IA Scholar Query: Legendre wavelets method for solving fractional partial differential equations with Dirichlet boundary conditions.
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Internet Archive Scholar query results feedeninfo@archive.orgTue, 28 Jun 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Trivariate Spectral Collocation Approach for the Numerical Solution of Three-Dimensional Elliptic Partial Differential Equations
https://scholar.archive.org/work/udc22tani5hw7blhuxkg5vhbqi
This article is concerned with the numerical solution of three-dimensional elliptic partial differential equations (PDEs) using the trivariate spectral collocation approach based on the Kronecker tensor product. By using the quasilinearization method, the nonlinear elliptic PDEs are simplified to a linear system of algebraic equations that can be discretized using the spectral collocation method. The method is based on approximating the solutions using the triple Lagrange interpolating polynomials, which interpolate the unknown functions at selected Chebyshev–Gauss–Lobatto (CGL) grid points. The CGL points are preferred to ensure simplicity in the conversion of continuous derivatives to discrete derivatives at the collocation points. The collocation process is carried out at the interior points to reduce the size of differentiation matrices. This work is aimed at verifying that the algorithm based on the method is simple and easily implemented in any scientific software to produce more accurate and stable results. The effectiveness and spectral accuracy of the numerical algorithm is checked through the determination and analysis of errors, condition numbers and computational time for various classes of single or system of elliptic PDEs including those with singular behavior. The communicated results indicate that the proposed method is more accurate, stable and effective for solving elliptic PDEs. This good accuracy becomes possible with the usage of few grid points and less memory requirements for numerical computation.Musawenkhosi Patson Mkhatshwa, Melusi Khumalowork_udc22tani5hw7blhuxkg5vhbqiTue, 28 Jun 2022 00:00:00 GMTComputational Wavelet Method for Multidimensional Integro-Partial Differential Equation of Distributed Order
https://scholar.archive.org/work/gjlooigqejci5pvjv7zu5nt6qi
This article provides an effective computational algorithm based on Legendre wavelet (LW) and standard tau approach to approximate the solution of multi-dimensional distributed order time-space fractional weakly singular integro-partial differential equation (DOT-SFWSIPDE). To the best of our understanding, the proposed computational algorithm is new and has not been previously applied for solving DOT-SFWSIPDE. The matrix representation of distributed order fractional derivatives, integer order derivatives and weakly singular kernel associated with the integral based on LWare established to find the numerical solutions of the proposed DOT-SFWSIPDE. Moreover, the association of standard tau rule and Legendre-Gauss quadrature (LGQ) techniques along with constructed matrix representation of differential and integral operators diminish DOT-SFWSIPDE into system of linear algebraic equations. Error bounds, convergence analysis, numerical algorithms and also error estimation of the DOT-SFWSIPDE are regorously investigated. For the reliability of the proposed computational algorithm, numerous test examples has been incorporated in the manuscript to ensure the robustness and theoretical results of proposed technique.Yashveer Kumawork_gjlooigqejci5pvjv7zu5nt6qiTue, 31 May 2022 00:00:00 GMTRegularity theory for a new class of fractional parabolic stochastic evolution equations
https://scholar.archive.org/work/4zx2vhcn45ekdofwmosqrzs7ci
A new class of fractional-order stochastic evolution equations of the form (∂_t + A)^γ X(t) = Ẇ^Q(t), t∈[0,T], γ∈ (0,∞), is introduced, where -A generates a C_0-semigroup on a separable Hilbert space H and the spatiotemporal driving noise Ẇ^Q is an H-valued cylindrical Q-Wiener process. Mild and weak solutions are defined; these concepts are shown to be equivalent and to lead to well-posed problems. Temporal and spatial regularity of the solution process X are investigated, the former being measured by mean-square or pathwise smoothness and the latter by using domains of fractional powers of A. In addition, the covariance of X and its long-time behavior are analyzed. These abstract results are applied to the cases when A := L^β and Q:=L^-α are fractional powers of symmetric, strongly elliptic second-order differential operators defined on (i) bounded Euclidean domains or (ii) smooth, compact surfaces. In these cases, the Gaussian solution processes can be seen as generalizations of merely spatial (Whittle-)Matérn fields to space-time.Kristin Kirchner, Joshua Willemswork_4zx2vhcn45ekdofwmosqrzs7ciSat, 30 Apr 2022 00:00:00 GMTOn Gaussian multiplicative chaos and conformal field theory
