scholarly journals Implicit method of high accuracy on hexagonal grids for approximating the solution to heat equation on rectangle

2021 ◽  
Author(s):  
Suzan C. Buranay ◽  
Nouman Arshad
Keyword(s):  
Heat Equation ◽  
High Accuracy ◽  
Implicit Method ◽  
Hexagonal Grids

scholarly journals Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

2021 ◽  
Vol 5 (1) ◽  
pp. 19
Author(s):  
Suzan Cival Buranay ◽  
Ahmed Hersi Matan ◽  
Nouman Arshad
Keyword(s):  
Heat Equation ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Implicit Method ◽  
Two Stage ◽  
Unconditionally Stable ◽  
Spatial Variables ◽  
Hexagonal Grids ◽  
Derivatives Of ◽  
Special Difference

The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

scholarly journals A Meshless Method to Determine a Source Term in Heat Equation with Radial Basis Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Baiyu Wang ◽  
Anping Liao
Keyword(s):  
Inverse Problem ◽  
Heat Equation ◽  
Radial Basis Functions ◽  
Meshless Method ◽  
Source Term ◽  
Initial Boundary ◽  
High Accuracy ◽  
Basis Functions ◽  
Unknown Source ◽  
Radial Basis

This paper considers a numerical method based on the radial basis functions for the inverse problem of heat equation; the inverse problem is determining an unknown source term subject to the overdetermination along with the usual initial boundary conditions, and the unknown source term is only time-dependent. The radial basis functions method is a meshless method with high accuracy for the inverse problem. Some numerical experiments using this method are presented and discussed.

Laplace inversion for the solution of an abstract heat equation without the forward transform of the source term

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Shu-Lin Wu
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Elliptic Equations ◽  
Source Term ◽  
Finite Interval ◽  
Quadrature Rule ◽  
High Accuracy ◽  
Laplace Inversion ◽  
Finite Set ◽  
Complex Coefficients

AbstractWe consider the discretization in time of inhomogeneous parabolic equations, using the technique of Laplace inversion along a contour located in the complex left half-plane which, after transformation to a finite interval, is then evaluated to high accuracy by a contour quadrature rule. This reduces the problem to a finite set of elliptic equations with complex coefficients, which may be solved in parallel. A serious problem is how to treat the source term

A New 1,000 kV Electron Microscope

1967 ◽  
Vol 25 ◽  
pp. 260-261
Author(s):  
M. Nishigaki ◽  
S. Katagiri ◽  
H. Kimura ◽  
B. Tadano
Keyword(s):  
Electron Microscope ◽  
High Voltage ◽  
Basic Research ◽  
Chromatic Aberration ◽  
High Accuracy ◽  
Biological Specimen ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
High Voltage Electron

The high voltage electron microscope has many advantageous features in comparison with the ordinary electron microscope. They are a higher penetrating efficiency of the electron, low chromatic aberration, high accuracy of the selected area diffraction and so on. Thus, the high voltage electron microscope becomes an indispensable instrument for the metallurgical, polymer and biological specimen studies. The application of the instrument involves today not only basic research but routine survey in the various fields. Particularly for the latter purpose, the performance, maintenance and reliability of the microscope should be same as those of commercial ones. The authors completed a 500 kV electron microscope in 1964 and a 1,000 kV one in 1966 taking these points into consideration. The construction of our 1,000 kV electron microscope is described below.

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