Journal articles: 'Geometric PDEs'

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Shardlow, Tony. "Geometric ergodicity for stochastic pdes." Stochastic Analysis and Applications 17, no. 5 (January 1999): 857–69. http://dx.doi.org/10.1080/07362999908809639.

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Krantz, Steven G., and Vicentiu D. Radulescu. "Perspectives of Geometric Analysis in PDEs." Journal of Geometric Analysis 30, no. 2 (November 1, 2019): 1411. http://dx.doi.org/10.1007/s12220-019-00303-2.

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Vitagliano, Luca. "Characteristics, bicharacteristics and geometric singularities of solutions of PDEs." International Journal of Geometric Methods in Modern Physics 11, no. 09 (October 2014): 1460039. http://dx.doi.org/10.1142/s0219887814600391.

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Many physical systems are described by partial differential equations (PDEs). Determinism then requires the Cauchy problem to be well-posed. Even when the Cauchy problem is well-posed for generic Cauchy data, there may exist characteristic Cauchy data. Characteristics of PDEs play an important role both in Mathematics and in Physics. I will review the theory of characteristics and bicharacteristics of PDEs, with a special emphasis on intrinsic aspects, i.e. those aspects which are invariant under general changes of coordinates. After a basically analytic introduction, I will pass to a modern, geometric point of view, presenting characteristics within the jet space approach to PDEs. In particular, I will discuss the relationship between characteristics and singularities of solutions and observe that: "wave-fronts are characteristic surfaces and propagate along bicharacteristics". This remark may be understood as a mathematical formulation of the wave/particle duality in optics and/or quantum mechanics. The content of the paper reflects the three-hour mini-course that I gave at the XXII International Fall Workshop on Geometry and Physics, September 2–5, 2013, Évora, Portugal.

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Bezerra Júnior, Elzon C., João Vitor da Silva, and Gleydson C. Ricarte. "Geometric estimates for doubly nonlinear parabolic PDEs." Nonlinearity 35, no. 5 (April 21, 2022): 2334–62. http://dx.doi.org/10.1088/1361-6544/ac636e.

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Abstract In this manuscript, we establish C loc α , α θ regularity estimates for bounded weak solutions of a certain class of doubly degenerate evolution PDEs, whose simplest model case is given by ∂ u ∂ t − d i v ( m | u | m − 1 | ∇ u | p − 2 ∇ u ) = f ( x , t ) in Ω T ≔ Ω × ( 0 , T ) , where m ⩾ 1, p ⩾ 2 and f belongs to a suitable anisotropic Lebesgue space. Employing intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In this scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence of our findings and approach, we address a Liouville type result for entire weak solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also present examples of degenerate PDEs where our results can be applied.

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Tehseen, Naghmana, and Geoff Prince. "Integration of PDEs by differential geometric means." Journal of Physics A: Mathematical and Theoretical 46, no. 10 (February 21, 2013): 105201. http://dx.doi.org/10.1088/1751-8113/46/10/105201.

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NODA, TAKAHIRO, and KAZUHIRO SHIBUYA. "ON IMPLICIT SECOND-ORDER PDE OF A SCALAR FUNCTION ON A PLANE VIA DIFFERENTIAL SYSTEMS." International Journal of Mathematics 22, no. 07 (July 2011): 907–24. http://dx.doi.org/10.1142/s0129167x11007069.

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In the present paper, we study implicit second-order PDEs (i.e. partial differential equations) of single type for one unknown function of two variables. In particular, by using the theory of differential systems, we give a geometric characterization of PDEs which have a certain singularity. Moreover, we provide a new invariant of PDEs under contact transformations.

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Udriste, Constantin, and Ionel Tevy. "Geometric Dynamics on Riemannian Manifolds." Mathematics 8, no. 1 (January 3, 2020): 79. http://dx.doi.org/10.3390/math8010079.

