Department of Mathematics

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IIT Jodhpur

The aim of this article is to find the approximate solution of the nonlinear Fredholm integro-differential equations of second kind with smooth kernels with less computational complexity and investigate the asymptotic behavior of convergence of the approximate solutions by using global polynomials based projection methods. We develop the theoretical framework for the nonlinear Fredholm integro-differential equations to obtain the superconvergence results by Legendre polynomial based projection methods and their iterated versions. Numerical examples are considered to demonstrate the theoretical results. © 2022 IMACS

World University root node

Applied Numerical Mathematics

Projection methods for approximate solution of a class of nonlinear Fredholm integro-differential equations

We prove the existence of a compactly supported, continuous (except at finitely many points) function gI,m:[0,∞)⟶R for all monomial prime ideals I of R of height one where (R,m) is the homogeneous coordinate ring associated to a projectively normal toric pair (X,D), such that ∫0∞gI,m(λ)dλ=β(I,m), where β(I,m) is the second coefficient of the Hilbert-Kunz function of I with respect to the maximal ideal m, as proved by Huneke-McDermott-Monsky [8]. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map τm:Cl(R)⟶R introduced in [8] to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function gI,m is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert plolynomial of the rings and the ideals. © 2021 Elsevier B.V.

Fulltext

Journal of Pure and Applied Algebra

β-Density function on the class group of projective toric varieties

Let p be a prime number, k a field of characteristic p and G a finite p-group. Let V be a finite-dimensional linear representation of G over k. Write S = Sym V ∗. For a class of p-groups which we call generalised Nakajima groups, we prove the following: (a) The Hilbert ideal is a complete intersection. As a consequence, for the case of generalised Nakajima groups, we prove a conjecture of Shank and Wehlau (reformulated by Broer) that asserts that if the invariant subring SG is a direct summand of S as SG-modules then SG is a polynomial ring. (b) The Hilbert ideal has a generating set with elements of degree at most |G|. This bound is conjectured by Derksen and Kemper. 2021 American Mathematical Society

Proceedings of the American Mathematical Society

ON HILBERT IDEALS for A CLASS of p-GROUPS in CHARACTERISTIC p

We consider reaction diffusion systems where components diffuse inside the domain and react on the surface through mass transport type boundary conditions. Under reasonable hypotheses, we establish the existence of component wise non-negative global solutions which are uniformly bounded in the sup norm. © 2020 American Institute of Mathematical Sciences. All rights reserved.

Communications on Pure and Applied Analysis

Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions

The multi-projection methods for solving the Fredholm-Hammerstein integral equation is proposed in this paper. We obtain the similar super-convergence results as in Mandal and Nelakanti (J Comput Appl Math 319:423–439, 2017) with a smooth kernel using piecewise polynomials of degree ≤ r- 1 , i.e., for both the multi-Galerkin and multi-collocation methods have order of convergence O(h3r) in uniform norm, where h is the norm of the partition. We have also considered iterated version of these methods and prove that both iterated multi-Galerkin and iterated multi-collocation methods have order of convergence O(h4r) in uniform norm. Numerical examples are given to illustrate the theoretical results. © 2020, Forum D'Analystes, Chennai.

Journal of Analysis

Superconvergence results for the nonlinear Fredholm–Hemmerstein integral equations of second kind

We propose the Jacobi spectral Galerkin and Jacobi spectral multi Galerkin methods with their iterated versions for obtaining the superconvergence results of a general class of nonlinear Volterra integral equations with a kernel xβ(z-x)-k, where 0 &lt; κ &lt; 1, β &gt; 0, which have an Abel-type and an endpoint singularity. The exact solutions for these types of integral equations are singular at the initial point of integration. First, we apply a transformation of independent variables to find a new integral equation with a sufficiently smooth solution. Then we discuss the superconvergence rates for the transformed equation in both uniform and weighted L2- norms. We obtain the order of convergence in Jacobi spectral Galerkin method O(N3/4-r) and O(N-r) in uniform and weighted L2- norms, respectively. Whereas iterated Jacobi spectral Galerkin method converges with the order of convergence O(N-2r) in both uniform and weighted L2- norms. We also show that iterated Jacobi spectral multi Galerkin method converges with the orders O(N-3rlogN) and O(N-3r) in uniform and weighted L2- norms, respectively. Theoretical results are verified by numerical illustrations. © 2021 Global Science Press. All rights reserved.

