Journal articles: 'Geometry of PDEs'

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Prástaro, Agostino. "Geometry of PDEs." Journal of Mathematical Analysis and Applications 319, no. 2 (July 2006): 547–66. http://dx.doi.org/10.1016/j.jmaa.2005.06.044.

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Pràstaro, Agostino. "Quantum geometry of super PDEs." Reports on Mathematical Physics 37, no. 1 (February 1996): 23–140. http://dx.doi.org/10.1016/0034-4877(96)88921-x.

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Gutt, Jan, Gianni Manno, and Giovanni Moreno. "Geometry of Lagrangian Grassmannians and nonlinear PDEs." Banach Center Publications 117 (2019): 9–44. http://dx.doi.org/10.4064/bc117-1.

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Marsden, Jerrold E., George W. Patrick, and Steve Shkoller. "Multisymplectic Geometry, Variational Integrators, and Nonlinear PDEs." Communications in Mathematical Physics 199, no. 2 (December 1, 1998): 351–95. http://dx.doi.org/10.1007/s002200050505.

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Savin, Ovidiu, and Enrico Valdinoci. "Elliptic PDEs with Fibered Nonlinearities." Journal of Geometric Analysis 19, no. 2 (January 14, 2009): 420–32. http://dx.doi.org/10.1007/s12220-008-9064-5.

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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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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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Vinogradov, A. M. "Some remarks on contact manifolds, Monge–Ampère equations and solution singularities." International Journal of Geometric Methods in Modern Physics 11, no. 07 (August 2014): 1460026. http://dx.doi.org/10.1142/s0219887814600263.

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We describe some natural relations connecting contact geometry, classical Monge–Ampère equations (MAEs) and theory of singularities of solutions to nonlinear PDEs. They reveal the hidden meaning of MAEs and sheds new light on some aspects of contact geometry.

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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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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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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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Lenglet, Christophe, Emmanuel Prados, Jean-Philippe Pons, Rachid Deriche, and Olivier Faugeras. "Brain Connectivity Mapping Using Riemannian Geometry, Control Theory, and PDEs." SIAM Journal on Imaging Sciences 2, no. 2 (January 2009): 285–322. http://dx.doi.org/10.1137/070710986.

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Ok Bayrakdar, Z., and T. Bayrakdar. "Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations." Advances in Mathematical Physics 2018 (2018): 1–8. http://dx.doi.org/10.1155/2018/7590847.

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We construct metric connection associated with a first-order differential equation by means of the generator set of a Pfaffian system on a submanifold of an appropriate first-order jet bundle. We firstly show that the inviscid and viscous Burgers’ equations describe surfaces attached to an ODE of the form dx/dt=u(t,x) with certain Gaussian curvatures. In the case of PDEs, we show that the scalar curvature of a three-dimensional manifold encoding a system of first-order PDEs is determined in terms of the integrability condition and the Gaussian curvatures of the surfaces corresponding to the integral curves of the vector fields which are annihilated by the contact form. We see that an integral manifold of any PDE defines intrinsically flat and totally geodesic submanifold.

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Hirica, Iulia, Constantin Udriste, Gabriel Pripoae, and Ionel Tevy. "Riccati PDEs That Imply Curvature-Flatness." Mathematics 9, no. 5 (March 4, 2021): 537. http://dx.doi.org/10.3390/math9050537.

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In this paper the following three goals are addressed. The first goal is to study some strong partial differential equations (PDEs) that imply curvature-flatness, in the cases of both symmetric and non-symmetric connection. Although the curvature-flatness idea is classic for symmetric connection, our main theorems about flatness solutions are completely new, leaving for a while the point of view of differential geometry and entering that of PDEs. The second goal is to introduce and study some strong partial differential relations associated to curvature-flatness. The third goal is to introduce and analyze some vector spaces of exotic objects that change the meaning of a generalized Kronecker delta projection operator, in order to discover new PDEs implying curvature-flatness. Significant examples clarify some ideas.

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Gutt, Jan, Gianni Manno, and Giovanni Moreno. "Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution." Journal of Geometry and Physics 113 (March 2017): 86–103. http://dx.doi.org/10.1016/j.geomphys.2016.04.021.