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The purpose of this paper is threefold: (i) to highlight the second order ordinary differential equations (ODEs) as generated by flows and Riemannian metrics (decomposable single-time dynamics); (ii) to analyze the second order partial differential equations (PDEs) as generated by multi-time flows and pairs of Riemannian metrics (decomposable multi-time dynamics); (iii) to emphasise second order PDEs as generated by m-distributions and pairs of Riemannian metrics (decomposable multi-time dynamics). We detail five significant decomposed dynamics: (i) the motion of the four outer planets relative to the sun fixed by a Hamiltonian, (ii) the motion in a closed Newmann economical system fixed by a Hamiltonian, (iii) electromagnetic geometric dynamics, (iv) Bessel motion generated by a flow together with an Euclidean metric (created motion), (v) sinh-Gordon bi-time motion generated by a bi-flow and two Euclidean metrics (created motion). Our analysis is based on some least squares Lagrangians and shows that there are dynamics that can be split into flows and motions transversal to the flows.

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Boyer, A. L., C. Cardenas, F. Gibou, and D. Levy. "Segmentation for radiotherapy treatment planning using geometric PDEs." International Journal of Radiation Oncology*Biology*Physics 54, no. 2 (October 2002): 82–83. http://dx.doi.org/10.1016/s0360-3016(02)03200-5.

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SURI, JASJIT, DEE WU, LAURA REDEN, JIANBO GAO, SAMEER SINGH, and SWAMY LAXMINARAYAN. "MODELING SEGMENTATION VIA GEOMETRIC DEFORMABLE REGULARIZERS, PDE AND LEVEL SETS IN STILL AND MOTION IMAGERY: A REVISIT." International Journal of Image and Graphics 01, no. 04 (October 2001): 681–734. http://dx.doi.org/10.1142/s0219467801000402.

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Partial Differential Equations (PDEs) have dominated image processing research recently. The three main reasons for their success are: first, their ability to transform a segmentation modeling problem into a partial differential equation framework and their ability to embed and integrate different regularizers into these models; second, their ability to solve PDEs in the level set framework using finite difference methods; and third, their easy extension to a higher dimensional space. This paper is an attempt to survey and understand the power of PDEs to incorporate into geometric deformable models for segmentation of objects in 2D and 3D in still and motion imagery. The paper first presents PDEs and their solutions applied to image diffusion. The main concentration of this paper is to demonstrate the usage of regularizers in PDEs and level set framework to achieve the image segmentation in still and motion imagery. Lastly, we cover miscellaneous applications such as: mathematical morphology, computation of missing boundaries for shape recovery and low pass filtering, all under the PDE framework. The paper concludes with the merits and the demerits of PDEs and level set-based framework for segmentation modeling. The paper presents a variety of examples covering both synthetic and real world images.

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Hirica, Iulia, Constantin Udriste, Gabriel Pripoae, and Ionel Tevy. "Least Squares Approximation of Flatness on Riemannian Manifolds." Mathematics 8, no. 10 (October 13, 2020): 1757. http://dx.doi.org/10.3390/math8101757.

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The purpose of this paper is fourfold: (i) to introduce and study the Euler–Lagrange prolongations of flatness PDEs solutions (best approximation of flatness) via associated least squares Lagrangian densities and integral functionals on Riemannian manifolds; (ii) to analyze some decomposable multivariate dynamics represented by Euler–Lagrange PDEs of least squares Lagrangians generated by flatness PDEs and Riemannian metrics; (iii) to give examples of explicit flat extremals and non-flat approximations; (iv) to find some relations between geometric least squares Lagrangian densities.

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Enciso, Alberto. "Geometric problems in PDEs with applications to mathematical physics." SeMA Journal 65, no. 1 (May 6, 2014): 1–11. http://dx.doi.org/10.1007/s40324-014-0015-8.

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Rosa, Márcio Antonio de Faria, Daniela Pereira Mendes Peres, and Rafael Peres. "ODEs together PDEs and Vector Fields in the SoftAge." International Journal on Engineering, Science and Technology 2, no. 2 (April 23, 2021): 77–83. http://dx.doi.org/10.46328/ijonest.25.

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At this moment when we can employ software, mathematics education has to be reviewed. In this article, we point out that ODEs should be taught from a geometric and qualitative point of view together with an introduction to PDEs and vector fields. This would increase the skills of the future mathematics user, not only to obtain explicit solutions from a straight command like DSolve, but also in the situations where this command does not help. The geometric interpretation and the concept of direction fields with images generated by software will give us a good understanding of possible system evolutions.

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Dziuk, Gerhard, and Charles M. Elliott. "Finite element methods for surface PDEs." Acta Numerica 22 (April 2, 2013): 289–396. http://dx.doi.org/10.1017/s0962492913000056.

