scholarly journals A Haar Wavelet Series Solution of Heat Equation with Involution

2021 ◽  
Vol 86 (2) ◽  
pp. 50-55
Author(s):  
Burma Saparova ◽  
Roza Mamytova ◽  
Nurjamal Kurbanbaeva ◽  
Anvarjon Akhatjonovich Ahmedov
Keyword(s):  
Heat Equation ◽  
Haar Wavelet ◽  
Wavelet Expansion ◽  
Heat Equations ◽  
Exact Representation ◽  
Numerical Techniques ◽  
New Methods ◽  
Mathematical Problems ◽  
Different Levels ◽  
Differential And Integral Equations

It is well known that the wavelets have widely applied to solve mathematical problems connected with the differential and integral equations. The application of the wavelets possess several important properties, such as orthogonality, compact support, exact representation of polynomials at certain degree and the ability to represent functions on different levels of resolution. In this paper, new methods based on wavelet expansion are considered to solve problems arising in approximation of the solution of heat equation with involution. We have developed new numerical techniques to solve heat equation with involution and obtained new approximative representation for solution of heat equations.

scholarly journals Green–Haar wavelets method for generalized fractional differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mujeeb ur Rehman ◽  
Dumitru Baleanu ◽  
Jehad Alzabut ◽  
Muhammad Ismail ◽  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Haar Wavelet ◽  
Haar Wavelets ◽  
Computational Time ◽  
Operational Matrices ◽  
Numerical Techniques ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems ◽  
Better Than

Abstract The objective of this paper is to present two numerical techniques for solving generalized fractional differential equations. We develop Haar wavelets operational matrices to approximate the solution of generalized Caputo–Katugampola fractional differential equations. Moreover, we introduce Green–Haar approach for a family of generalized fractional boundary value problems and compare the method with the classical Haar wavelets technique. In the context of error analysis, an upper bound for error is established to show the convergence of the method. Results of numerical experiments have been documented in a tabular and graphical format to elaborate the accuracy and efficiency of addressed methods. Further, we conclude that accuracy-wise Green–Haar approach is better than the conventional Haar wavelets approach as it takes less computational time compared to the Haar wavelet method.

scholarly journals Geomorphometric analysis of cave ceiling channels mapped with 3-D terrestrial laser scanning

2016 ◽  
Vol 20 (5) ◽  
pp. 1827-1849 ◽  
Author(s):  
Michal Gallay ◽  
Zdenko Hochmuth ◽  
Ján Kaňuk ◽  
Jaroslav Hofierka
Keyword(s):  
Laser Scanning ◽  
Methodological Approach ◽  
Terrestrial Laser Scanning ◽  
Scalar Fields ◽  
Complex Environments ◽  
New Methods ◽  
Digital Elevation ◽  
Unique Manner ◽  
Elevation Model ◽  
Different Levels

Abstract. The change of hydrological conditions during the evolution of caves in carbonate rocks often results in a complex subterranean geomorphology, which comprises specific landforms such as ceiling channels, anastomosing half tubes, or speleothems organized vertically in different levels. Studying such complex environments traditionally requires tedious mapping; however, this is being replaced with terrestrial laser scanning technology. Laser scanning overcomes the problem of reaching high ceilings, providing new options to map underground landscapes with unprecedented level of detail and accuracy. The acquired point cloud can be handled conveniently with dedicated software, but applying traditional geomorphometry to analyse the cave surface is limited. This is because geomorphometry has been focused on parameterization and analysis of surficial terrain. The theoretical and methodological concept has been based on two-dimensional (2-D) scalar fields, which are sufficient for most cases of the surficial terrain. The terrain surface is modelled with a bivariate function of altitude (elevation) and represented by a raster digital elevation model. However, the cave is a 3-D entity; therefore, a different approach is required for geomorphometric analysis. In this paper, we demonstrate the benefits of high-resolution cave mapping and 3-D modelling to better understand the palaeohydrography of the Domica cave in Slovakia. This methodological approach adopted traditional geomorphometric methods in a unique manner and also new methods used in 3-D computer graphics, which can be applied to study other 3-D geomorphological forms.

scholarly journals No local L1 solutions for semilinear fractional heat equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Kexue Li
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Heat Equations ◽  
Fractional Heat Equation ◽  
The Cauchy Problem