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De Vecchi, Francesco C., and Paola Morando. "The geometry of differential constraints for a class of evolution PDEs." Journal of Geometry and Physics 156 (October 2020): 103771. http://dx.doi.org/10.1016/j.geomphys.2020.103771.

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Hammerl, M., P. Somberg, V. Souček, and J. Šilhan. "Invariant prolongation of overdetermined PDEs in projective, conformal, and Grassmannian geometry." Annals of Global Analysis and Geometry 42, no. 1 (December 6, 2011): 121–45. http://dx.doi.org/10.1007/s10455-011-9306-9.

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Sawhney, Rohan, Dario Seyb, Wojciech Jarosz, and Keenan Crane. "Grid-free Monte Carlo for PDEs with spatially varying coefficients." ACM Transactions on Graphics 41, no. 4 (July 2022): 1–17. http://dx.doi.org/10.1145/3528223.3530134.

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Partial differential equations (PDEs) with spatially varying coefficients arise throughout science and engineering, modeling rich heterogeneous material behavior. Yet conventional PDE solvers struggle with the immense complexity found in nature, since they must first discretize the problem---leading to spatial aliasing, and global meshing/sampling that is costly and error-prone. We describe a method that approximates neither the domain geometry, the problem data, nor the solution space, providing the exact solution (in expectation) even for problems with extremely detailed geometry and intricate coefficients. Our main contribution is to extend the walk on spheres (WoS) algorithm from constant- to variable-coefficient problems, by drawing on techniques from volumetric rendering. In particular, an approach inspired by null-scattering yields unbiased Monte Carlo estimators for a large class of 2nd order elliptic PDEs, which share many attractive features with Monte Carlo rendering: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations.

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Lu, Dianchen, Aly R. Seadawy, and M. Arshad. "Solitary wave and elliptic function solutions of sinh-Gordon equation and its applications." Modern Physics Letters B 33, no. 35 (December 16, 2019): 1950436. http://dx.doi.org/10.1142/s0217984919504360.

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The [Formula: see text]-Gordon model is an important model in special nonlinear partial differential equations (PDEs) which is arising in solid-state physics, mathematical physics, fluid dynamics, fluid flow, differential geometry, quantum theory, etc. The exact solutions in the type of solitary wave and elliptic functions solutions are created of [Formula: see text]-Gordon model by employing modified direct algebraic scheme. Moments of a few solutions are also depicted graphically. These solutions helps the physicians and mathematicians to understand the physical phenomena of this model. This technique can be utilized on other models to launch further exclusively novel solutions for other categories of nonlinear PDEs occurring in mathematical Physics.

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Ahmad, Imtiaz, Muhammad Ahsan, Zaheer-ud Din, Ahmad Masood, and Poom Kumam. "An Efficient Local Formulation for Time–Dependent PDEs." Mathematics 7, no. 3 (February 26, 2019): 216. http://dx.doi.org/10.3390/math7030216.

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In this paper, a local meshless method (LMM) based on radial basis functions (RBFs) is utilized for the numerical solution of various types of PDEs. This local approach has flexibility with respect to geometry along with high order of convergence rate. In case of global meshless methods, the two major deficiencies are the computational cost and the optimum value of shape parameter. Therefore, research is currently focused towards localized RBFs approximations, as proposed here. The proposed local meshless procedure is used for spatial discretization, whereas for temporal discretization, different time integrators are employed. The proposed local meshless method is testified in terms of efficiency, accuracy and ease of implementation on regular and irregular domains.

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Xenos, M. A. "An Euler–Lagrange approach for studying blood flow in an aneurysmal geometry." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 473, no. 2199 (March 2017): 20160774. http://dx.doi.org/10.1098/rspa.2016.0774.

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To numerically study blood flow in an aneurysm, the development of an approach that tracks the moving tissue and accounts for its interaction with the fluid is required. This study presents a mathematical approach that expands fluid mechanics principles, taking into consideration the domain’s motion. The initial fluid equations, derived in Euler form, are expanded to a mixed Euler–Lagrange formulation to study blood flow in the aneurysm during the cardiac cycle. Transport equations are transformed into a moving body-fitted reference frame using generalized curvilinear coordinates. The equations of motion consist of a coupled and nonlinear system of partial differential equations (PDEs). The PDEs are discretized using the finite volume method. Owing to strong coupling and nonlinear terms, a simultaneous solution approach is applied. The results show that velocity is substantially influenced by the pulsating wall. Intensification of polymorphic flow patterns is observed. Increments of Reynolds and Womersley numbers are evident as pulsatility increases. The pressure field reveals areas of a lateral pressure gradient at the aneurysm. As pulsatility increases, the diastolic flow vortex shifts towards the aortic wall, distal to the aneurysmal neck. Wall shear stress is amplified at the shoulders of the moving wall compared with that of the rigid one.