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In this article we consider finite element methods for approximating the solution of partial differential equations on surfaces. We focus on surface finite elements on triangulated surfaces, implicit surface methods using level set descriptions of the surface, unfitted finite element methods and diffuse interface methods. In order to formulate the methods we present the necessary geometric analysis and, in the context of evolving surfaces, the necessary transport formulae. A wide variety of equations and applications are covered. Some ideas of the numerical analysis are presented along with illustrative numerical examples.

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Du, Haixia, and Hong Qin. "Free-Form Geometric Modeling by Integrating Parametric and Implicit PDEs." IEEE Transactions on Visualization and Computer Graphics 13, no. 3 (May 2007): 549–61. http://dx.doi.org/10.1109/tvcg.2007.1004.

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Moore, Brian E., Laura Noreña, and Constance M. Schober. "Conformal conservation laws and geometric integration for damped Hamiltonian PDEs." Journal of Computational Physics 232, no. 1 (January 2013): 214–33. http://dx.doi.org/10.1016/j.jcp.2012.08.010.

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Eshkobilov, Olimjon, Gianni Manno, Giovanni Moreno, and Katja Sagerschnig. "Contact manifolds, Lagrangian Grassmannians and PDEs." Complex Manifolds 5, no. 1 (February 2, 2018): 26–88. http://dx.doi.org/10.1515/coma-2018-0003.

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Abstract In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

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Tünger, Çetin, and Şule Taşlı Pektaş. "A comparison of the cognitive actions of designers in geometry-based and parametric design environments." Open House International 45, no. 1/2 (June 17, 2020): 87–101. http://dx.doi.org/10.1108/ohi-04-2020-0008.

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Purpose This paper aims to compare designers’ cognitive behaviors in geometry-based modeling environments (GMEs) and parametric design environments (PDEs). Design/methodology/approach This study used Rhinoceros as the geometric and Grasshopper as the parametric design tool in an experimental setting. Designers’ cognitive behaviors were investigated by using the retrospective protocol analysis method with a content-oriented approach. Findings The results indicated that the participants performed more cognitive actions per minute in the PDE because of the extra algorithmic space that such environments include. On the other hand, the students viewed their designs more and focused more on product–user relation in the geometric modeling environment. While the students followed a top-down process and produced less number of topologically different design alternatives with the parametric design tool, they had more goal setting activities and higher number of alternative designs in the geometric modeling environment. Originality/value This study indicates that cognitive behaviors of designers in GMEs and PDEs differ significantly and these differences entail further attention from researchers and educators.

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Valizadeh, Navid, and Timon Rabczuk. "Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces." Computer Methods in Applied Mechanics and Engineering 351 (July 2019): 599–642. http://dx.doi.org/10.1016/j.cma.2019.03.043.

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Liu, Hanze. "Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/156965.

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The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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Krstic, Miroslav, and Rafael Vazquez. "NONLINEAR CONTROL OF PDES: ARE FEEDBACK LINEARIZATION AND GEOMETRIC METHODS APPLICABLE?" IFAC Proceedings Volumes 40, no. 12 (2007): 20–27. http://dx.doi.org/10.3182/20070822-3-za-2920.00004.

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Bunge, Astrid, Philipp Herholz, Olga Sorkine-Hornung, Mario Botsch, and Michael Kazhdan. "Variational quadratic shape functions for polygons and polyhedra." ACM Transactions on Graphics 41, no. 4 (July 2022): 1–14. http://dx.doi.org/10.1145/3528223.3530137.

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Solving partial differential equations (PDEs) on geometric domains is an important component of computer graphics, geometry processing, and many other fields. Typically, the given discrete mesh is the geometric representation and should not be altered for simulation purposes. Hence, accurately solving PDEs on general meshes is a central goal and has been considered for various differential operators over the last years. While it is known that using higher-order basis functions on simplicial meshes can substantially improve accuracy and convergence, extending these benefits to general surface or volume tessellations in an efficient fashion remains an open problem. Our work proposes variationally optimized piecewise quadratic shape functions for polygons and polyhedra, which generalize quadratic P 2 elements, exactly reproduce them on simplices, and inherit their beneficial numerical properties. To mitigate the associated cost of increased computation time, particularly for volumetric meshes, we introduce a custom two-level multigrid solver which significantly improves computational performance.