AbstractWe study the Cauchy problem for the semilinear fractional heat equation

scholarly journals Numerical Techniques for Solving Linear Volterra Fractional Integral Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Safa’ Hamdan ◽  
Naji Qatanani ◽  
Adnan Daraghmeh
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Fractional Integral ◽  
Integration Method ◽  
Haar Wavelet ◽  
Numerical Techniques ◽  
Product Integration ◽  
Fractional Integral Equation ◽  
Product Integration Method ◽  
Good Agreement

Two numerical techniques, namely, Haar Wavelet and the product integration methods, have been employed to give an approximate solution of the fractional Volterra integral equation of the second kind. To test the applicability and efficiency of the numerical method, two illustrative examples with known exact solution are presented. Numerical results show clearly that the accuracy of these methods are in a good agreement with the exact solution. A comparison between these methods shows that the product integration method provides more accurate results than its counterpart.

scholarly journals Genomic Insights into the Different Layers of Gene Regulation in Yeast

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
José E. Pérez-Ortín ◽  
Daniel A. Medina ◽  
Antonio Jordán-Pla
Keyword(s):  
Gene Regulation ◽  
Transcription Initiation ◽  
Model Organism ◽  
New Methods ◽  
Control Gene Expression ◽  
Gene Sets ◽  
Initiation Step ◽  
Technical Features ◽  
Trans Regulation ◽  
Different Levels

The model organism Saccharomyces cerevisiae has allowed the development of new functional genomics techniques devoted to the study of transcription in all its stages. With these techniques, it has been possible to find interesting new mechanisms to control gene expression that act at different levels and for different gene sets apart from the known cis-trans regulation in the transcription initiation step. Here we discuss a method developed in our laboratory, Genomic Run-On, and other new methods that have recently appeared with distinct technical features. A comparison between the datasets generated by them provides interesting genomic insights into the different layers of gene regulation in eukaryotes.

A new difference scheme for time fractional heat equations based on the Crank-Nicholson method

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Ibrahim Karatay ◽  
Nurdane Kale ◽  
Serife Bayramoglu
Keyword(s):  
Diffusion Equation ◽  
Heat Equation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Caputo Derivative ◽  
Numerical Experiments ◽  
Time Derivative ◽  
Heat Equations ◽  
First Order ◽  
Fractional Heat Equation

AbstractIn this paper, we consider the numerical solution of a time-fractional heat equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with the Caputo derivative of order α, where 0 < α < 1. The main purpose of this work is to extend the idea on the Crank-Nicholson method to the time-fractional heat equations. By the method of the Fourier analysis, we prove that the proposed method is stable and the numerical solution converges to the exact one with the order O(τ 2-α + h 2), conditionally. Numerical experiments are carried out to support the theoretical claims.

scholarly journals A standardized, extensible framework for optimizing classification improves marker-gene taxonomic assignments

2015 ◽  
Author(s):  
Nicholas A Bokulich ◽  
Jai Ram Rideout ◽  
Evguenia Kopylova ◽  
Evan Bolyen ◽  
Jessica Patnode ◽  
...  
Keyword(s):  
Marker Gene ◽  
Amplicon Sequencing ◽  
Evaluation Framework ◽  
Taxonomic Resolution ◽  
Marker Genes ◽  
Classification Methods ◽  
New Methods ◽  
Taxonomic Assignments ◽  
Different Levels

Background: Taxonomic classification of marker-gene (i.e., amplicon) sequences represents an important step for molecular identification of microorganisms. Results: We present three advances in our ability to assign and interpret taxonomic classifications of short marker gene sequences: two new methods for taxonomy assignment, which reduce runtime up to two-fold and achieve high precision genus-level assignments; an evaluation of classification methods that highlights differences in performance with different marker genes and at different levels of taxonomic resolution; and an extensible framework for evaluating and optimizing new classification methods, which we hope will serve as a model for standardized and reproducible bioinformatics methods evaluations. Conclusions: Our new methods are accessible in QIIME 1.9.0, and our evaluation framework will support ongoing optimization of classification methods to complement rapidly evolving short-amplicon sequencing and bioinformatics technologies. Static versions of all of the analysis notebooks generated with this framework, which contain all code and analysis results, can be viewed at http://bit.ly/srta-010.

