary value problem is the method of variational iteration.

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Javeed, Shumaila, Dumitru Baleanu, Asif Waheed, Mansoor Shaukat Khan et Hira Affan. « Analysis of Homotopy Perturbation Method for Solving Fractional Order Differential Equations ». Mathematics 7, n^o 1 (3 janvier 2019) : 40. http://dx.doi.org/10.3390/math7010040.

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The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order.

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Rafiq, Muhammad, Muhammad Aslam Noor, Shabieh Farwa, Muhammad Kamran, Faisal Saeed, Khaled A. Gepreel, Shao-Wen Yao et Hijaz Ahmad. « Series solution to fractional contact problem using Caputo’s derivative ». Open Physics 19, n^o 1 (1 janvier 2021) : 402–12. http://dx.doi.org/10.1515/phys-2021-0046.

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Abstract In this article, contact problem with fractional derivatives is studied. We use fractional derivative in the sense of Caputo. We deploy penalty function method to degenerate the obstacle problem into a system of fractional boundary value problems (FBVPs). The series solution of this system of FBVPs is acquired by using the variational iteration method (VIM). The performance as well as precision of the applied method is gauged by means of significant numerical tests. We further study the convergence and residual errors of the solutions by giving variation to the fractional parameter, and graphically present the solutions and residual errors accordingly. The outcomes thus obtained witness the high effectiveness of VIM for solving FBVPs.

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Bonito, Andrea, Wenyu Lei et Abner J. Salgado. « Finite element approximation of an obstacle problem for a class of integro–differential operators ». ESAIM : Mathematical Modelling and Numerical Analysis 54, n^o 1 (janvier 2020) : 229–53. http://dx.doi.org/10.1051/m2an/2019058.

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We study the regularity of the solution to an obstacle problem for a class of integro–differential operators. The differential part is a second order elliptic operator, whereas the nonlocal part is given by the integral fractional Laplacian. The obtained smoothness is then used to design and analyze a finite element scheme.

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Korvenpää, Janne, Tuomo Kuusi et Giampiero Palatucci. « Hölder continuity up to the boundary for a class of fractional obstacle problems ». Rendiconti Lincei - Matematica e Applicazioni 27, n^o 3 (2016) : 355–67. http://dx.doi.org/10.4171/rlm/739.

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Rafiq, Muhammad, Muhammad Aslam Noor, Madeeha Tahir, Muhammad Kamran, Muhammad Amer Qureshi et Shabieh Farwa. « Efficient analytical approach to solve system of BVPs associated with fractional obstacle problem ». AIP Advances 9, n^o 9 (septembre 2019) : 095007. http://dx.doi.org/10.1063/1.5111900.

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Silvestre, Luis. « Regularity of the obstacle problem for a fractional power of the laplace operator ». Communications on Pure and Applied Mathematics 60, n^o 1 (2006) : 67–112. http://dx.doi.org/10.1002/cpa.20153.

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Cheng, Zhiwei. « Corrigendum to : “Fractional optimal maximization problem and the unstable fractional obstacle problem” [J. Math. Anal. Appl. 495 (1) (2021) 124686] ». Journal of Mathematical Analysis and Applications 510, n^o 1 (juin 2022) : 126014. http://dx.doi.org/10.1016/j.jmaa.2022.126014.

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Zigic, Miodrag, et Nenad Grahovac. « Application of Fractional Calculus to Frontal Crash Modeling ». Mathematical Problems in Engineering 2017 (2017) : 1–10. http://dx.doi.org/10.1155/2017/7419602.

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Fractional derivative viscoelastic model is used in the analysis of a frontal impact of a vehicle against a rigid obstacle. The frontal part of the vehicle is first modeled as a viscoelastic fractional rod and then it is modeled as two different viscoelastic fractional rods with a different length. In the second model also the friction is taken into account. A motion is analyzed during several phases because of both different lengths of the rods and the presence of a dry friction force in the later model. Governing systems of differential equations together with the corresponding initial conditions are derived. Parameter identification is done on the basis of the existing experimental results using the solution of a posed impact problem. What makes the problem more complex, regarding the second model, is the fact that it belongs to the class of nonsmooth fractional order systems, which require special treatment when dealing with deformation history during different motion phases.

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Barrios, Begoña, Alessio Figalli et Xavier Ros-Oton. « Global regularity for the free boundary in the obstacle problem for the fractional Laplacian ». American Journal of Mathematics 140, n^o 2 (2018) : 415–47. http://dx.doi.org/10.1353/ajm.2018.0010.

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Waheed, Asif, Syed Tauseef Mohyud-Din et Iqra Naz. « On analytical solution of system of nonlinear fractional boundary value problems associated with obstacle ». Journal of Ocean Engineering and Science 3, n^o 1 (mars 2018) : 49–55. http://dx.doi.org/10.1016/j.joes.2017.12.001.