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Ghayesh, Mergen H. "Resonant dynamics of axially functionally graded imperfect tapered Timoshenko beams." Journal of Vibration and Control 25, no. 2 (August 21, 2018): 336–50. http://dx.doi.org/10.1177/1077546318777591.

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This paper addresses the nonlinear resonant dynamics of axially functionally graded (AFG) tapered beams subjected to initial geometric imperfections, based on the Timoshenko beam theory. A rigorous coupled axial–transverse–rotational nonlinear model is developed taking into account the geometric nonlinearities due to the large deformations coupled with an initial imperfection along the length of the beam as well as nonlinear expressions accounting for nonuniform tapered geometry and mechanical properties. The Hamilton’s energy principle is used to balance the kinetic and potential energies of the AFG imperfect tapered Timoshenko beam with the work done by damping and the external excitation load. This results in a set of strongly nonlinear partial differential equations (PDEs). The Galerkin decomposition scheme involving an adequate number of both symmetric and asymmetric modes is utilized to reduce the PDEs to a set of nonlinear ordinary differential equations. A well-optimized numerical scheme is developed to handle the high-dimensional discretized model.

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CIEGIS, R., F. GASPAR, and C. RODRIGO. "On The Parallel Multiblock Geometric Multigrid Algorithm." Computational Methods in Applied Mathematics 8, no. 3 (2008): 223–36. http://dx.doi.org/10.2478/cmam-2008-0016.

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Abstract The application of a parallel multiblock geometric multigrid is consid-ered. It is applied to solve a two-dimensional poroelastic model. This system of PDEs is approximated by a special stabilized monotone finite-difference scheme. The obtained system of linear algebraic equations is solved by a multigrid method, when a domain is partitioned into structured blocks. A new strategy for the solution of the discrete problem on the coarsest grid is proposed and the efficiency of the obtained algorithm is investigated. The geometrical structure of the sequential multigrid method is used to develop a parallel version of the multigrid algorithm. The convergence properties of several smoothers are investigated and some computational results are presented.

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Mitsopoulos, Antonios, and Michael Tsamparlis. "Integrable and Superintegrable 3D Newtonian Potentials Using Quadratic First Integrals: A Review." Universe 9, no. 1 (December 29, 2022): 22. http://dx.doi.org/10.3390/universe9010022.

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The determination of the first integrals (FIs) of a dynamical system and the subsequent assessment of their integrability or superintegrability in a systematic way is still an open subject. One method which has been developed along these lines for holonomic autonomous dynamical systems with dynamical equations q¨a=−Γbca(q)q˙bq˙c−Qa(q), where Γbca(q) are the coefficients of the Riemannian connection defined by the kinetic metric of the system and −Qa(q) are the generalized forces, is the so-called direct method. According to this method, one assumes a general functional form for the FI I and requires the condition dIdt=0 along the dynamical equations. This results in a system of partial differential equations (PDEs) to which one adds the necessary integrability conditions of the involved scalar quantities. It is found that the final system of PDEs breaks into two sets: a. One set containing geometric elements only and b. A second set with geometric and dynamical quantities. Then, provided the geometric quantities are known or can be found, one uses the second set to compute the FIs and, accordingly, assess the integrability of the dynamical system. The `solution’ of the system of PDEs for quadratic FIs (QFIs) has been given in a recent paper (M. Tsamparlis and A. Mitsopoulos, J. Math. Phys. 61, 122701 (2020)). In the present work, we consider the application of this `solution’ to Newtonian autonomous conservative dynamical systems with three degrees of freedom, and compute integrable and superintegrable potentials V(x,y,z) whose integrability is determined via autonomous and/or time-dependent QFIs. The geometric elements of these systems are the ones of the Euclidean space E3, which are known. Setting various values for the parameters determining the geometric elements, we determine in a systematic way all known integrable and superintegrable potentials in E3 together with new ones obtained in this work. For easy reference, the results are collected in tables so that the present work may act as an updated review of the QFIs of Newtonian autonomous conservative dynamical systems with three degrees of freedom. It is emphasized that, by assuming different values for the parameters, other authors may find more integrable potentials of this type of system.