scholarly journals A State-of-the-Art Review on Integral Transform Technique in Laser–Material Interaction: Fourier and Non-Fourier Heat Equations

Materials ◽  
2021 ◽  
Vol 14 (16) ◽  
pp. 4733
Author(s):  
Mihai Oane ◽  
Muhammad Arif Mahmood ◽  
Andrei C. Popescu
Keyword(s):  
Electron Beam ◽  
Differential Equations ◽  
Heat Equation ◽  
Material Properties ◽  
Integral Transform ◽  
Generalized Solutions ◽  
Heat Equations ◽  
Temperature Model ◽  
Two Temperature ◽  
Integral Transform Technique

Heat equations can estimate the thermal distribution and phase transformation in real-time based on the operating conditions and material properties. Such wonderful features have enabled heat equations in various fields, including laser and electron beam processing. The integral transform technique (ITT) is a powerful general-purpose semi-analytical/numerical method that transforms partial differential equations into a coupled system of ordinary differential equations. Under this category, Fourier and non-Fourier heat equations can be implemented on both equilibrium and non-equilibrium thermo-dynamical processes, including a wide range of processes such as the Two-Temperature Model, ultra-fast laser irradiation, and biological processes. This review article focuses on heat equation models, including Fourier and non-Fourier heat equations. A comparison between Fourier and non-Fourier heat equations and their generalized solutions have been discussed. Various components of heat equations and their implementation in multiple processes have been illustrated. Besides, literature has been collected based on ITT implementation in various materials. Furthermore, a future outlook has been provided for Fourier and non-Fourier heat equations. It was found that the Fourier heat equation is simple to use but involves infinite speed heat propagation in comparison to the non-Fourier heat equation and can be linked with the Two-Temperature Model in a natural way. On the other hand, the non-Fourier heat equation is complex and involves various unknowns compared to the Fourier heat equation. Fourier and Non-Fourier heat equations have proved their reliability in the case of laser–metallic materials, electron beam–biological and –inorganic materials, laser–semiconducting materials, and laser–graphene material interactions. It has been identified that the material properties, electron–phonon relaxation time, and Eigen Values play an essential role in defining the precise results of Fourier and non-Fourier heat equations. In the case of laser–graphene interaction, a restriction has been identified from ITT. When computations are carried out for attosecond pulse durations, the laser wavelength approaches the nucleus-first electron separation distance, resulting in meaningless results.

scholarly journals Finite approximate controllability for semilinear heat equations in noncylindrical domains

2004 ◽  
Vol 76 (3) ◽  
pp. 475-487
Author(s):  
Silvano B. de Menezes ◽  
Juan Limaco ◽  
Luis A. Medeiros
Keyword(s):  
Fixed Point ◽  
Heat Equation ◽  
Approximate Controllability ◽  
State Equation ◽  
Heat Equations ◽  
Semilinear Heat Equation ◽  
Fixed Point Result ◽  
Linearized Problem

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

scholarly journals Insensitizing control for linear and semi-linear heat equations with partially unknown domain

2019 ◽  
Vol 25 ◽  
pp. 50 ◽  
Author(s):  
Pierre Lissy ◽  
Yannick Privat ◽  
Yacouba Simporé
Keyword(s):  
Heat Equation ◽  
Control Problem ◽  
Duality Theory ◽  
Unique Continuation ◽  
Dirichlet Boundary Conditions ◽  
Linear Case ◽  
Heat Equations ◽  
Geometrical Approach ◽  
Linear Heat Equation ◽  
Linear Heat

We consider a semi-linear heat equation with Dirichlet boundary conditions and globally Lipschitz nonlinearity, posed on a bounded domain of ℝN (N ∈ ℕ*), assumed to be an unknown perturbation of a reference domain. We are interested in an insensitizing control problem, which consists in finding a distributed control such that some functional of the state is insensitive at the first order to the perturbations of the domain. Our first result consists of an approximate insensitization property on the semi-linear heat equation. It rests upon a linearization procedure together with the use of an appropriate fixed point theorem. For the linear case, an appropriate duality theory is developed, so that the problem can be seen as a consequence of well-known unique continuation theorems. Our second result is specific to the linear case. We show a property of exact insensitization for some families of deformation given by one or two parameters. Due to the nonlinearity of the intrinsic control problem, no duality theory is available, so that our proof relies on a geometrical approach and direct computations.

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