Abstract
We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles alpha _{j}pi , j=1,2,ldots,M, where alpha _{j}in { frac{1}{2},frac{1}{3},frac{2}{3} } . To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order O ( h^{2}+tau ^{2} ) and O ( h^{4}+tau ) respectively, where h and frac{sqrt{3}}{2}h are the step sizes in space variables x_{1} and x_{2} respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.
Highlights
The use of differential equations in modeling of physical phenomenon is inevitable
To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points
For the numerical solution of the Boundary Value Problem (BVP) we propose the following difference problem
Summary
The use of differential equations in modeling of physical phenomenon is inevitable. Some modeling examples include but are not limited to the modeling of musical instruments such as string and wind instruments using digital waveguides, addressed by Smith [1]. By Iqbal et al [3] an Buranay and Arshad Advances in Difference Equations (2020) 2020:309 unconditionally stable and structure preserving computational technique for fractional order Schnakenberg model has been given. [18] investigated a fourth order block-hexagonal grid approximation for the solution of Laplace’s equation on special polygons with singularities. 3, a two layer implicit difference scheme with 14 points on hexagonal grids is proposed and it is proved that this scheme is unconditionally stable and the solution of the constructed. 4, we give a two layer implicit unconditionally stable scheme with 14 points on hexagonal grids and showed that the solution of the constructed Difference.
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