Abstract
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.
Highlights
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The Brownian motion problem in statistics is modeled by heat equation via the Fokker–Planck equation
Azzam and Kreyszig studied the smoothness of solutions of parabolic equations for the Dirichlet problem in [35] and for the mixed boundary value problem in [36]
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. A study on fractional diffusion equation-based image denoising model using Crank–Nicholson and Grünwald Letnikov difference schemes (CN–GL) have been given in Abirami et al [17]. Another example is the most recent investigation by Buranay and Nouman [18] in which computation of the solution to heat equation. Numerical methods using implicit schemes with hexagonal grids approximating the derivatives of the solution of (2) on a rectangle has been given. The motivation of the contributions of this research is the need of highly accurate and time-efficient numerical algorithms that compute the derivatives of the solution u( x1 , x2 , t) to the heat Equation (2). Concus et al [30], Axelsson [31]) is applied for the preconditioning
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