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Engwer, Christian, and Sebastian Westerheide. "An Unfitted dG Scheme for Coupled Bulk-Surface PDEs on Complex Geometries." Computational Methods in Applied Mathematics 21, no. 3 (June 1, 2021): 569–91. http://dx.doi.org/10.1515/cmam-2020-0056.

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Abstract The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.

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Reid, Gregory J. "Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution." European Journal of Applied Mathematics 2, no. 4 (December 1991): 293–318. http://dx.doi.org/10.1017/s0956792500000577.

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We present several algorithms, executable in a finite number of steps, which have been implemented in the symbolic language maple. The standard form algorithm reduces a system of PDEs to a simplified standard form which has all of its integrability conditions satisfied (i.e. is involutive). The initial data algorithm uses a system's standard form to calculate a set of initial data that uniquely determines a local solution to the system without needing to solve the system. The number of arbitrary constants and arbitrary functions in the general solution to the system is directly calculable from this set. The taylor algorithm uses a system's standard form and initial data set to determine the Taylor series expansion of its solution about any point to any given finite degree. All systems of linear PDEs and many systems of nonlinear PDEs can be reduced to standard form in a finite number of steps. Our algorithms have simple geometric interpretations which are illustrated through the use of diagrams. The standard form algorithm is generally more efficient than the classical methods due to Janet and Cartan for reducing systems of PDEs to involutive form.

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Lashkarian, Elham, Elaheh Saberi, and S. Reza Hejazi. "Symmetry reductions and exact solutions for a class of nonlinear PDEs." Asian-European Journal of Mathematics 09, no. 03 (August 2, 2016): 1650061. http://dx.doi.org/10.1142/s1793557116500613.

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This paper uses Lie symmetry group method to study a special kind of PDE. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions are obtained.

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D’Onofrio, Luigi, Carlo Sbordone, and Roberta Schiattarella. "Grand Sobolev spaces and their applications in geometric function theory and PDEs." Journal of Fixed Point Theory and Applications 13, no. 2 (June 2013): 309–40. http://dx.doi.org/10.1007/s11784-013-0140-5.

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Deckelnick, Klaus, Gerhard Dziuk, and Charles M. Elliott. "Computation of geometric partial differential equations and mean curvature flow." Acta Numerica 14 (April 19, 2005): 139–232. http://dx.doi.org/10.1017/s0962492904000224.

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This review concerns the computation of curvature-dependent interface motion governed by geometric partial differential equations. The canonical problem of mean curvature flow is that of finding a surface which evolves so that, at every point on the surface, the normal velocity is given by the mean curvature. In recent years the interest in geometric PDEs involving curvature has burgeoned. Examples of applications are, amongst others, the motion of grain boundaries in alloys, phase transitions and image processing. The methods of analysis, discretization and numerical analysis depend on how the surface is represented. The simplest approach is when the surface is a graph over a base domain. This is an example of a sharp interface approach which, in the general parametric approach, involves seeking a parametrization of the surface over a base surface, such as a sphere. On the other hand an interface can be represented implicitly as a level surface of a function, and this idea gives rise to the so-called level set method. Another implicit approach is the phase field method, which approximates the interface by a zero level set of a phase field satisfying a PDE depending on a new parameter. Each approach has its own advantages and disadvantages. In the article we describe the mathematical formulations of these approaches and their discretizations. Algorithms are set out for each approach, convergence results are given and are supported by computational results and numerous graphical figures. Besides mean curvature flow, the topics of anisotropy and the higher order geometric PDEs for Willmore flow and surface diffusion are covered.

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Wang, Naige, Guohua Cao, Lu Yan, and Lei Wang. "Modeling and Control for a Multi-Rope Parallel Suspension Lifting System under Spatial Distributed Tensions and Multiple Constraints." Symmetry 10, no. 9 (September 18, 2018): 412. http://dx.doi.org/10.3390/sym10090412.

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The modeling and control of the multi-rope parallel suspension lifting system (MPSLS) are investigated in the presence of different and spatial distributed tensions; unknown boundary disturbances; and multiple constraints, including time varying geometric constraint, input saturation, and output constraint. To describe the system dynamics more accurately, the MPSLS is modelled by a set of partial differential equations and ordinary differential equations (PDEs-ODEs) with multiple constraints, which is a nonhomogeneous and coupled PDEs-ODEs, and makes its control more difficult. Adaptive boundary control is a recommended method for position regulation and vibration degradation of the MPSLS, where adaptation laws and a boundary disturbance observer are formulated to handle system uncertainties. The system stability is rigorously proved by using Lyapunov’s direct method, and the position and vibration eventually diminish to a bounded neighborhood of origin. The original PDEs-ODEs are solved by finite difference method, and the multiple constraints problem is processed simultaneously. Finally, the performance of the proposed control is demonstrated by both the results of ADAMS simulation and numerical calculation.

