Journal articles: 'Diffusion Operator'

1

Franců, Jan. "Homogenization of diffusion equation with scalar hysteresis operator." Mathematica Bohemica 126, no. 2 (2001): 363–77. http://dx.doi.org/10.21136/mb.2001.134031.

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Antoniou, I., I. Prigogine, V. Sadovnichii, and S. A. Shkarin. "Time operator for diffusion." Chaos, Solitons & Fractals 11, no. 4 (March 2000): 465–77. http://dx.doi.org/10.1016/s0960-0779(99)00052-1.

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Igbida, Noureddine, and Thi Nguyet Nga Ta. "Sub-gradient diffusion operator." Journal of Differential Equations 262, no. 7 (April 2017): 3837–63. http://dx.doi.org/10.1016/j.jde.2016.11.034.

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Cantin, Pierre, and Alexandre Ern. "Vertex-Based Compatible Discrete Operator Schemes on Polyhedral Meshes for Advection-Diffusion Equations." Computational Methods in Applied Mathematics 16, no. 2 (April 1, 2016): 187–212. http://dx.doi.org/10.1515/cmam-2016-0007.

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AbstractWe devise and analyze vertex-based, Péclet-robust, lowest-order schemes for advection-diffusion equations that support polyhedral meshes. The schemes are formulated using Compatible Discrete Operators (CDO), namely, primal and dual discrete differential operators, a discrete contraction operator for advection, and a discrete Hodge operator for diffusion. Moreover, discrete boundary operators are devised to weakly enforce Dirichlet boundary conditions. The analysis sheds new light on the theory of Friedrichs' operators at the purely algebraic level. Moreover, an extension of the stability analysis hinging on inf-sup conditions is presented to incorporate divergence-free velocity fields under some assumptions. Error bounds and convergence rates for smooth solutions are derived and numerical results are presented on three-dimensional polyhedral meshes.

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Vizilter, Y. V., O. V. Vygolov, and S. Y. Zheltov. "Comparison of statistical properties for various morphological filters based on mosaic image shape models." Computer Optics 45, no. 3 (June 2021): 449–60. http://dx.doi.org/10.18287/2412-6179-co-842.

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We consider the statistical properties of different mosaic filters. We demonstrate that in Pitiev's morphology, the measure of shape complexity is directly related to the shape simplicity measure based on morphological correlation coefficient (MCC). Based on MCC, we introduce the normalized morphological simplification index (NMSI). Using NMSI, we show that the simpler the mosaic shape, the more shape simplification is provided by the corresponding Pyt'ev projector. For the examples of mean and median mosaic filters, we address the problem of different operator comparison. In this context we introduce the concept of statistically simplifying morphological operators. Morphological correlation of mosaic shape and diffusion mosaic operator is considered. We prove that the NMSI for the diffusion mosaic operator is not related to the complexity for the corresponding diffusion shape kernel. Thus, a principal qualitative difference in the relationship between relational and operator models for diffuse and projective mosaic linear filters is demonstrated.

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Пененко, А. В., and A. V. Penenko. "Numerical Algorithms for Diffusion Coefficient Identification in Problems of Tissue Engineering." Mathematical Biology and Bioinformatics 11, no. 2 (December 22, 2016): 426–44. http://dx.doi.org/10.17537/2016.11.426.

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Identification algorithms of diffusion coefficients in a specimen with tomographic images of the solution penetration dynamics are considered. With the sensitivity operator, built on the basis of adjoint equations for diffusion process model, the corresponding coefficient inverse problem is reduced to the quasilinear operator equation which is then solved by the Newton-type method with successive evaluation of r-pseudo inverse operators of increasing dimensionality. The efficiency of the constructed algorithm is tested in numerical experiments. For comparison, a gradient-based algorithm for the inverse problem solution is considered.

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Ahmed, Nauman, Tahira S.S., M. Rafiq, M. A. Rehman, Mubasher Ali, and M. O. Ahmad. "Positivity preserving operator splitting nonstandard finite difference methods for SEIR reaction diffusion model." Open Mathematics 17, no. 1 (April 29, 2019): 313–30. http://dx.doi.org/10.1515/math-2019-0027.

