Legendre spectral projection methods for Fredholm integral equations of first kind

Abstract

In this paper, we discuss the Legendre spectral projection method for solving Fredholm integral equations of the first kind using Tikhonov regularization. First, we discuss the convergence analysis under an a priori parameter strategy for the Tikhonov regularization using Legendre polynomial basis functions, and we obtain the optimal convergence rates in the uniform norm. Next, we discuss Arcangeli’s discrepancy principle to find a suitable regularization parameter and obtain the optimal order of convergence in uniform norm. We present numerical examples to illustrate the theoretical results.

Numerical solution of the first kind Fredholm integral equations using a regularized projection method.

“…By consolidating the aforementioned estimates, we derive the expression (16), which completes the proof.…”

“…According to Remark 3, the parameter selection strategy presented in (30) is distinguished by the fact that it is not specific to any particular numerical method, such as the methods investigated in previous works ( see, [14,17]). This characteristic contributes to its superiority in terms of computational efficiency compared to many other methods, resulting in time savings during implementation.…”

“…In recent years, several researchers have made significant advancements in the field of solving integral equations of the first kind. These advancements encompass a range of regularization techniques, such as collocation methods [5][6][7], multilevel methods [8,9], multiscale methods [10][11][12], projection methods [13][14][15][16][17], and other approaches that have been explored in previous works such as [18,19].…”

“…The proposed approach combines Tikhonov regularization and the projection method to achieve rapid convergence of the discrete equation solutions. It is distinct from the projection methods introduced in [13][14][15][16][17], which primarily utilize projection operators to approximate the compact operator A into discrete operator with a range in R n . Additionally, they employ Tikhonov regularization to address the ill-posedness of the discrete equation and obtain a stable approximate solution.…”