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Papaioannou, Panagiotis G., Ronen Talmon, Ioannis G. Kevrekidis, and Constantinos Siettos. "Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 8 (August 2022): 083113. http://dx.doi.org/10.1063/5.0094887.

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We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the “curse of dimensionality” related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). We assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001–29 October 2020.

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W. Hess, Martin, and Peter Benner. "A reduced basis method for microwave semiconductor devices with geometric variations." COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 33, no. 4 (July 1, 2014): 1071–81. http://dx.doi.org/10.1108/compel-12-2012-0377.

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Purpose – The Reduced Basis Method (RBM) generates low-order models of parametrized PDEs to allow for efficient evaluation of parametrized models in many-query and real-time contexts. The purpose of this paper is to investigate the performance of the RBM in microwave semiconductor devices, governed by Maxwell's equations. Design/methodology/approach – The paper shows the theoretical framework in which the RBM is applied to Maxwell's equations and present numerical results for model reduction under geometry variation. Findings – The RBM reduces model order by a factor of $1,000 and more with guaranteed error bounds. Originality/value – Exponential convergence speed can be observed by numerical experiments, which makes the RBM a suitable method for parametric model reduction (PMOR).

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Dong, Guozhi, Michael Hintermueller, and Ye Zhang. "A Class of Second-Order Geometric Quasilinear Hyperbolic PDEs and Their Application in Imaging." SIAM Journal on Imaging Sciences 14, no. 2 (January 2021): 645–88. http://dx.doi.org/10.1137/20m1366277.

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Smith, Abraham D. "Involutive tableaux, characteristic varieties, and rank-one varieties in the geometric study of PDEs." Banach Center Publications 117 (2019): 57–112. http://dx.doi.org/10.4064/bc117-3.

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Harrell, Evans M. "Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part." Journal of Computational and Applied Mathematics 194, no. 1 (September 2006): 26–35. http://dx.doi.org/10.1016/j.cam.2005.06.012.

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Bin Zubair, H., S. P. MacLachlan, and C. W. Oosterlee. "A geometric multigrid method based on L-shaped coarsening for PDEs on stretched grids." Numerical Linear Algebra with Applications 17, no. 6 (November 26, 2010): 871–94. http://dx.doi.org/10.1002/nla.665.

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Endtmayer, Bernhard, Ulrich Langer, and Thomas Wick. "Multigoal-oriented error estimates for non-linear problems." Journal of Numerical Mathematics 27, no. 4 (December 18, 2019): 215–36. http://dx.doi.org/10.1515/jnma-2018-0038.

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Abstract In this work, we further develop multigoal-oriented a posteriori error estimation with two objectives in mind. First, we formulate goal-oriented mesh adaptivity for multiple functionals of interest for nonlinear problems in which both the Partial Differential Equation (PDE) and the goal functionals may be nonlinear. Our method is based on a posteriori error estimates in which the adjoint problem is used and a partition-of-unity is employed for the error localization that allows us to formulate the error estimator in the weak form. We provide a careful derivation of the primal and adjoint parts of the error estimator. The second objective is concerned with balancing the nonlinear iteration error with the discretization error yielding adaptive stopping rules for Newton’s method. Our techniques are substantiated with several numerical examples including scalar PDEs and PDE systems, geometric singularities, and both nonlinear PDEs and nonlinear goal functionals. In these tests, up to six goal functionals are simultaneously controlled.

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Vargas, Arturo, Jesse Chan, Thomas Hagstrom, and Timothy Warburton. "Variations on Hermite Methods for Wave Propagation." Communications in Computational Physics 22, no. 2 (June 21, 2017): 303–37. http://dx.doi.org/10.4208/cicp.260915.281116a.

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AbstractHermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.

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Astala, Kari, Tadeusz Iwaniec, István Prause, and Eero Saksman. "A hunt for sharp Lp-estimates and rank-one convex variational integrals." Filomat 29, no. 2 (2015): 245–61. http://dx.doi.org/10.2298/fil1502245a.