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Abstract In this work, we will introduce two novel positivity preserving operator splitting nonstandard finite difference (NSFD) schemes for the numerical solution of SEIR reaction diffusion epidemic model. In epidemic model of infection diseases, positivity is an important property of the continuous system because negative value of a subpopulation is meaningless. The proposed operator splitting NSFD schemes are dynamically consistent with the solution of the continuous model. First scheme is conditionally stable while second operator splitting scheme is unconditionally stable. The stability of the diffusive SEIR model is also verified numerically with the help of Routh-Hurwitz stability condition. Bifurcation value of transmission coefficient is also carried out with and without diffusion. The proposed operator splitting NSFD schemes are compared with the well-known operator splitting finite difference (FD) schemes.

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NATAF, F., and F. ROGIER. "FACTORIZATION OF THE CONVECTION-DIFFUSION OPERATOR AND THE SCHWARZ ALGORITHM." Mathematical Models and Methods in Applied Sciences 05, no. 01 (February 1995): 67–93. http://dx.doi.org/10.1142/s021820259500005x.

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In the original Schwarz algorithm, Dirichlet boundary conditions are used as interface conditions. We consider the use of the operators arising from the factorization of the convection-diffusion operator as transmission conditions. The rate of convergence is then significantly higher. Theoretical results are proven and numerical tests are shown.

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Sat, Murat, and Etibar S. Panakhov. "Spectral problem for diffusion operator." Applicable Analysis 93, no. 6 (July 23, 2013): 1178–86. http://dx.doi.org/10.1080/00036811.2013.821113.

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10

Davies, E. B. "An Indefinite Convection-Diffusion Operator." LMS Journal of Computation and Mathematics 10 (2007): 288–306. http://dx.doi.org/10.1112/s1461157000001418.

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AbstractWe give a mathematically rigorous analysis which confirms the surprising results in a recent paper of Benilov, O‘Brien and Sazonov [J. Fluid Mech. 497 (2003) 201-224] about the spectrum of a highly singular non-self-adjoint operator that arises in a problem in fluid mechanics. We also show that the set of eigenvectors does not form a basis for the operator.

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11

dos Santos, Maike. "Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting." Physics 1, no. 1 (March 6, 2019): 40–58. http://dx.doi.org/10.3390/physics1010005.

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In this work, we investigate a series of mathematical aspects for the fractional diffusion equation with stochastic resetting. The stochastic resetting process in Evans–Majumdar sense has several applications in science, with a particular emphasis on non-equilibrium physics and biological systems. We propose a version of the stochastic resetting theory for systems in which the reset point is in motion, so the walker does not return to the initial position as in the standard model, but returns to a point that moves in space. In addition, we investigate the proposed stochastic resetting model for diffusion with the fractional operator of Prabhakar. The derivative of Prabhakar consists of an integro-differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes a series of fractional operators such as Riemann–Liouville–Caputo. We present how the generalized model of stochastic resetting for fractional diffusion implies a rich class of anomalous diffusive processes, i.e., ⟨ ( Δ x ) 2 ⟩ ∝ t α , which includes sub-super-hyper-diffusive regimes. In the sequence, we generalize these ideas to the fractional Fokker–Planck equation for quadratic potential U ( x ) = a x 2 + b x + c . This work aims to present the generalized model of Evans–Majumdar’s theory for stochastic resetting under a new perspective of non-static restart points.

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12

Hanyga, Andrzej, and Richard L. Magin. "A new anisotropic fractional model of diffusion suitable for applications of diffusion tensor imaging in biological tissues." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2170 (October 8, 2014): 20140319. http://dx.doi.org/10.1098/rspa.2014.0319.

