Partial Differential Equation
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Raffaele Grande

AbstractThe evolution by horizontal mean curvature flow (HMCF) is a partial differential equation in a sub-Riemannian setting with applications in IT and neurogeometry [see Citti et al. (SIAM J Imag Sci 9(1):212–237, 2016)]. Unfortunately this equation is difficult to study, since the horizontal normal is not always well defined. To overcome this problem the Riemannian approximation was introduced. In this article we obtain a stochastic representation of the solution of the approximated Riemannian mean curvature using the Riemannian approximation and we will prove that it is a solution in the viscosity sense of the approximated mean curvature flow, generalizing the result of Dirr et al. (Commun Pure Appl Math 9(2):307–326, 2010).

2022 ◽  
Vol 6 (1) ◽  
pp. 24
Muhammad Shakeel ◽  
Nehad Ali Shah ◽  
Jae Dong Chung

In this article, a new method for obtaining closed-form solutions of the simplified modified Camassa-Holm (MCH) equation, a nonlinear fractional partial differential equation, is suggested. The modified Riemann-Liouville fractional derivative and the wave transformation are used to convert the fractional order partial differential equation into an integer order ordinary differential equation. Using the novel (G’/G2)-expansion method, several exact solutions with extra free parameters are found in the form of hyperbolic, trigonometric, and rational function solutions. When parameters are given appropriate values along with distinct values of fractional order α travelling wave solutions such as singular periodic waves, singular kink wave soliton solutions are formed which are forms of soliton solutions. Also, the solutions obtained by the proposed method depend on the value of the arbitrary parameters H. Previous results are re-derived when parameters are given special values. Furthermore, for numerical presentations in the form of 3D and 2D graphics, the commercial software Mathematica 10 is incorporated. The method is accurately depicted, and it provides extra general exact solutions.

П.А. Вельмисов ◽  
А.В. Анкилов ◽  
Г.А. Анкилов

ва подхода к решению аэрогидродинамической части задачи, основанные на методах теории функций комплексного переменного и методе Фурье. В результате применения каждого подхода решение исходной задачи сведено к исследованию дифференциального уравнения с частными производными для деформации элемента, позволяющего изучать его динамику. На основе метода Галеркина произведены численные эксперименты для конкретных примеров механической системы, подтверждающие идентичность решений, найденных для каждого дифференциального уравнения с частными производными. The dynamics of an elastic element of a vibration device, simulated by a channel, inside which a stream of a liquid flows, is investigated. Two approaches to solving the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable and the Fourier method, are given. As a result of applying each approach, the solution to the original problem is reduced to the study of a partial differential equation for the deformation of an element, which makes it possible to study its dynamics. Based on the Galerkin method, the numerical experiments were carried out for specific examples of mechanical system, confirming the identity of the solutions found for each partial differential equation.

2021 ◽  
Vol 56 (2) ◽  
pp. 195-223
Igoris Belovas ◽  

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 349
Joël Chaskalovic

A probabilistic approach is developed for the exact solution u to a deterministic partial differential equation as well as for its associated approximation uh(k) performed by Pk Lagrange finite element. Two limitations motivated our approach: On the one hand, the inability to determine the exact solution u relative to a given partial differential equation (which initially motivates one to approximating it) and, on the other hand, the existence of uncertainties associated with the numerical approximation uh(k). We, thus, fill this knowledge gap by considering the exact solution u together with its corresponding approximation uh(k) as random variables. By a method of consequence, any function where u and uh(k) are involved are modeled as random variables as well. In this paper, we focus our analysis on a variational formulation defined on Wm,p Sobolev spaces and the corresponding a priori estimates of the exact solution u and its approximation uh(k) in order to consider their respective Wm,p-norm as a random variable, as well as the Wm,p approximation error with regards to Pk finite elements. This will enable us to derive a new probability distribution to evaluate the relative accuracy between two Lagrange finite elements Pk1 and Pk2,(k1<k2).

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Jinghua Ning ◽  
Song Yu

Financial statements are the basis of financial analysis. Most of them are still using traditional financial statement software, and financial data cannot achieve better information sharing. Using a barcode to improve the efficiency of data sharing is undoubtedly the most convenient and fast way to realize financial statement information sharing. This paper studies the problem of the barcode image location and recognition in the financial statement system and tries to apply the partial differential equation image recognition algorithm to the barcode location in the financial statement system. In this paper, image technology is used to preprocess the code 39 barcode label image in the captured image, including color space conversion, image enhancement, threshold segmentation, binarization, and edge detection. The barcode position location is realized by using the method based on integral projection peak analysis and Hough transform line detection. It is proved that the positioning function of the barcode makes the data in financial statements realize resource sharing. Then, the performance of the completed barcode positioning and recognition algorithm is tested. The reliability and effectiveness of the algorithm are verified on the manually made test set.

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Phillip Baumann ◽  
Kevin Sturm

PurposeThe goal of this paper is to give a comprehensive and short review on how to compute the first- and second-order topological derivatives and potentially higher-order topological derivatives for partial differential equation (PDE) constrained shape functionals.Design/methodology/approachThe authors employ the adjoint and averaged adjoint variable within the Lagrangian framework and compare three different adjoint-based methods to compute higher-order topological derivatives. To illustrate the methodology proposed in this paper, the authors then apply the methods to a linear elasticity model.FindingsThe authors compute the first- and second-order topological derivatives of the linear elasticity model for various shape functionals in dimension two and three using Amstutz' method, the averaged adjoint method and Delfour's method.Originality/valueIn contrast to other contributions regarding this subject, the authors not only compute the first- and second-order topological derivatives, but additionally give some insight on various methods and compare their applicability and efficiency with respect to the underlying problem formulation.

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