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Learning how to figure out sharp Lp-estimates of nonlinear differential expressions, to prove and use them, is a fundamental part of the development of PDEs and Geometric Function Theory (GFT). Our survey presents, among what is known to date, some notable recent efforts and novelties made in this direction. We focus attention here on the historic Morrey?s Conjecture and Burkholder martingale inequalities for stochastic integrals. Some of these topics have already been discussed by the present authors [5] and by Rodrigo Ba?uelos [10]. Nevertheless, there is always something new to add.

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Lehrenfeld, Christoph, and Maxim Olshanskii. "An Eulerian finite element method for PDEs in time-dependent domains." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 2 (March 2019): 585–614. http://dx.doi.org/10.1051/m2an/2018068.

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The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.

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Martynenko, S. I. "Potentialities of the Robust Multigrid Technique." Computational Methods in Applied Mathematics 10, no. 1 (2010): 87–94. http://dx.doi.org/10.2478/cmam-2010-0004.

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AbstractThe present paper discusses the parallelization of the robust multigrid technique (RMT) and the possible way of applying this to unstructured grids. As opposed to the classical multigrid methods, the RMT is a trivial method of parallelization on coarse grids independent of the smoothing iterations. Estimates of the minimum speed-up and parallelism efficiency are given. An almost perfect load balance is demonstrated in a 3D illustrative test. To overcome the geometric nature of the technique, the RMT is used as a preconditioner in solving PDEs on unstructured grids. The procedure of auxiliary structured grids generation is considered in details.

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Boateng, Francis Ohene, Joseph Ackora-Prah, Benedict Barnes, and John Amoah-Mensah. "A Finite Difference Fictitious Domain Wavelet Method for Solving Dirichlet Boundary Value Problem." European Journal of Pure and Applied Mathematics 14, no. 3 (August 5, 2021): 706–22. http://dx.doi.org/10.29020/nybg.ejpam.v14i3.3893.

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In this paper, we introduce a Finite Difference Fictitious Domain Wavelet Method (FDFDWM) for solving two dimensional (2D) linear ellipticÂ partial differential equations (PDEs) with Dirichlet boundary conditions on regular geometric domain. The method reduces the 2D PDE into a 1D system of ordinary differential equations and applies a compactly supported wavelet to approximate the solution. The problem is embedded in a fictitious domain to aid the enforcement of the Dirichlet boundary conditions. We present numerical analysis and show that our method yields better approximation to the solution of the Dirichlet problem than traditional methods like the finite element and finite difference methods.

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Avey, Mahmure, Nicholas Fantuzzi, and Abdullah Sofiyev. "Mathematical Modeling and Analytical Solution of Thermoelastic Stability Problem of Functionally Graded Nanocomposite Cylinders within Different Theories." Mathematics 10, no. 7 (March 28, 2022): 1081. http://dx.doi.org/10.3390/math10071081.

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Revolutionary advances in technology have led to the use of functionally graded nanocomposite structural elements that operate at high temperatures and whose properties depend on position, such as cylindrical shells designed as load-bearing elements. These advances in technology require new mathematical modeling and updated numerical calculations to be performed using improved theories at design time to reliably apply such elements. The main goal of this study is to model, mathematically and within an analytical solution, the thermoelastic stability problem of composite cylinders reinforced by carbon nanotubes (CNTs) under a uniform thermal loading within the shear deformation theory (ST). The influence of transverse shear deformations is considered when forming the fundamental relations of CNT-patterned cylindrical shells and the basic partial differential equations (PDEs) are derived within the modified Donnell-type shell theory. The PDEs are solved by the Galerkin method, and the formula is found for the eigenvalue (critical temperature) of the functionally graded nanocomposite cylindrical shells. The influences of CNT patterns, volume fraction, and geometric parameters on the critical temperature within the ST are estimated by comparing the results within classical theory (CT).

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Jha, Navnit, Venu Gopal, and Bhagat Singh. "Geometric grid network and third-order compact scheme for solving nonlinear variable coefficients 3D elliptic PDEs." International Journal of Modeling, Simulation, and Scientific Computing 09, no. 06 (December 2018): 1850053. http://dx.doi.org/10.1142/s1793962318500538.