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BRIDGES, THOMAS J., PETER E. HYDON, and JEFFREY K. LAWSON. "Multisymplectic structures and the variational bicomplex." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 1 (August 4, 2009): 159–78. http://dx.doi.org/10.1017/s0305004109990259.

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AbstractMultisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.

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Owens, Z. C., D. W. Mattison, E. A. Barbour, C. I. Morris, and Ronald K. Hanson. "Flowfield characterization and simulation validation of multiple-geometry PDEs using cesium-based velocimetry." Proceedings of the Combustion Institute 30, no. 2 (January 2005): 2791–98. http://dx.doi.org/10.1016/j.proci.2004.08.050.

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Huru, Hilja, Iosif Krasil’shchik, Boris Kruglikov, and Vladimir Roubtsov. "Editors’ preface to the Topical Issue on the Geometry and Algebra of PDEs." Journal of Geometry and Physics 141 (July 2019): 159–60. http://dx.doi.org/10.1016/j.geomphys.2019.04.011.

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Kuksin, S. B. "Spectral Properties of Solutions for Nonlinear PDEs in the Turbulent Regime." Geometric And Functional Analysis 9, no. 1 (March 1, 1999): 141–84. http://dx.doi.org/10.1007/s000390050083.

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Bashingwa, Jean J. H., Ashfaque H. Bokhari, A. H. Kara, and F. D. Zaman. "The geometry and invariance properties for certain classes of metrics with neutral signature." International Journal of Geometric Methods in Modern Physics 13, no. 06 (June 15, 2016): 1650080. http://dx.doi.org/10.1142/s0219887816500808.

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In this paper, we study anti-self dual manifolds endowed with metrics of neutral signature. Since the metrics depend on solutions of, in some cases, well-known partial differential equations (PDEs), we determine exact solutions using Lie group methods. This concludes specific forms of the metrics. We then determine the isometries and the variational symmetries of the underlying metrics and corresponding Euler–Lagrange (geodesic) equations, respectively, and establish relationships between the resultant Lie algebras. In some cases, we construct conservation laws via these symmetries or the “multiplier approach”.

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Kuksin, Sergei, Vahagn Nersesyan, and Armen Shirikyan. "Exponential mixing for a class of dissipative PDEs with bounded degenerate noise." Geometric and Functional Analysis 30, no. 1 (February 2020): 126–87. http://dx.doi.org/10.1007/s00039-020-00525-5.

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Bertola, Marco, and Bertrand Eynard. "The PDEs of Biorthogonal Polynomials Arising in the Two-Matrix Model." Mathematical Physics, Analysis and Geometry 9, no. 1 (February 2006): 23–52. http://dx.doi.org/10.1007/s11040-005-9000-x.

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Adil, Nazakat, Xufeng Xiao, and Xinlong Feng. "Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces." Entropy 24, no. 7 (June 22, 2022): 857. http://dx.doi.org/10.3390/e24070857.

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In this paper, we present a fully Lagrangian method based on the radial basis function (RBF) finite difference (FD) method for solving convection–diffusion partial differential equations (PDEs) on evolving surfaces. Surface differential operators are discretized by the tangent plane approach using Gaussian RBFs augmented with two-dimensional (2D) polynomials. The main advantage of our method is the simplicity of calculating differentiation weights. Additionally, we couple the method with anisotropic RBFs (ARBFs) to obtain more accurate numerical solutions for the anisotropic growth of surfaces. In the ARBF interpolation, the Euclidean distance is replaced with a suitable metric that matches the anisotropic surface geometry. Therefore, it will lead to a good result on the aspects of stability and accuracy of the RBF-FD method for this type of problem. The performance of this method is shown for various convection–diffusion equations on evolving surfaces, which include the anisotropic growth of surfaces and growth coupled with the solutions of PDEs.