https://scholar.archive.org/work/bhwea4esx5ci7euvz3qishl4ii
This thesis is concerned with conformally invariant stochastic processes in two dimensions and their applications to conformal field theory (CFT). The main probabilistic objects are the Gaussian free field (GFF) and the random geometries associated to it. Especially, we are interested in Gaussian multiplicative chaos (GMC), Schramm-Loewner evolution (SLE) and Liouville CFT, which can be understood as theories of random surfaces. From the point of view of physics, the idea of a "summing over surfaces" can be traced back to Polyakov's work on bosonic string theory. Indeed, the starting point of string theory is to replace a point particle by a one dimensional manifold (a string), so that one must replace the worldline by a worldsheet, i.e. an embedding of a surface into space-time. The path integral that Polyakov wrote down features a random conformal factor that should be described by the quantisation of the Liouville action. Therefore, this probability measure should describe random fluctuations around the uniform metric. Polyakov also suggested that the resulting quantum field theory should exhibit conformal invariance. This means that the Hilbert space of the theory should carry a projective unitary representation of the group of local conformal transformations, i.e. a unitary representation of the Virasoro algebra. Since it is an infinite dimensional Lie algebra, this is a huge constraint to put on a system and this led Belavin, Polyakov & Zamolodchikov to give an axiomatic framework for CFT based on the representation theory of the Virasoro algebra. Here, the game is somehow reversed: one tries to exhibit and classify all theories fitting in this framework. In particular, it is not even clear in the first place that such algebraic structures exist. In this context, Liouville theory is a success story in the interaction of algebra, geometry and probability. On the one hand, the algebraic point of view was successful in finding a theory fitting in the BPZ framework. On the other hand, it was unclear that t [...]Guillaume Baverez, Apollo-University Of Cambridge Repository, Jason Millerwork_bhwea4esx5ci7euvz3qishl4iiTue, 05 Apr 2022 00:00:00 GMTLagging Heat Models in Thermodynamics and Bioheat Transfer: a Critical Review
https://scholar.archive.org/work/dqaj6xlunbawjabkamavgpzr5m
The accuracy of the classical heat conduction model, known as Fourier's law, is highly questioned, dealing with the micro and nanosystems and biological tissues. In other words, the results obtained from the classical equations deviate from the available experimental data. It means that the continuum heat diffusion equation is insufficient and inappropriate for modeling heat transport in these cases. There are several techniques for modeling non-Fourier heat conduction. In the present paper, we place our focus on the dual-phase-lag (DPL) approach. The DPL model, as a popular modification of Fourier's law, has already been utilized in numerous situations, such as simulating ultrafast laser heating and heat conduction in carbon nanotubes. There has been a sharp increase in research on non-Fourier heat conduction in recent years. Several studies have been performed in the fields of thermoelasticity, thermodynamics, transistor modeling, and bioheat transport. This review presents the most recent non-Fourier bioheat conduction works and the related thermodynamics background. The various mathematical tools, modeling different thermal therapies, and relevant criticisms and disputes are discussed. Finally, the novel and other possible studies are also presented to provide a better overview, and the roadmap to the future research and challenges ahead is drawn up.Zahra Shomali, Robert Kovács, Peter Ván, Igor Vasilievich Kudinov, Jafar Ghazanfarianwork_dqaj6xlunbawjabkamavgpzr5mSun, 27 Mar 2022 00:00:00 GMTExponential Convergence of hp-Time-Stepping in Space-Time Discretizations of Parabolic PDEs
