We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initial-boundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders, non-trivial variable coefficients, and initialboundary conditions. The method uses an orthogonal basis that we refer to as Jacobi fractional polynomials, which are obtained from an appropriate change of variable in weighted classical Jacobi polynomials. New algorithms for building the matrices used to represent fractional integration operators are presented and compared. Even though these algorithms are unstable and require the use of high-precision computations, the spectral method nonetheless yields well-conditioned linear systems and is therefore stable and efficient. For time-fractional heat and wave equations, we show that our method (which is not sparse but uses an orthogonal basis) outperforms a sparse spectral method (which uses a basis that is not orthogonal) due to its superior stability.

Camellia tetracocca, a kind of tea with local popularity, is unique to southwest China, where it has an important natural heritage and cultural heritage. However, the tea plant and its sprout are frequently attacked on a large-scale by Empoasca onukii. The metabolic mechanisms of the unique plant for defending against these pest insects are unclear. Therefore, we used untargeted gas chromatography–mass spectrometry (GC-MS) and high performance liquid chromatography–mass spectrometry (LC-MS) to compare the metabolite profiles between E. onukii-attacked leaves and healthy leaves. Using GC-MS, 56 metabolites (24 up-regulated, 32 down-regulated) were preliminarily identified. Additionally, 576 metabolites (280 up-regulated, 287 down-regulated) were rudimentarily identified with LC-MS. Differentially abundant metabolites were mainly enriched in the biosynthesis of specialized metabolites. Fourteen accumulated specialized metabolites are related to insect resistance. Mainly, momordicin I and arabidopside B are reportedly involved in the resistance to the insect. Therefore, we conjectured that the accumulation of momordicin I and arabidopside B is involved in the C. tetracocca’s resistance to E. onukii. Our results indicate that these specialized metabolites may be served as candidate biocontrol agents against the pest of E. onukii of C. tetracocca located in the State-owned Pubai Forest Farm.