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Petrosyan, Arshak, et Camelia A. Pop. « Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift ». Journal of Functional Analysis 268, n^o 2 (janvier 2015) : 417–72. http://dx.doi.org/10.1016/j.jfa.2014.10.009.

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Jhaveri, Yash, et Robin Neumayer. « Higher regularity of the free boundary in the obstacle problem for the fractional Laplacian ». Advances in Mathematics 311 (avril 2017) : 748–95. http://dx.doi.org/10.1016/j.aim.2017.03.006.

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Sun, HongGuang, Wen Chen et K. Y. Sze. « A semi-discrete finite element method for a class of time-fractional diffusion equations ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 371, n^o 1990 (13 mai 2013) : 20120268. http://dx.doi.org/10.1098/rsta.2012.0268.

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As fractional diffusion equations can describe the early breakthrough and the heavy-tail decay features observed in anomalous transport of contaminants in groundwater and porous soil, they have been commonly used in the related mathematical descriptions. These models usually involve long-time-range computation, which is a critical obstacle for their application; improvement of computational efficiency is of great significance. In this paper, a semi-discrete method is presented for solving a class of time-fractional diffusion equations that overcome the critical long-time-range computation problem. In the procedure, the spatial domain is discretized by the finite element method, which reduces the fractional diffusion equations to approximate fractional relaxation equations. As analytical solutions exist for the latter equations, the burden arising from long-time-range computation can effectively be minimized. To illustrate its efficiency and simplicity, four examples are presented. In addition, the method is used to solve the time-fractional advection–diffusion equation characterizing the bromide transport process in a fractured granite aquifer. The prediction closely agrees with the experimental data, and the heavy-tail decay of the anomalous transport process is well represented.

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Piccinini, Mirco. « The obstacle problem and the Perron Method for nonlinear fractional equations in the Heisenberg group ». Nonlinear Analysis 222 (septembre 2022) : 112966. http://dx.doi.org/10.1016/j.na.2022.112966.

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Garofalo, Nicola, Arshak Petrosyan, Camelia A. Pop et Mariana Smit Vega Garcia. « Regularity of the free boundary for the obstacle problem for the fractional Laplacian with drift ». Annales de l'Institut Henri Poincaré C, Analyse non linéaire 34, n^o 3 (mai 2017) : 533–70. http://dx.doi.org/10.1016/j.anihpc.2016.03.001.

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Garofalo, Nicola, et Xavier Ros-Oton. « Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian ». Revista Matemática Iberoamericana 35, n^o 5 (4 juin 2019) : 1309–65. http://dx.doi.org/10.4171/rmi/1087.

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Zhang, Y. P., Y. M. Chen, J. K. Liu et G. Meng. « Highly Accurate Solution of Limit Cycle Oscillation of an Airfoil in Subsonic Flow ». Advances in Acoustics and Vibration 2011 (23 juin 2011) : 1–10. http://dx.doi.org/10.1155/2011/926271.

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The homotopy analysis method (HAM) is employed to propose a highly accurate technique for solving strongly nonlinear aeroelastic systems of airfoils in subsonic flow. The frequencies and amplitudes of limit cycle oscillations (LCOs) arising in the considered systems are expanded as series of an embedding parameter. A series of algebraic equations are then derived, which determine the coefficients of the series. Importantly, all these equations are linear except the first one. Using some routine procedures to deduce these equations, an obstacle would arise in expanding some fractional functions as series in the embedding parameter. To this end, an approach is proposed for the expansion of fractional function. This provides us with a simple yet efficient iteration scheme to seek very-high-order approximations. Numerical examples show that the HAM solutions are obtained very precisely. At the same time, the CPU time needed can be significantly reduced by using the presented approach rather than by the usual procedure in expanding fractional functions.

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Borthagaray, Juan Pablo, Ricardo H. Nochetto et Abner J. Salgado. « Weighted Sobolev regularity and rate of approximation of the obstacle problem for the integral fractional Laplacian ». Mathematical Models and Methods in Applied Sciences 29, n^o 14 (19 décembre 2019) : 2679–717. http://dx.doi.org/10.1142/s021820251950057x.

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We obtain regularity results in weighted Sobolev spaces for the solution of the obstacle problem for the integral fractional Laplacian [Formula: see text] in a Lipschitz bounded domain [Formula: see text] satisfying the exterior ball condition. The weight is a power of the distance to the boundary [Formula: see text] of [Formula: see text] that accounts for the singular boundary behavior of the solution for any [Formula: see text]. These bounds then serve us as a guide in the design and analysis of a finite element scheme over graded meshes for any dimension [Formula: see text], which is optimal for [Formula: see text].