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By using nonuniform (geometric) grid network, a new high-order finite-difference compact scheme has been obtained for the numerical solution of three-space dimensions partial differential equations of elliptic type. Single cell discretization to the elliptic equation makes it easier to compute and exhibit stability of the numerical solutions. The monotone and irreducible property of the Jacobian matrix to the system of difference equations analyses the converging behavior of the numerical solution values. As an experiment, applications of the compact scheme to Schrödinger equations, sine-Gordon equations, elliptic Allen–Cahn equation and Poisson’s equation have been presented with root mean squared errors of exact and approximate solution values. The results corroborate the reliability and efficiency of the scheme.

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Bessa, G. Pacelli, Stefano Pigola, and Alberto Setti. "Spectral and stochastic properties of the $f$-Laplacian, solutions of PDEs at infinity and geometric applications." Revista Matemática Iberoamericana 29, no. 2 (2013): 579–610. http://dx.doi.org/10.4171/rmi/731.

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Kabelitz, C., and S. J. Linz. "The dynamics of geometric PDEs: Surface evolution equations and a comparison with their small gradient approximations." Chaos: An Interdisciplinary Journal of Nonlinear Science 29, no. 10 (October 2019): 103119. http://dx.doi.org/10.1063/1.5112833.

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Theljani, Anis. "Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising." CSIAM Transactions on Applied Mathematics 2, no. 4 (June 2021): 652–79. http://dx.doi.org/10.4208/csiam-am.so-2020-0007.

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Xu, Guoliang, and Qing Pan. "Design of Loop’s Subdivision Surfaces by Fourth-Order Geometric PDEs with $$G^1$$ G 1 Boundary Conditions." Journal of Scientific Computing 62, no. 3 (June 4, 2014): 674–92. http://dx.doi.org/10.1007/s10915-014-9872-7.

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BESSE, NICOLAS. "ON THE CAUCHY PROBLEM FOR THE GYRO-WATER-BAG MODEL." Mathematical Models and Methods in Applied Sciences 21, no. 09 (September 2011): 1839–69. http://dx.doi.org/10.1142/s0218202511005623.

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In this paper we prove the existence and uniqueness of classical solution for a system of PDEs recently developed in Refs. 60, 8, 10 and 11 to modelize the nonlinear gyrokinetic turbulence in magnetized plasma. From the analytical and numerical point of view this model is very promising because it allows to recover kinetic features (wave–particle interaction, Landau resonance) of the dynamic flow with the complexity of a multi-fluid model. This model, called the gyro-water-bag model, is derived from two-phase space variable reductions of the Vlasov equation through the existence of two underlying invariants. The first one, the magnetic moment, is adiabatic and the second, a geometric invariant named "water-bag", is exact and is just the direct consequence of the Liouville theorem.

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Chawda, Denil, and Senthil Murugan. "Dynamic Response of a Cantilevered Beam Under Combined Moving Moment, Torque and Force." International Journal of Structural Stability and Dynamics 20, no. 05 (May 2020): 2050065. http://dx.doi.org/10.1142/s0219455420500650.

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This paper studies the dynamic response of a cantilevered beam subjected to a moving moment and torque, and combination of them with a moving force. The moving loads are considered to traverse along the length of the beam either from fixed-to-free end or free-to-fixed end. The beam is considered to have constant material and geometric properties. The beam is modeled using the Rayleigh beam theory considering the rotary inertia effects. The Dirac-delta function used to model the moving loads in the governing partial differential equations (PDEs) has complicated the solution of the problem. The Eigenfunction expansions coupled with the Laplace transformation method is used to find the semi-analytical solution for the resulting governing PDEs. The effects of moving loads on the dynamic response are studied. The dynamic effects are quantified based on the number of oscillations per unit travel time of the moving load and the Dynamic Amplification Factor (DAF) of the beam’s tip response. Numerical results are also analyzed for the two-speed regimes, namely high-speed and low-speed regimes, defined with respect to the critical speed of the moving loads. The accuracy of the analytical solutions are verified by the finite element analysis. The numerical results show that the loads moving with low speeds have significant impact on the dynamic response compared to high speeds. Also, the moving moment has significant impact on the amplitude of dynamic response compared with the moving force case.

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Journal articles on the topic 'Geometric PDEs'

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