Advances in Applied Mathematics and Mechanics

Jacobi spectral galerkin methods for a class of nonlinear weakly singular volterra integral equations

In this article, the discrete version of Legendre projection and iterated Legendre projection methods are considered to find the approximate eigenfunctions (eigenvalues and eigenvectors) of a weakly singular compact integral operator. Making use of a sufficiently accurate numerical quadrature rule, we establish the error bounds of the approximated eigenvalues and eigenvectors by discrete Legendre projection and iterated discrete Legendre projection methods in both L2 and uniform norm. In particular, we obtain the optimal convergence rates O(n −m) for the eigenfunctions in iterated discrete Legendre projection method in L2 and uniform norms, where n is the highest degree of the Legendre polynomial employed in the approximation and m is the smoothness of the eigenvectors. Numerical examples are presented to illustrate the theoretical results. © 2021, Wilmington Scientific Publisher. All rights reserved.

Journal of Applied Analysis and Computation

Eigenvalue problem of a weakly singular compact integral operator by discrete legendre projection methods

The effect of a floating elastic membrane on a gravity-driven flow down an inclined porous plane is examined. The present work extends the earlier work of Anjalaiah et al. [Phys. Fluid, Vol. 25, 022101 (2013)] by including the thin membrane and excluding the insoluble surfactant at the top surface of the flow. The membrane tension emergence from the elasticity, is used to monitoring the surface tension on the top surface of the fluid layer. The fluid flow below the elastic membrane is governed by the Navier–Stokes equations and flow through the porous medium is governed by the Darcy–Brickmann equations. Using the normal mode analysis, two Orr–Sommerfeld systems are derived for each layers and solved by using the spectral collocation method to capture the linear instability. Further, the temporal growth rate and marginal stability curve are analyzed for different set of structural parameters. In general, the instability of free surface flow is more due to the presence of the unstable Yih mode. Both, the membrane tension and the uniform mass distribution rate along the length of the membrane assure to stabilize the surface waves by lowering the temporal growth rate of the disturbances. Consequently, the floating elastic membrane and the porosity of the inclined substrate pretend to suppress the unstable Yih mode and stabilize the flow anatomy. Hence, the floating elastic membrane can be disposed for the lethargic control of the free-surface flows' instability. © 2020 Wiley-VCH GmbH

ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

Effect of a floating elastic membrane for stabilizing the film flow down a porous inclined plane

Image captioning is a widely known problem in the area of AI. Caption generation from floor plan images has applications in indoor path planning, real estate, and providing architectural solutions. Several methods have been explored in the literature for generating captions or semi-structured descriptions from floor plan images. Since only the caption is insufficient to capture fine-grained details, researchers also proposed descriptive paragraphs from images. However, these descriptions have a rigid structure and lack flexibility, making it difficult to use them in real-time scenarios. This paper offers two models, description synthesis from image cue (DSIC) and transformer-based description generation (TBDG), for text generation from floor plan images. These two models take advantage of modern deep neural networks for visual feature extraction and text generation. The difference between both models is in the way they take input from the floor plan image. The DSIC model takes only visual features automatically extracted by a deep neural network, while the TBDG model learns textual captions extracted from input floor plan images with paragraphs. The specific keywords generated in TBDG and understanding them with paragraphs make it more robust in a general floor plan image. Experiments were carried out on a large-scale publicly available dataset and compared with state-of-the-art techniques to show the proposed model’s superiority. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

Department of Computer Science & Engineering

International Journal on Document Analysis and Recognition

Knowledge-driven description synthesis for floor plan interpretation

In this paper, a novel hashing framework is proposed using edge map and image normalization. In the proposed method, the input image is normalized using geometric moments and an edge map is then employed using the canny edge detector. The estimated binary image is divided into non-overlapping blocks and a chaotic map is used for random block selection. The singular value decomposition is carried out on the selected blocks for extracting the significant features followed by generation of hash value. The simulation results support the contention that the proposed technique is secure and considerately robust against a variety of image manipulations. © 2021, Springer Nature Singapore Pte Ltd.