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Kristály, Alexandru, and Dušan Repovš. "Quantitative Rellich inequalities on Finsler—Hadamard manifolds." Communications in Contemporary Mathematics 18, no. 06 (September 14, 2016): 1650020. http://dx.doi.org/10.1142/s0219199716500206.

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In this paper, we are dealing with quantitative Rellich inequalities on Finsler–Hadamard manifolds where the remainder terms are expressed by means of the flag curvature. By exploring various arguments from Finsler geometry and PDEs on manifolds, we show that more weighty curvature implies more powerful improvements in Rellich inequalities. The sharpness of the involved constants is also studied. Our results complement those of Yang, Su and Kong [Hardy inequalities on Riemannian manifolds with negative curvature, Commun. Contemp. Math. 16 (2014), Article ID: 1350043, 24 pp.].

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Ayub, Assad, Tanveer Sajid, Wasim Jamshed, William Rolando Miranda Zamora, Leandro Alonso Vallejos More, Luz Marina Galván Talledo, Nélida Isabel Rodríguez Ortega de Peña, Syed M. Hussain, Muhammad Bilal Hafeez, and Marek Krawczuk. "Activation Energy and Inclination Magnetic Dipole Influences on Carreau Nanofluid Flowing via Cylindrical Channel with an Infinite Shearing Rate." Applied Sciences 12, no. 17 (August 31, 2022): 8779. http://dx.doi.org/10.3390/app12178779.

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Background: The infinite shear viscosity model of Carreau fluid characterizes the attitude of fluid flow at a very high/very low shear rate. This model has the capacity for interpretation of fluid at both extreme levels, and an inclined magnetic dipole in fluid mechanics has its valuable applications such as magnetic drug engineering, cold treatments to destroy tumors, drug targeting, bio preservation, cryosurgery, astrophysics, reaction kinetics, geophysics, machinery efficiency, sensors, material selection and cosmology. Novelty: This study investigates and interprets the infinite shear rate of Carreau nanofluid over the geometry of a cylindrical channel. The velocity is assumed to be investigated through imposing an inclined magnetic field onto cylindrical geometry. Activation energy is utilized because it helps with chemical reactions and mass transport. Furthermore, the effects of thermophoresis, the binary chemical process and the Brownian movement of nanoparticles are included in this attempt. Formulation: The mathematics of the assumed Carreau model is derived from Cauchy stress tensor, and partial differential equations (PDEs) are obtained. Similarity transformation variables converted these PDEs into a system of ordinary differential equations (ODEs). Passing this system under the bvp4c scheme, we reached at numerical results of this research attempt. Findings: Graphical debate and statistical analysis are launched on the basis of the obtained computed numerical results. The infinite shear rate aspect of Carreau nanofluid gives a lower velocity. The inclined magnetic dipole effect shows a lower velocity but high energy. A positive variation in activation energy amplifies the concentration field.

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SOLE, ALBERTO DE, MAMUKA JIBLADZE, VICTOR G. KAC, and DANIELE VALERI. "INTEGRABILITY OF CLASSICAL AFFINE W-ALGEBRAS." Transformation Groups 26, no. 2 (April 15, 2021): 479–500. http://dx.doi.org/10.1007/s00031-021-09645-0.

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AbstractWe prove that all classical affine W-algebras 𝒲(𝔤; f), where g is a simple Lie algebra and f is its non-zero nilpotent element, admit an integrable hierarchy of bi-Hamiltonian PDEs, except possibly for one nilpotent conjugacy class in G2, one in F4, and five in E8.

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Georgiev Georgiev, Svetlin, and Karima Mebarki. "On fixed point index theory for the sum of operators and applications to a class of ODEs and PDEs." Applied General Topology 22, no. 2 (October 1, 2021): 259. http://dx.doi.org/10.4995/agt.2021.13248.

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The aim of this work is two fold: first we extend some results concerning the computation of the fixed point index for the sum of an expansive mapping and a $k$-set contraction obtained in \cite{DjebaMeb, Svet-Meb}, to the case of the sum $T+F$, where $T$ is a mapping such that $(I-T)$ is Lipschitz invertible and $F$ is a $k$-set contraction. Secondly, as illustration of some our theoretical results, we study the existence of positive solutions for two classes of differential equations, covering a class of first-order ordinary differential equations (ODEs for short) posed on the positive half-line as well as a class of partial differential equations (PDEs for short).

