An efficient collocation method for a Caputo two-point boundary value problem

peer-reviewedA two-point boundary value problem is considered on the interval , where the leading term in the differential operator is a Caputo fractional-order derivative of order with . The problem is reformulated as a Volterra integral equation of the second kind in terms of the quantity , where is the solution of the original problem. A collocation method that uses piecewise polynomials of arbitrary order is developed and analysed for this Volterra problem; then by postprocessing an approximate solution of is computed. Error bounds in the maximum norm are proved for and . Numerical results are presented to demonstrate the sharpness of these bounds.ACCEPTEDpeer-reviewe

Systems-wide analysis of manganese deficiency-induced changes in gene activity of Arabidopsis roots

Manganese (Mn) is pivotal for plant growth and development, but little information is available regarding the strategies that evolved to improve Mn acquisition and cellular homeostasis of Mn. Using an integrated RNA-based transcriptomic and high-throughput shotgun proteomics approach, we generated a comprehensive inventory of transcripts and proteins that showed altered abundance in response to Mn deficiency in roots of the model plant Arabidopsis. A suite of 22,385 transcripts was consistently detected in three RNA-seq runs; LC-MS/MS-based iTRAQ proteomics allowed the unambiguous determination of 11,606 proteins. While high concordance between mRNA and protein expression (R = 0.87) was observed for transcript/protein pairs in which both gene products accumulated differentially upon Mn deficiency, only approximately 10% of the total alterations in the abundance of proteins could be attributed to transcription, indicating a large impact of protein-level regulation. Differentially expressed genes spanned a wide range of biological functions, including the maturation, translation, and transport of mRNAs, as well as primary and secondary metabolic processes. Metabolic analysis by UPLC-qTOF-MS revealed that the steady-state levels of several major glucosinolates were significantly altered upon Mn deficiency in both roots and leaves, possibly as a compensation for increased pathogen susceptibility under conditions of Mn deficiency

Competition between uptake of ammonium and potassium in barley and Arabidopsis roots: molecular mechanisms and physiological consequences

Plants can use ammonium (NH4+) as the sole nitrogen source, but at high NH4+ concentrations in the root medium, particularly in combination with a low availability of K+, plants suffer from NH4+ toxicity. To understand the role of K+ transporters and non-selective cation channels in K+/NH4+ interactions better, growth, NH4+ and K+ accumulation and the specific fluxes of NH4+, K+, and H+ were examined in roots of barley (Hordeum vulgare L.) and Arabidopsis seedlings. Net fluxes of K+ and NH4+ were negatively correlated, as were their tissue concentrations, suggesting that there is direct competition during uptake. Pharmacological treatments with the K+ transport inhibitors tetraethyl ammonium (TEA+) and gadolinium (Gd3+) reduced NH4+ influx, and the addition of TEA+ alleviated the NH4+-induced depression of root growth in germinating Arabidopsis plants. Screening of a barley root cDNA library in a yeast mutant lacking all NH4+ and K+ uptake proteins through the deletion of MEP1–3 and TRK1 and TRK2 resulted in the cloning of the barley K+ transporter HvHKT2;1. Further analysis in yeast suggested that HvHKT2;1, AtAKT1, and AtHAK5 transported NH4+, and that K+ supplied at increasing concentrations competed with this NH4+ transport. On the other hand, uptake of K+ by AtHAK5, and to a lesser extent via HvHKT2;1 and AtAKT1, was inhibited by increasing concentrations of NH4+. Together, the results of this study show that plant K+ transporters and channels are able to transport NH4+. Unregulated NH4+ uptake via these transporters may contribute to NH4+ toxicity at low K+ levels, and may explain the alleviation of NH4+ toxicity by K+

A numerical method to solve higher-order fractional differential equations

In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order derivatives only. With this, we can rewrite FDEs in terms of a classical one and then apply any known technique. With some examples, we show the accuracy of the method

Detailed error analysis for a fractional adams method with graded meshes

The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-017-0419-5We consider a fractional Adams method for solving the nonlinear fractional differential equation \, ^{C}_{0}D^{\alpha}_{t} y(t) = f(t, y(t)), \, \alpha >0, equipped with the initial conditions $y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots, \lceil \alpha \rceil -1$. Here $\alpha$ may be an arbitrary positive number and $\lceil \alpha \rceil$ denotes the smallest integer no less than $\alpha$ and the differential operator is the Caputo derivative. Under the assumption \, ^{C}_{0}D^{\alpha}_{t} y \in C^{2}[0, T], Diethelm et al. \cite[Theorem 3.2]{dieforfre} introduced a fractional Adams method with the uniform meshes $t_{n}= T (n/N), n=0, 1, 2, \dots, N$ and proved that this method has the optimal convergence order uniformly in $t_{n}$, that is $O(N^{-2})$ if $\alpha > 1$ and $O(N^{-1-\alpha})$ if $\alpha \leq 1$. They also showed that if \, ^{C}_{0}D^{\alpha}_{t} y(t) \notin C^{2}[0, T], the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well known that for $y \in C^{m} [0, T]$ for some $m \in \mathbb{N}$ and $0 < \alpha 1$, we show that the optimal convergence order of this method can be recovered uniformly in $t_{n}$ even if \, ^{C}_{0}D^{\alpha}_{t} y behaves as $t^{\sigma}, 0< \sigma <1$. Numerical examples are given to show that the numerical results are consistent with the theoretical results