Abstract
This article mainly studies a collocation spectral method for two-dimensional (2D) Sobolev equations. To this end, a collocation spectral model based on the Chebyshev polynomials for the 2D Sobolev equations is first established. And then, the existence, uniqueness, stability, and convergence of the collocation spectral numerical solutions are discussed. Finally, some numerical experiments are provided to verify the corrections of theoretical results. This implies that the collocation spectral model is very effective for solving the 2D Sobolev equations.
Highlights
1 Introduction Because any bounded closed domain in R2 can be approximately filled with several rectangles [ai, bi] × [ci, di] (i = 1, 2, . . . , I), for convenience and without losing universality, let us just assume that = [a, b] × [c, d] ⊂ R2, whose boundary is denoted by ∂, consider the following two-dimensional (2D) Sobolev equations:
The Sobolev equations hold very significant physical background so that they have become a class of important evolution partial differential equations (PDEs) and have been successfully used to many numerical simulations in mathematical and physical problems, such as the exchange in different media and the moisture migration in soil
The Sobolev equations can be used to depict the porous phenomena saturated into rocks with cracks
Summary
3, we build the CSM scheme for the 2D Sobolev equations and analyze the existence, uniqueness, stability, and convergence of the CSM solutions. For Problem 3, we have the following result of the existence, uniqueness, and stability of the generalized solution. Theorem 4 If f ∈ L2(0, T; L2ω( )) and u0 ∈ Hω1 ( ), there exists a unique generalized solution for the variational formulation (8) satisfying the following stability: u 1,ω ≤ cu0 1,ω + f L2(L2ω) , (9)
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