https://scholar.archive.org/work/twsw6vlkdzhvfivjzilptcf5bq
For linear parabolic initial-boundary value problems with self-adjoint, time-homogeneous elliptic spatial operator in divergence form with Lipschitz-continuous coefficients, and for incompatible, time-analytic forcing term in polygonal/polyhedral domains D, we prove time-analyticity of solutions. Temporal analyticity is quantified in terms of weighted, analytic function classes, for data with finite, low spatial regularity and without boundary compatibility. Leveraging this result, we prove exponential convergence of a conforming, semi-discrete hp-time-stepping approach. We combine this semi-discretization in time with first-order, so-called "h-version" Lagrangian Finite Elements with corner-refinements in space into a tensor-product, conforming discretization of a space-time formulation. We prove that, under appropriate corner- and corner-edge mesh-refinement of D, error vs. number of degrees of freedom in space-time behaves essentially (up to logarithmic terms), to what standard FEM provide for one elliptic boundary value problem solve in D. We focus on two-dimensional spatial domains and comment on the one- and the three-dimensional case.Ilaria Perugia, Christoph Schwab, Marco Zankwork_twsw6vlkdzhvfivjzilptcf5bqTue, 22 Mar 2022 00:00:00 GMTPeridynamic Galerkin methods for nonlinear solid mechanics
https://scholar.archive.org/work/z3tm5mnsfbgczk66o7rqgw5eqe
Simulation-driven product development is nowadays an essential part in the industrial digitalization. Notably, there is an increasing interest in realistic high-fidelity simulation methods in the fast-growing field of additive and ablative manufacturing processes. Thanks to their flexibility, meshfree solution methods are particularly suitable for simulating the stated processes, often accompanied by large deformations, variable discontinuities, or phase changes. Furthermore, in the industrial domain, the meshing of complex geometries represents a significant workload, which is usually minor for meshfree methods. Over the years, several meshfree schemes have been developed. Nevertheless, along with their flexibility in discretization, meshfree methods often endure a decrease in accuracy, efficiency and stability or suffer from a significantly increased computation time. Peridynamics is an alternative theory to local continuum mechanics for describing partial differential equations in a non-local integro-differential form. The combination of the so-called peridynamic correspondence formulation with a particle discretization yields a flexible meshfree simulation method, though does not lead to reliable results without further treatment.\newline In order to develop a reliable, robust and still flexible meshfree simulation method, the classical correspondence formulation is generalized into the Peridynamic Galerkin (PG) methods in this work. On this basis, conditions on the meshfree shape functions of virtual and actual displacement are presented, which allow an accurate imposition of force and displacement boundary conditions and lead to stability and optimal convergence rates. Based on Taylor expansions moving with the evaluation point, special shape functions are introduced that satisfy all the previously mentioned requirements employing correction schemes. In addition to displacement-based formulations, a variety of stabilized, mixed and enriched variants are developed, which are tailored in their application to [...]Tobias Bode, University, My, Institut Für Kontinuumsmechanikwork_z3tm5mnsfbgczk66o7rqgw5eqeWed, 02 Feb 2022 00:00:00 GMTThe SPDE approach for Gaussian and non-Gaussian fields: 10 years and still running
https://scholar.archive.org/work/wrgdz7oszzeilifz3so45nl2wm
Gaussian processes and random fields have a long history, covering multiple approaches to representing spatial and spatio-temporal dependence structures, such as covariance functions, spectral representations, reproducing kernel Hilbert spaces, and graph based models. This article describes how the stochastic partial differential equation approach to generalising Mat\'ern covariance models via Hilbert space projections connects with several of these approaches, with each connection being useful in different situations. In addition to an overview of the main ideas, some important extensions, theory, applications, and other recent developments are discussed. The methods include both Markovian and non-Markovian models, non-Gaussian random fields, non-stationary fields and space-time fields on arbitrary manifolds, and practical computational considerations.Finn Lindgren and David Bolin and Håvard Ruework_wrgdz7oszzeilifz3so45nl2wmTue, 04 Jan 2022 00:00:00 GMTApplication of Lane-Emden differential equation numerical method in fair value analysis of financial accounting