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Zaris, P., J. Wood, H. Pillai, and E. Rogers. "On invariant zeros of linear systems of PDEs." Linear Algebra and its Applications 417, no. 1 (August 2006): 275–97. http://dx.doi.org/10.1016/j.laa.2005.10.025.

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Li, Benling, and Zhongmin Shen. "On a class of locally projectively flat Finsler metrics." International Journal of Mathematics 27, no. 06 (June 2016): 1650052. http://dx.doi.org/10.1142/s0129167x1650052x.

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Locally projectively flat Finsler metrics compose an important group of metrics in Finsler geometry. The characterization of these metrics is the regular case of the Hilbert’s Fourth Problem. In this paper, we study a class of Finsler metrics composed by a Riemann metric [Formula: see text] and a [Formula: see text]-form [Formula: see text] called general ([Formula: see text], [Formula: see text])-metrics. We classify those locally projectively flat when [Formula: see text] is projectively flat. By solving the corresponding nonlinear PDEs, the metrics in this class are totally determined. Then a new group of locally projectively flat Finsler metrics is found.

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Agarwal, Praveen, Jochen Merker, and Gregor Schuldt. "Singular Integral Neumann Boundary Conditions for Semilinear Elliptic PDEs." Axioms 10, no. 2 (April 24, 2021): 74. http://dx.doi.org/10.3390/axioms10020074.

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In this article, we discuss semilinear elliptic partial differential equations with singular integral Neumann boundary conditions. Such boundary value problems occur in applications as mathematical models of nonlocal interaction between interior points and boundary points. Particularly, we are interested in the uniqueness of solutions to such problems. For the sublinear and subcritical case, we calculate, on the one hand, illustrative, rather explicit solutions in the one-dimensional case. On the other hand, we prove in the general case the existence and—via the strong solution of an integro-PDE with a kind of fractional divergence as a lower order term—uniqueness up to a constant.

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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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Gao, Han, Luning Sun, and Jian-Xun Wang. "PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain." Journal of Computational Physics 428 (March 2021): 110079. http://dx.doi.org/10.1016/j.jcp.2020.110079.

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Al-Mufti, A. Wesam, Uda Hashim, Md Mijanur Rahman, and Tijjani Adam. "An Investigation of the Distribution of Electric Potential and Space Charge in a Silicon Nanowire." Zeitschrift für Naturforschung A 69, no. 10-11 (November 1, 2014): 597–605. http://dx.doi.org/10.5560/zna.2013-0087.

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AbstractThe distribution of electric potential and space charge in a silicon nanowire has been investigated. First, a model of the nanowire is generated taking into consideration the geometry and physics of the nanowire. The physics of the nanowire was modelled by a set of partial differential equations (PDEs) which were solved using the finite element method (FEM). Comprehensive simulation experiments were performed on the model in order to compute the distribution of potential and space charge. We also determined, through simulation, how the characteristic of the nanowire is affected by its dimensions. The characterization of the resulting nanowire, calculated by COMSOL Multiphysics, shows different dimensions and their effect on space charge and electrical potential

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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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Ali, Ahmad T. "Invariant Inhomogeneous Bianchi Type-I Cosmological Models with Electromagnetic Fields Using Lie Group Analysis in Lyra Geometry." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/918927.

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We find a new class of invariant inhomogeneous Bianchi type-I cosmological models in electromagnetic field with variable magnetic permeability. For this, Lie group analysis method is used to identify the generators that leave the given system of nonlinear partial differential equations (NLPDEs) (Einstein field equations) invariant. With the help of canonical variables associated with these generators, the assigned system of PDEs is reduced to ordinary differential equations (ODEs) whose simple solutions provide nontrivial solutions of the original system. A new class of exact (invariant-similarity) solutions have been obtained by considering the potentials of metric and displacement field as functions of coordinatesxandt. We have assumed thatF12is only nonvanishing component of electromagnetic field tensorFij. The Maxwell equations show thatF12is the function ofxalone whereas the magnetic permeabilityμ¯is the function ofxandtboth. The physical behavior of the obtained model is discussed.