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An anomalous anisotropic diffusion equation is constructed in which the order of the spatial pseudo-differential operator is generalized to be distributed with a directionally dependent distribution. A time fractional version of this equation is also considered. First, it is proved that the equation is positivity-preserving and properly normalized. Second, the existence of a smooth Green's function solution is proved. Finally, an expression for the diffusive flux density for this new fractional order process is calculated. This approach may find utility in modelling diffusion tensor imaging data in the white matter of the human brain where both the apparent diffusion coefficient and the order of the pseudo-differential operator are anisotropic.

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13

Wang, Yi-Xiang J., Xian-Lun Zhu, Min Deng, Deyond Y. W. Siu, Jason C. S. Leung, Queenie Chan, Danny T. M. Chan, Calvin Hoi Kwan Mak, and Wai S. Poon. "The use of diffusion tensor tractography to measure the distance between the anterior tip of the Meyer loop and the temporal pole in a cohort from Southern China." Journal of Neurosurgery 113, no. 6 (December 2010): 1144–51. http://dx.doi.org/10.3171/2010.7.jns10393.

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Object Anterior temporal lobe resection plus amygdalohippocampectomy can cause damage to the anterior portion of the optic radiation, also known as the Meyer loop, resulting in homonymous superior quadrantanopia. Magnetic resonance diffusion tensor tractography (DTT) of the Meyer loop can help in surgical planning. In this study, the distance of the anterior tip of the Meyer loop to the temporal lobe pole (ML-TP) in the Southern Chinese population was assessed. Methods The authors studied 16 Southern Chinese individuals (8 men and 8 women; mean age 45.6 years, range 21–60 years). Diffusion tensor images were obtained with a 3-T MR imaging system using a single-shot spin echo echo planar imaging sequence. Two trained operators, one neurosurgeon (Operator A) and one radiologist (Operator B), carried out the DTT analysis with software iPlan (BrainLAB) and FiberTrak (Philips). Results For the 32 temporal lobes, the intraclass correlation coefficient (ICC) of the 2 operators' results using iPlan was 0.96, while that of Operator A using iPlan and Operator B using FiberTrak was 0.75. The ICC of Operator B using iPlan and FiberTrak was 0.81. The ML-TP distance of normal lobes (30 lobes [2 lobes that previously underwent surgery were excluded]) was 36.3 ± 5.5 mm (range 26.6–48.9 mm), 36.3 ± 5.3 mm (range 26.8–48.2 mm), and 35.9 ± 6.4 mm (range 20.8–48.4 mm) for Operator A using iPlan, Operator B using iPlan, and Operator B using FiberTrak, respectively (p > 0.05). Conclusions The 2 operators reached good agreement on ML-TP distance measurement using DTT. The DDT results can be more software dependent than operator dependent. The measurement with FiberTrak demonstrated larger range and standard deviation than measurement with iPlan.

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14

Nambu, Takao. "The Ljapunov equation and an application to stabilisation of one-dimensional diffusion equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 104, no. 1-2 (1986): 39–52. http://dx.doi.org/10.1017/s0308210500019053.

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SynopsisA Ljapunov equation XL − BX = C appears in stabilisation studies of linear systems. Here, the operators L, B, and C are given linear operators working in infinite-dimensional Hilbert spaces, which are derived from a specific control system. We have so far considered the case where L is a general elliptic operator of order 2 in a bounded domain of an Euclidean space. When L is instead a self-adjoint elliptic operator working in an interval of ℝ1, we derive here a stronger geometrical character of the solution X to the Ljapunov equation. The result is applied to stabilisation of one-dimensional diffusion equations.

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Klus, Stefan, Feliks Nüske, and Boumediene Hamzi. "Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator." Entropy 22, no. 7 (June 30, 2020): 722. http://dx.doi.org/10.3390/e22070722.

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Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

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16

Klemp, Joseph B. "Damping Characteristics of Horizontal Laplacian Diffusion Filters." Monthly Weather Review 145, no. 11 (November 2017): 4365–79. http://dx.doi.org/10.1175/mwr-d-17-0015.1.

