Numerical Solution of the One-Dimensional Heat Equation by Using Chebyshev Wavelets Method

Abstract

In this paper, we develop an efficient Chebyshev wavelet method for well-known one-dimensional heat equation. In the proposed method we applied operational matrices of integration to get numerical solution of the onedimensional heat equation with Dirichlet boundary conditions. The power of this manageable method is confirmed. Moreover the use of Chebyshev wavelet is found to be accurate, simple and fast.

Generalized Lucas Tau Method for the Numerical Treatment of the One and Two-Dimensional Partial Differential Heat Equation

“…In Table 1, the maximum absolute errors (MAEs) obtained 2, we compare the errors resulted from the application of the GLTM for the case corresponding to M = 4 and a = b = 1 with the best errors resulted from the application of the methods developed in [9,12]. It is noticed from the obtained results in Table 2 that GLTM is more accurate than the two methods that developed in [9,12]. Figure 1 displays the exact solution.…”

“…Many authors have researched theoretically and numerically the heat equations. For example, the authors in [9] obtained a numerical solution of the one-dimensional heat equation by using the Chebyshev wavelets method. In [10], the authors treated the same equation using a high-order compact boundary value method.…”

Numerical Solution for the Heat Equation Using Forward Time Centered Space Difference Method

“…Hooshmandasl M.R., et al have applied operational matrices of integration to get numerical solution of the one dimensional heat equation with Dirichlet boundary conditions. They have concluded that the use of Chebyshev wavelets is found to be accurate, simple and fast [7]. Tahrich N.A.Shahid et al have investigated that modified difference equation in specific problems are more convenient for discussing the solution behavior including physical interpretation of accuracy, stability and consistency [8].…”

Construction and investigation of new numerical algorithms for the heat equation : Part 1

“…As we want to construct explicit methods, we exclude the implicit formula at the first stage, thus 6 possibilities remain. However, at the second stage, the new values of the neighbours are already calculated at the first stage, therefore the UPFD scheme (6) coincides with the implicit scheme(5). We remind that if the old values n 1 i u − and n 1 i u + were used, the original UPFD method would be obtained.…”

“…In fact even the numerical solution of Eq. (1) is still investigated [4], [5], and, on the other hand, even the nonlinear PDEs have some analytical solutions [6]- [10]. However, these latter ones are always valid in special circumstances only, e.g.…”