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Liu, Si-Qi, and Youjin Zhang. "On quasi-triviality and integrability of a class of scalar evolutionary PDEs." Journal of Geometry and Physics 57, no. 1 (December 2006): 101–19. http://dx.doi.org/10.1016/j.geomphys.2006.02.005.

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Ma, Wen-Xiu. "Abundant lumps and their interaction solutions of (3+1)-dimensional linear PDEs." Journal of Geometry and Physics 133 (November 2018): 10–16. http://dx.doi.org/10.1016/j.geomphys.2018.07.003.

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YANG, XIAO-JUN, J. A. TENREIRO MACHADO, and DUMITRU BALEANU. "EXACT TRAVELING-WAVE SOLUTION FOR LOCAL FRACTIONAL BOUSSINESQ EQUATION IN FRACTAL DOMAIN." Fractals 25, no. 04 (July 25, 2017): 1740006. http://dx.doi.org/10.1142/s0218348x17400060.

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The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains.

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Qiu, Kaixuan, and Heng Li. "An Analytical Model for Production Analysis of Hydraulically Fractured Shale Gas Reservoirs Considering Irregular Stimulated Regions." Energies 13, no. 22 (November 12, 2020): 5899. http://dx.doi.org/10.3390/en13225899.

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Shale gas reservoirs are typically developed by multistage, propped hydraulic fractures. The induced fractures have a complex geometry and can be represented by a high permeability region near each fracture, also called stimulated region. In this paper, a new integrative analytical solution coupled with gas adsorption, non-Darcy flow effect is derived for shale gas reservoirs. The modified pseudo-pressure and pseudo-time are defined to linearize the nonlinear partial differential equations (PDEs) and thus the governing PDEs are transformed into ordinary differential equations (ODEs) by integration, instead of the Laplace transform. The rate vs. pseudo-time solution in real-time space can be obtained, instead of using the numerical inversion for Laplace transform. The analytical model is validated by comparison with the numerical model. According to the fitting results, the calculation accuracy of analytic solution is almost 99%. Besides the computational convenience, another advantage of the model is that it has been validated to be feasible to estimate the pore volume of hydraulic region, stimulated region, and matrix region, and even the shape of regions is irregular and asymmetrical for multifractured horizontal wells. The relative error between calculated volume and given volume is less than 10%, which meets the engineering requirements. The model is finally applied to field production data for history matching and forecasting.

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Markakis, M. P. "Approximate Ad Hoc Parametric Solutions for Nonlinear First-Order PDEs Governing Two-Dimensional Steady Vector Fields." Mathematical Problems in Engineering 2010 (2010): 1–23. http://dx.doi.org/10.1155/2010/874540.

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Through a suitable ad hoc assumption, a nonlinear PDE governing a three-dimensional weak, irrotational, steady vector field is reduced to a system of two nonlinear ODEs: the first of which corresponds to the two-dimensional case, while the second involves also the third field component. By using several analytical tools as well as linear approximations based on the weakness of the field, the first equation is transformed to an Abel differential equation which is solved parametrically. Thus, we obtain the two components of the field as explicit functions of a parameter. The derived solution is applied to the two-dimensional small perturbation frictionless flow past solid surfaces with either sinusoidal or parabolic geometry, where the plane velocities are evaluated over the body's surface in the case of a subsonic flow.

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Grundland, A. M., and J. de Lucas. "On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications." Differential Geometry and its Applications 67 (December 2019): 101557. http://dx.doi.org/10.1016/j.difgeo.2019.101557.

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Gorgone, Matteo, and Francesco Oliveri. "Nonlinear first order PDEs reducible to autonomous form polynomially homogeneous in the derivatives." Journal of Geometry and Physics 113 (March 2017): 53–64. http://dx.doi.org/10.1016/j.geomphys.2016.07.005.

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Igonin, Sergei, and Gianni Manno. "On Lie algebras responsible for integrability of (1+1)-dimensional scalar evolution PDEs." Journal of Geometry and Physics 150 (April 2020): 103596. http://dx.doi.org/10.1016/j.geomphys.2020.103596.

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Journal articles on the topic 'Geometry of PDEs'

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