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Horizontally diffusive computational damping terms are frequently employed in 3D atmospheric simulation models to enhance stability and to suppress small-scale noise. In configuring these filters, it is desirable that damping effects are concentrated on the smaller-scale disturbances close to the grid scale and that the dissipation is spatially isotropic. On Cartesian meshes, the isotropy of the damping can vary greatly depending on the numerical formulation of the horizontal filter. The most isotropic behavior appears to result from recursive application of a 2D Laplacian that combines both along-axis and diagonal contributions. Also, the recursive application of 1D Laplacians in each coordinate direction provides better isotropy than the recursive application of the 2D Laplacian represented with a five-point operator. Increased isotropy also permits a larger maximum diffusivity, which may be beneficial in certain filter applications. On hexagonal and triangular meshes, Laplacian operators exhibit excellent isotropy, owing to the more isotropic nature of the meshes. However, previous research has established that straightforward application of the Laplacian may yield a diffusion operator that damps both resolved physical modes and unresolved high-wavenumber (aliased) modes, but it does not converge to the proper analytic behavior. Special averaging is then required to recover an accurate representation for the Laplacian. A consequence of this averaging is that the resulting filters do not act on the aliased modes (the checkerboard mode in particular) and thus employing the unaveraged diffusion operators may be preferable. The damping characteristics and stability constraints are derived for both the unaveraged and averaged Laplacian filters for C-grid staggering on these meshes.

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17

Harlamov, B. P. "Characteristic Operator of a Diffusion Process." Journal of Mathematical Sciences 128, no. 1 (July 2005): 2625–39. http://dx.doi.org/10.1007/s10958-005-0211-2.

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Reddy, Satish C., and Lloyd N. Trefethen. "Pseudospectra of the Convection-Diffusion Operator." SIAM Journal on Applied Mathematics 54, no. 6 (December 1994): 1634–49. http://dx.doi.org/10.1137/s0036139993246982.

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Nagel, Rainer. "Operator matrices and reaction-diffusion systems." Rendiconti del Seminario Matematico e Fisico di Milano 59, no. 1 (December 1989): 185–96. http://dx.doi.org/10.1007/bf02925301.

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20

Ergün, Abdullah. "Inverse problem for singular diffusion operator." Miskolc Mathematical Notes 22, no. 1 (2021): 173. http://dx.doi.org/10.18514/mmn.2021.3377.

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Vabishchevich, Nikolay, and Petr Vabishchevich. "VAGO METHOD FOR THE SOLUTION OF ELLIPTIC SECOND‐ORDER BOUNDARY VALUE PROBLEMS." Mathematical Modelling and Analysis 15, no. 4 (November 15, 2010): 533–45. http://dx.doi.org/10.3846/1392-6292.2010.15.533-545.

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Mathematical physics problems are often formulated using differential operators of vector analysis, i.e. invariant operators of first order, namely, divergence, gradient and rotor (curl) operators. In approximation of such problems it is natural to employ similar operator formulations for grid problems. The VAGO (Vector Analysis Grid Operators) method is based on such a methodology. In this paper the vector analysis difference operators are constructed using the Delaunay triangulation and the Voronoi diagrams. Further the VAGO method is used to solve approximately boundary value problems for the general elliptic equation of second order. In the convection‐diffusion‐reaction equation the diffusion coefficient is a symmetric tensor of second order.

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Borisov, Leonid A., Yuriy N. Orlov, and Vsevolod Zh Sakbaev. "Feynman averaging of semigroups generated by Schrödinger operators." Infinite Dimensional Analysis, Quantum Probability and Related Topics 21, no. 02 (June 2018): 1850010. http://dx.doi.org/10.1142/s0219025718500108.

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The extension of averaging procedure for operator-valued function is defined by means of the integration of measurable map with respect to complex-valued measure or pseudomeasure. The averaging procedure of one-parametric semigroups of linear operators based on Chernoff equivalence for operator-valued functions is constructed. The initial value problem solutions are investigated for fractional diffusion equation and for Schrödinger equation with relativistic Hamiltonian of free motion. It is established that in these examples the solution of evolutionary equation can be obtained by applying the constructed averaging procedure to the random translation operators in classical coordinate space.