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Banerjee, A., D. Danielli, N. Garofalo et A. Petrosyan. « The regular free boundary in the thin obstacle problem for degenerate parabolic equations ». St. Petersburg Mathematical Journal 32, n^o 3 (11 mai 2021) : 449–80. http://dx.doi.org/10.1090/spmj/1656.

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This paper is devoted to the existence, the optimal regularity of solutions, and the regularity of the free boundary near the so-called regular points in a thin obstacle problem that arises as the local extension of the obstacle problem for the fractional heat operator ( ∂ t − Δ x ) s (\partial _t - \Delta _x)^s for s ∈ ( 0 , 1 ) s \in (0,1) . The regularity estimates are completely local in nature. This aspect is of crucial importance in our forthcoming work on the blowup analysis of the free boundary, including the study of the singular set. The approach is based on first establishing the boundedness of the time-derivative of the solution. This allows reduction to an elliptic problem at every fixed time level. By using several results from the elliptic theory, including the epiperimetric inequality, the optimal regularity is established for solutions as well as the H 1 + γ , 1 + γ 2 H^{1+\gamma ,\frac {1+\gamma }{2}} regularity of the free boundary near such regular points.

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Tantiparimongkol, Lalida, et Pattarapong Phasukkit. « IR-UWB Pulse Generation Using FPGA Scheme for through Obstacle Human Detection ». Sensors 20, n^o 13 (4 juillet 2020) : 3750. http://dx.doi.org/10.3390/s20133750.

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This research proposes a scheme of field programmable gate array (FPGA) to generate an impulse-radio ultra-wideband (IR-UWB) pulse. The FPGA scheme consists of three parts: digital clock manager, four-delay-paths stratagem, and edge combiner. The IR-UWB radar system is designed to detect human subjects from their respiration underneath the rubble in the aftermath of an earthquake and to locate the human subjects based on range estimation. The proposed IR-UWB radar system is experimented with human subjects lying underneath layers of stacked clay bricks in supine and prone position. The results reveal that the IR-UWB radar system achieves a pulse duration of 540 ps with a bandwidth of 2.073 GHz (fractional bandwidth of 1.797). In addition, the IR-UWB technology can detect human subjects underneath the rubble from respiration and identify the location of human subjects by range estimation. The novelty of this research lies in the use of the FPGA scheme to achieve an IR-UWB pulse with a 2.073 GHz (117 MHz–2.19 GHz) bandwidth, thereby rendering the technology suitable for a wide range of applications, in addition to through-obstacle detection.

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Caffarelli, Luis A., Sandro Salsa et Luis Silvestre. « Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian ». Inventiones mathematicae 171, n^o 2 (27 octobre 2007) : 425–61. http://dx.doi.org/10.1007/s00222-007-0086-6.

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Fernández-Real, Xavier, et Xavier Ros-Oton. « Free Boundary Regularity for Almost Every Solution to the Signorini Problem ». Archive for Rational Mechanics and Analysis 240, n^o 1 (11 février 2021) : 419–66. http://dx.doi.org/10.1007/s00205-021-01617-8.

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AbstractWe investigate the regularity of the free boundary for the Signorini problem in $${\mathbb {R}}^{n+1}$$ R n + 1 . It is known that regular points are $$(n-1)$$ ( n - 1 ) -dimensional and $$C^\infty $$ C ∞ . However, even for $$C^\infty $$ C ∞ obstacles $$\varphi $$ φ , the set of non-regular (or degenerate) points could be very large—e.g. with infinite $${\mathcal {H}}^{n-1}$$ H n - 1 measure. The only two assumptions under which a nice structure result for degenerate points has been established are when $$\varphi $$ φ is analytic, and when $$\Delta \varphi < 0$$ Δ φ < 0 . However, even in these cases, the set of degenerate points is in general $$(n-1)$$ ( n - 1 ) -dimensional—as large as the set of regular points. In this work, we show for the first time that, “usually”, the set of degenerate points is small. Namely, we prove that, given any $$C^\infty $$ C ∞ obstacle, for almost every solution the non-regular part of the free boundary is at most $$(n-2)$$ ( n - 2 ) -dimensional. This is the first result in this direction for the Signorini problem. Furthermore, we prove analogous results for the obstacle problem for the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s , and for the parabolic Signorini problem. In the parabolic Signorini problem, our main result establishes that the non-regular part of the free boundary is $$(n-1-\alpha _\circ )$$ ( n - 1 - α ∘ ) -dimensional for almost all times t, for some $$\alpha _\circ > 0$$ α ∘ > 0 . Finally, we construct some new examples of free boundaries with degenerate points.

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Colli, Pierluigi, Gianni Gilardi et Jürgen Sprekels. « Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials ». Discrete & ; Continuous Dynamical Systems - S 14, n^o 1 (2021) : 243–71. http://dx.doi.org/10.3934/dcdss.2020213.

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