In this paper, we propose an efficient B-spline finite element method for a class of fourth order nonlinear differential equations with variable coefficient. For the temporal discretization, we choose the Crank–Nicolson scheme. Boundedness and error estimates are rigorously derived for both semi-discrete and fully discrete schemes. A numerical experiment confirms our theoretical analysis.

In the original publication of this article [1] the name of the second author is incorrect. The correct name of the second author is Jiawei Tan rather than John Jiawei Tan. The error in this Correction has been updated in the original article.

Background: Lung adenocarcinoma (LUAD) is the predominant type of lung cancer, and metastasis is a major cause of poor prognosis and death. Metabolic activation is a crucial factor driving tumor metastasis; however, the metabolic heterogeneity at the single-cell level presents significant challenges in targeting metabolism-related genes for treatment. This study aimed to decode the metabolic drivers in tumor metastasis progression to optimize LUAD prognosis prediction and screen specific targeted drugs. Methods: In this study, we determined that the metabolic activation of tumor and immune cells in the microenvironment is significantly altered during LUAD metastasis. Simultaneously, we identify pivotal metabolic driver genes (MDGs) based on single-cell RNA-sequencing (scRNA-seq) data, which could serve as targets for targeted therapy. We then constructed a novel prognostic risk model based on MDGs and validated its excellent predictive performance in independent datasets. Using the non-negative matrix factorization (NMF) algorithm, we classify LUAD molecular subtypes into three clusters according to MDGs and evaluate their association with prognosis and clinical characteristics. Results: We screened a panel of 307 drugs targeting MDGs and confirmed the efficacy of cholic acid, as a representative compound from the screened panel, in inhibiting the migration of LUAD cells. Conclusions: Our research provides potential targets and candidate drug for targeting metabolic-related genes in metastatic LUAD treatment.