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Jurado, Francisco, and Andrés A. Ramírez. "State Feedback Regulation Problem to the Reaction-Diffusion Equation." Mathematics 8, no. 11 (November 6, 2020): 1983. http://dx.doi.org/10.3390/math8111983.

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In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator.

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Xu, Xing-Lei, Xiu-Xia Wang, and Hong-Yi Fan. "Energy variation of mesoscopic L–C electric circuit in external electromagnetic field." International Journal of Modern Physics B 29, no. 23 (September 17, 2015): 1550169. http://dx.doi.org/10.1142/s0217979215501696.

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In this paper, we investigate energy variation of mesoscopic L–C electric circuit in external electromagnetic field (EMF) due to the energy flow of EMF, we consider this a diffusion process governed by the master equation for diffusion channel with the diffusion rate being determined by the energy flow of EMF. By using the entangled state representation and the method of integration within ordered product of operators we derive time evolution law of the initial density operator of the circuit and the energy variation formula.

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Malyk, Igor V., Mykola Gorbatenko, Arun Chaudhary, Shivani Sharma, and Ravi Shanker Dubey. "Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator." Fractal and Fractional 5, no. 3 (July 2, 2021): 64. http://dx.doi.org/10.3390/fractalfract5030064.

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In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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Almady, Wasif. "Analytical Solution for Boltzmann Collision Operator for the1-D Diffusion equation." International Journal for Research in Applied Science and Engineering Technology 9, no. 9 (September 30, 2021): 1514–17. http://dx.doi.org/10.22214/ijraset.2021.38189.

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Abstract: In this paper, we have presented the analytical solution of the collision operator for the Boltzmann equation of onedimensional diffusion equation using the analytical solution of the one-dimensional Navier Stoke diffusion equation. Keywords: Boltzmann equation; analytical collision operator; one-dimensional diffusion equation.

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FAN, HONG-YI, and LI-YUN HU. "OPERATOR-SUM REPRESENTATION OF DENSITY OPERATORS AS SOLUTIONS TO MASTER EQUATIONS OBTAINED VIA THE ENTANGLED STATE APPROACH." Modern Physics Letters B 22, no. 25 (October 10, 2008): 2435–68. http://dx.doi.org/10.1142/s0217984908017072.

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We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.

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Li, Qiang, Guotao Wang, and Mei Wei. "Monotone iterative technique for time-space fractional diffusion equations involving delay." Nonlinear Analysis: Modelling and Control 26, no. 2 (March 1, 2021): 241–58. http://dx.doi.org/10.15388/namc.2021.26.21656.

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This paper considers the initial boundary value problem for the time-space fractional delayed diffusion equation with fractional Laplacian. By using the semigroup theory of operators and the monotone iterative technique, the existence and uniqueness of mild solutions for the abstract time-space evolution equation with delay under some quasimonotone conditions are obtained. Finally, the abstract results are applied to the time-space fractional delayed diffusion equation with fractional Laplacian operator, which improve and generalize the recent results of this issue.

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SOKOLOV, I. M., and A. V. CHECHKIN. "ANOMALOUS DIFFUSION AND GENERALIZED DIFFUSION EQUATIONS." Fluctuation and Noise Letters 05, no. 02 (June 2005): L275—L282. http://dx.doi.org/10.1142/s0219477505002653.

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Fractional diffusion equations are widely used to describe anomalous diffusion processes where the characteristic displacement scales as a power of time. The forms of such equations might differ with respect to the position of the corresponding fractional operator in addition to or instead of the whole-number derivative in the Fick's equation. For processes lacking simple scaling the corresponding description may be given by distributed-order equations. In the present paper different forms of distributed-order diffusion equations are considered. The properties of their solutions are discussed for a simple special case.