https://scholar.archive.org/work/ia7yu3eivvgexcitzpw4pdx4uq
In order to study the fair value analysis of financial accounting, the Euler wavelet method is proposed to solve the numerical solutions of a class of Lane-Emden type differential equations with Dirichlet, Neumann and Neumann-Robin boundary conditions. The results show that the fractional integral formula of Euler wavelet function under the Riemann-Liouville fractional order definition and the L∞ and L2 errors of Haar wavelet are derived by the analytic form of Euler polynomial. By fixing M=4 and increasing the resolution scale k of Euler wavelet, a stable convergence solution can be obtained. The Lane-Emden equation with boundary conditions is transformed into algebraic equations by Euler wavelet collocation method, and the numerical results are compared with the results and exact solutions of other methods. The application advantages of fair value can be exerted through financial accounting to promote the transformation and upgrading of enterprises and realise the stable economic growth.Linying Xu, Marwan Aouadwork_ia7yu3eivvgexcitzpw4pdx4uqThu, 30 Dec 2021 00:00:00 GMTMultiwavelet-based Operator Learning for Differential Equations
https://scholar.archive.org/work/rnlbxmmspvhanfavjfjrzetkqi
The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a multiwavelet-based neural operator learning scheme that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a (2X-10X) improvement (0.0018 (0.0033) relative L2 error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.Gaurav Gupta, Xiongye Xiao, Paul Bogdanwork_rnlbxmmspvhanfavjfjrzetkqiSun, 10 Oct 2021 00:00:00 GMTImproved Galactic Foreground Removal for B-Modes Detection with Clustering Methods
https://scholar.archive.org/work/zmnnxzmvwzccdgde4uvewqlssq
Characterizing the sub-mm Galactic emission has become increasingly critical especially in identifying and removing its polarized contribution from the one emitted by the Cosmic Microwave Background (CMB). In this work, we present a parametric foreground removal performed onto sub-patches identified in the celestial sphere by means of spectral clustering. Our approach takes into account efficiently both the geometrical affinity and the similarity induced by the measurements and the accompanying errors. The optimal partition is then used to parametrically separate the Galactic emission encoding thermal dust and synchrotron from the CMB one applied on two nominal observations of forthcoming experiments from the ground and from the space. Performing the parametric fit singularly on each of the clustering derived regions results in an overall improvement: both controlling the bias and the uncertainties in the CMB B-mode recovered maps. We finally apply this technique using the map of the number of clouds along the line of sight, 𝒩_c, as estimated from HI emission data and perform parametric fitting onto patches derived by clustering on this map. We show that adopting the 𝒩_c map as a tracer for the patches related to the thermal dust emission, results in reducing the B-mode residuals post-component separation. The code is made publicly available.Giuseppe Puglisi, Gueorgui Mihaylov, Georgia V. Panopoulou, Davide Poletti, Josquin Errard, Paola A. Puglisi, Giacomo Vianellowork_zmnnxzmvwzccdgde4uvewqlssqMon, 04 Oct 2021 00:00:00 GMTFast Solvers and Simulation Data Compression Algorithms for Granular Media and Complex Fluid Flows
https://scholar.archive.org/work/unnrbtyu7rgghdrq3syiyzk6sm
