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 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 GAUGED LIFSHITZ MODEL WITH CHERN–SIMONS TERM

2013 ◽  
Vol 28 (09) ◽  
pp. 1350025 ◽  
Author(s):  
GUSTAVO S. LOZANO ◽  
FIDEL A. SCHAPOSNIK ◽  
GIANNI TALLARITA
Keyword(s):  
Fourth Order ◽  
Order Term ◽  
Equations Of Motion ◽  
Nontrivial Solutions ◽  
Oscillatory Solutions ◽  
Modulated Phases ◽  
Fourth Order Term ◽  
Spatial Derivatives ◽  
Chern Simons ◽  
Derivatives Of

We present a gauged Lifshitz Lagrangian including second- and fourth-order spatial derivatives of the scalar field and a Chern–Simons term, and study nontrivial solutions of the classical equations of motion. While the coefficient β of the fourth-order term should be positive in order to guarantee positivity of the energy, the coefficient α of the quadratic one need not be. We investigate the parameter domains and find significant differences in the field behaviors. Apart from the usual vortex field behavior of the ordinary relativistic Chern–Simons–Higgs model, we find in certain parameter domains oscillatory solutions reminiscent of the modulated phases of Lifshitz systems.

scholarly journals Successive Iteration of Positive Solutions for Fourth-Order Two-Point Boundary Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Yongping Sun ◽  
Xiaoping Zhang ◽  
Min Zhao
Keyword(s):  
Positive Solutions ◽  
Fourth Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Point Boundary ◽  
Interesting Point ◽  
Zero Function ◽  
Existence Of Positive Solutions ◽  
Derivatives Of ◽  
Cubic Function

We are concerned with a fourth-order two-point boundary value problem. We prove the existence of positive solutions and establish iterative schemes for approximating the solutions. The interesting point of our method is that the nonlinear term is involved with all lower-order derivatives of unknown function, and the iterative scheme starts off with a known cubic function or the zero function. Finally we give two examples to verify the effectiveness of the main results.

scholarly journals Four Point Implicit Methods for the Second Derivatives of the Solution of First Type Boundary Value Problem for One Dimensional Heat Equation

2018 ◽  
Vol 22 ◽  
pp. 01011
Author(s):  
Suzan Cival Buranay ◽  
Lawrence Adedayo Farinola
Keyword(s):  
Boundary Value Problem ◽  
Heat Equation ◽  
Boundary Value ◽  
Step Size ◽  
Holder Space ◽  
One Dimensional ◽  
Second Derivatives ◽  
Boundary Functions ◽  
Type Boundary ◽  
Derivatives Of

We construct four-point implicit difference boundary value problem for the first derivative of the solution u(x,t) of the first type boundary value problem for one dimensional heat equation with respect to the time variable t. Also, for the second derivatives of u(x,t) special four-point implicit difference boundary value problems are proposed. It is assumed that the initial function belongs to the Hölder space C8+α,0 < α < 1, the heat source function given in the heat equation is from the Hölder space [see formula in PDF], the boundary functions are from [see formula in PDF], and between the initial and the boundary functions the conjugation conditions of orders q = 0,1,2,3,4 are satisfied. We prove that the solution of the proposed difference schemes converge uniformly on the grids of the order O(h2+τ) (second order accurate in the spatial variable x and first order accurate in time t) where, h is the step size in x and τ is the step size in time. Theoretical results are justified by numerical examples.

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