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YAN, JUN, and GUOLIANG SHI. "MULTIPLICITIES OF EIGENVALUES OF THE DIFFUSION OPERATOR WITH RANDOM JUMPS FROM THE BOUNDARY." Bulletin of the Australian Mathematical Society 99, no. 1 (November 28, 2018): 101–13. http://dx.doi.org/10.1017/s0004972718001120.

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This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\unicode[STIX]{x1D6E5}(\unicode[STIX]{x1D706})$. This is a new criterion for determining the multiplicities of eigenvalues for concrete operators.

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Gárriz, A., and L. I. Ignat. "A non-local coupling model involving three fractional Laplacians." Bulletin of Mathematical Sciences 11, no. 02 (June 29, 2021): 2150007. http://dx.doi.org/10.1142/s1664360721500077.

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In this paper, we study a non-local diffusion problem that involves three different fractional Laplacian operators acting on two domains. Each domain has an associated operator that governs the diffusion on it, and the third operator serves as a coupling mechanism between the two of them. The model proposed is the gradient flow of a non-local energy functional. In the first part of the paper, we provide results about existence of solutions and the conservation of mass. The second part encompasses results about the [Formula: see text] decay of the solutions. The third part is devoted to study, the asymptotic behavior of the solutions of the problem when the two domains are a ball and its complementary. Exterior fractional Sobolev and Nash inequalities of independent interest are also provided in Appendix A.

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Babaoglu, Mine, and Etibar S. Panakhov. "On Some Spectral Problems for Diffusion Operator." ITM Web of Conferences 13 (2017): 01036. http://dx.doi.org/10.1051/itmconf/20171301036.

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Schulz, Michael, and Alice L. Newman. "Eigenfunctions of the magnetospheric radial-diffusion operator." Physica Scripta 37, no. 4 (April 1, 1988): 632–39. http://dx.doi.org/10.1088/0031-8949/37/4/023.

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Vera, M., E. Gonzalez, Y. Huérfano, E. Gelvez, and O. Valbuena. "New anisotropic diffusion operator in images filtering." Journal of Physics: Conference Series 1448 (January 2020): 012019. http://dx.doi.org/10.1088/1742-6596/1448/1/012019.

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Koyunbakan, Hikmet. "Reconstruction of Potential Function for Diffusion Operator." Numerical Functional Analysis and Optimization 30, no. 1-2 (February 20, 2009): 111–20. http://dx.doi.org/10.1080/01630560802279256.

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Ramos, J. I. "Hermitian operator methods for reaction-diffusion equations." Numerical Methods for Partial Differential Equations 3, no. 4 (1987): 241–87. http://dx.doi.org/10.1002/num.1690030402.

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Guebbai, Hamza, and Alain Largillier. "Spectra and pseudospectra of convection-diffusion operator." Lobachevskii Journal of Mathematics 33, no. 3 (July 2012): 274–83. http://dx.doi.org/10.1134/s1995080212030079.

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Akhtyamov, A. M. "Degenerate boundary conditions for the diffusion operator." Differential Equations 53, no. 11 (November 2017): 1515–18. http://dx.doi.org/10.1134/s0012266117110143.

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Du, Feng, Qiaoling Wang, Levi Adriano, and Rosane Gomes Pereira. "Eigenvalue inequalities for the Markov diffusion operator." Monatshefte für Mathematik 185, no. 2 (December 26, 2017): 207–30. http://dx.doi.org/10.1007/s00605-017-1146-7.

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Sene, Ndolane. "Fractional diffusion equation with new fractional operator." Alexandria Engineering Journal 59, no. 5 (October 2020): 2921–26. http://dx.doi.org/10.1016/j.aej.2020.03.027.

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Fan, Hong-Yi, and Qian-Fan Chen. "Density operator for describing driven damped harmonic oscillator in the diffusion-limited channel." Canadian Journal of Physics 92, no. 10 (October 2014): 1069–73. http://dx.doi.org/10.1139/cjp-2013-0488.