Granular and particulate flows are common forms of materials used in various physical and industrial applications. For instance, we model the soil as a collection of rigid particles with frictional contact in soil-vehicle simulations, and we simulate bacterial colonies as active rigid particles immersed in a viscous fluid. Due to the complex interactions in-between the particles and/or the particles and the fluid, numerical simulations are often the only way to study these systems apart from typically expensive physical experiments. A standard method for simulating these systems is to apply simple physical laws to each of the particles using the discrete element method (DEM) and evolve the resulting multibody system in time. However, due to the sheer number of particles in even a moderate-scale real-world system, it quickly becomes expensive to timestep these systems unless we exploit fast algorithms and high-performance computing techniques. For instance, a big challenge in granular media simulations is resolving contact between the constituent particles. We use a cone-complementarity formulation of frictional contact to resolve collisions; this approach leads to a quadratic optimization problem whose solution gives us the contact forces between particles at each timestep. In this thesis, we introduce strategies for solving these optimization problems on distributed memory machines. In particular, by imposing a locality-preserving partitioning of the rigid bodies among the computing nodes, we minimize the communication cost and construct a scalable framework for collision detecting and resolution that can be easily scaled to handle hundreds of millions of particles. For robust and efficient simulation of axisymmetric particles in viscous fluids, we introduce a fast method for solving Stokes boundary integral equations (BIEs) on surfaces of revolution. By first transforming the Stokes integral kernels into a rotationally invariant form and then decoupling the transformed kernels using the Fourier series, we reduc [...]Saibal De, University, Mywork_unnrbtyu7rgghdrq3syiyzk6smFri, 24 Sep 2021 00:00:00 GMTEfficient Representations of Signals in Nonlinear Signal Processing with Applications to Inverse Problems
https://scholar.archive.org/work/t6ohs4z4wvdtfnrklnafyuo7j4
The focus of this thesis is the construction and analysis of efficient representations in nonlinear signal processing, and the applications of these structures to inverse problems in a variety of fields. The work is composed of three major sections, each associated with a different form of data: - Regression and Distance Estimation on Graphs and Riemannian Manifolds. - Instantaneous Time-Frequency Analysis via Synchrosqueezing. - Multiscale Dictionaries of Slepian Functions on the Sphere.Eugene Brevdowork_t6ohs4z4wvdtfnrklnafyuo7j4Mon, 20 Sep 2021 00:00:00 GMTMathematical modeling of cancer treatments with fractional derivatives: An Overview
https://scholar.archive.org/work/oguh67zcavesfe3ielo6kxs6em
This review article presents fractional derivative cancer treatment models to show the importance of fractional derivatives in modeling cancer treatments. Cancer treatment is a significant research area with many mathematical models developed by mathematicians to represent the cancer treatment processes like hyperthermia, immunotherapy, chemotherapy, and radiotherapy. However, many of these models were based on ordinary derivatives and the use of fractional derivatives is still new to many mathematicians. Therefore, it is imperative to review fractional cancer treatment models. The review was done by first presenting 22 various definitions of fractional derivative. Thereafter, 11 articles were selected from different online databases which included Scopus, EBSCOHost, ScienceDirect Journal, SpringerLink Journal, Wiley Online Library, and Google Scholar. These articles were summarized, and the used fractional derivative models were analyzed. Based on this analysis, the merit of modeling with fractional derivative, the most used fractional derivative definition, and the future direction for cancer treatment modeling were presented. From the results of the analysis, it was shown that fractional derivatives incorporated memory effects which gave it an advantage over ordinary derivative for cancer treatment modeling. Moreover, the fractional derivative is a general definition of all derivatives. Also, the fractional models can be applied to different cancer treatment procedures and the most used fractional derivative is the Caputo as well as its non-singular kernel versions. Finally, it was concluded that the future direction for cancer treatment modeling is the adoption of fractional derivative models corroborated with experimental or clinical data.Musiliu Folarin Farayola, Sharidan Shafie, Fuaada Mohd Siam, Rozi Mahmud, Suraju Olusegun Ajadiwork_oguh67zcavesfe3ielo6kxs6emTue, 31 Aug 2021 00:00:00 GMT