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In the present work, the Kraus-operator sum representation of the exact solution to the master equation describing a harmonic oscillator (driven by an external source) damping in the diffusion-limited channel is obtained by virtue of the thermo-entangled state representation and the technique of integration within an ordered product of operators. According to this solution, the initial coherent state will evolve into a displaced chaotic state that manifestly exhibits quantum decoherence.

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Santos, Maike A. F. dos. "Mittag–Leffler Memory Kernel in Lévy Flights." Mathematics 7, no. 9 (August 21, 2019): 766. http://dx.doi.org/10.3390/math7090766.

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In this article, we make a detailed study of some mathematical aspects associated with a generalized Lévy process using fractional diffusion equation with Mittag–Leffler kernel in the context of Atangana–Baleanu operator. The Lévy process has several applications in science, with a particular emphasis on statistical physics and biological systems. Using the continuous time random walk, we constructed a fractional diffusion equation that includes two fractional operators, the Riesz operator to Laplacian term and the Atangana–Baleanu in time derivative, i.e., a A B D t α ρ ( x , t ) = K α , μ ∂ x μ ρ ( x , t ) . We present the exact solution to model and discuss how the Mittag–Leffler kernel brings a new point of view to Lévy process. Moreover, we discuss a series of scenarios where the present model can be useful in the description of real systems.

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Esquinas, J., and J. López-Gómez. "Multiparameter bifurcation for some particular reaction–diffusion systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 112, no. 1-2 (1989): 135–43. http://dx.doi.org/10.1017/s0308210500028201.

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SynopsisIn some cases, a reaction–diffusion system can be transformed into an abstract equation where the linear part is given by a polynomial of a linear operator, say Multiparameter bifurcation for this equation is considered as the coefficients of the operator polynomial in are varied.

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Sin, Chung-Sik. "Cauchy problem for general time fractional diffusion equation." Fractional Calculus and Applied Analysis 23, no. 5 (October 1, 2020): 1545–59. http://dx.doi.org/10.1515/fca-2020-0077.

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Abstract In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis technique. Then, based on the representation of the solution of the inhomogenous linear ordinary differential equation with the general Caputo-type operator, the general diffusion equation with source term is studied.

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Yang, Chuan-Fu. "Reconstruction of the Diffusion Operator from Nodal Data." Zeitschrift für Naturforschung A 65, no. 1-2 (January 1, 2010): 100–106. http://dx.doi.org/10.1515/zna-2010-1-211.

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AbstractIn this paper, we deal with the inverse problem of reconstructing the diffusion equation on a finite interval. We prove that a dense subset of nodal points uniquely determine the boundary conditions and the coefficients of the diffusion equation. We also provide constructive procedure for them.

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Ropp, David L., and John N. Shadid. "Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems." Journal of Computational Physics 203, no. 2 (March 2005): 449–66. http://dx.doi.org/10.1016/j.jcp.2004.09.004.

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Zhao, Dong Hong. "Adaptive Fourth Order Partial Differential Equation Filter from the Webers Total Variation for Image Restoration." Applied Mechanics and Materials 475-476 (December 2013): 394–400. http://dx.doi.org/10.4028/www.scientific.net/amm.475-476.394.

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By substituting an anisotropic diffusion operator for the isotropic Laplace operator in the directed diffusion equation, an improvd directed diffusion equation model is proposed. To overcome the staircasing effects and simultaneously avoid edge blurring, this paper proposed an adaptive fourth order partial differential equation from the Webers Total Variation for Image Restoration. This functional is not only to use Laplace operator but also to add the human psychology system, this paper show numerical evidence of the power of resolution of the model with respect to other known models as the Perona-Malik model. Compared results disctincly demonstrate the superiority of our proposed scheme , in terms of removing noise while sharply maintaining the edge features.

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Yu, Jimin, Rumeng Zhai, Shangbo Zhou, and LiJian Tan. "Image Denoising Based on Adaptive Fractional Order with Improved PM Model." Mathematical Problems in Engineering 2018 (2018): 1–11. http://dx.doi.org/10.1155/2018/9620754.

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In order to improve the image quality, in this paper, we propose an improved PM model. In the proposed model, we introduce two novel diffusion coefficients and a residual error term and replace the integer differential operator with the fractional differential operator in the PM model. The diffusion coefficients can be used effectively for edge detection and noise removal. The residual error term can help to prevent image distortion. Fractional order differential operator has a good characteristic that it can enhance image texture information while removing image noise. Additionally, in the two new diffusion coefficients, a novel method is proposed for automatically setting parameter k, and it does not need to do any experiments to get the value of k. For the computing fractional order diffusion coefficient, we employ the discrete Fourier transform, and an iterative scheme is carried out in the frequency domain. In the proposed model, not only is the integer differential operator replaced with the fractional differential operator, but also the order of the fractional differentiation is determined adaptively with the local variance. Comparing with some existing models, the experimental results show that the proposed algorithm can not only better suppress noise, but also better preserve edge and texture information. Moreover, the running time is greatly reduced.

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DE FALCO, I., R. DEL BALIO, E. TARANTINO, and R. VACCARO. "SIMULATION OF GENETIC ALGORITHMS ON MIMD MULTICOMPUTERS." Parallel Processing Letters 02, no. 04 (December 1992): 381–89. http://dx.doi.org/10.1142/s0129626492000532.

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In this paper, a Parallel Genetic Algorithm has been developed and mapped onto a coarse grain MIMD multicomputer whose processors have been configured in a fully connected chordal ring topology. In this way, parallel diffusion processes of good local information among processors have been carried out. The Parallel Genetic Algorithm has been applied, specifically, to the Travelling Salesman Problem. Many experiments have been performed with different combinations of genetic operators; the test results suggest that PMX crossover can be avoided by using only the inversion genetic operator and that a diffusion process leads to improved search in Parallel Genetic Algorithms.

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Keller, Sarah, Zhiyue J. Wang, Amir Golsari, Anne Catherine Kim, Hendrik Kooijman, Gerhard Adam, and Jin Yamamura. "Feasibility of peripheral nerve MR neurography using diffusion tensor imaging adapted to skeletal muscle disease." Acta Radiologica 59, no. 5 (August 10, 2017): 560–68. http://dx.doi.org/10.1177/0284185117726100.

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Background Diffusion tensor imaging (DTI) of peripheral nerves may provide additional information about nerve involvement in muscular disorders, but is considered difficult due to different optimal scan parameters tailored to magnetic resonance (MR) signal properties of muscle and neural tissues. Purpose To assess the feasibility of sciatic nerve DTI using two different approaches of region of interest (ROI)-localization in DTI scans with b-values 500 s/mm2, in participants with muscular disorders and in controls. Material and Methods DTI of the thigh was conducted on a 3T system in ten patients (6 men, 4 women; mean age =54 ± 15 years) with neuromuscular disorders and ten controls. T1-weighted (T1W) images were co-registered to fractional anisotropy (FA) color-encoded images. The apparent diffusion coefficient (ADC), FA, and fiber track length (FTL) were analyzed by two operators using a freehand ROI and a single-point ROI covering the sciatic nerve. Interclass correlation coefficient (ICC) and Bland–Altman analysis were used for evaluation of inter-operator and inter-technical agreement, respectively. Results Three-dimensional visualization of sciatic nerve fiber was achievable using both techniques. The ICC of DTI metrics showed excellent inter-operator agreement both in patients and controls. Bland–Altman analysis revealed good agreement of both techniques. A maximum FTL was achieved using the single-point ROI technique, but with a lower inter-operator agreement (ICC = 0.99 vs. 0.83). The ADC and maximum FTL were significantly decreased in patients compared to controls. Conclusion Both ROI localization techniques are feasible to analyze the sciatic nerve in the setting of muscular disease. A maximum FTL is reached using the single-point ROI, however, at the cost of lower inter-operator agreement.

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Journal articles on the topic 'Diffusion Operator'

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