Mathematical studies of the solution of Burgers' equations by Adomian decomposition method

In this paper, a novel Adomian decomposition method (ADM) is developed for the solution of Burgers' equation. While high level of this method for differential equations are found in the literature, this work covers most of the necessary details required to apply ADM for partial differential equations. The present ADM has the capability to produce three different types of solutions, namely, explicit exact solution, analytic solution, and semi-analytic solution. In the best cases, when a closed-form solution exists, ADM is able to capture this exact solution, while most of the numerical methods can only provide an approximation solution. The proposed ADM is validated using different test cases dealing with inviscid and viscous Burgers' equations. Satisfactory results are obtained for all test cases, and, particularly, results reported in this paper agree well with those reported by other researchers.     

Comparative structural analysis of a histone-like protein from Spiroplasma melliferum in the crystalline state and in solution

A solution of a histone-like protein from Spiroplasma melliferum (HUSpm) was examined by small-angle X-ray scattering (SAXS). The experimental SAXS curve was compared with those calculated for the HUSpm structures from the PDB databank obtained by both X-ray diffraction analysis and nuclear magnetic resonance spectroscopy. The model of the HUSpm structure in solution, which best agrees with the experimental SAXS data, has a shorter distance between the centers of mass of the HUSpm monomers compared to the crystal structure, indicating that the HUSpm monomers can be located closer to each other in solution than in the crystalline state.     

Boussinesq方程行波解的存在性

利用平面动力系统方法的分支理论，研究了Boussinesq方程，通过对Boussinesq方程进行行波变换，得到了相应行波系统的首次积分和平衡点，给出了不同参数条件下的相图，证实了Boussinesq方程存在孤立波解和周期波解。     

一致分数阶非线性微分方程的精确解

同时考虑了Kudryashov方法和Khalil一致分数阶变换,构造了求解一致分数阶非线性微分方程精确解的新方法,并将其用于求解时间-空间一致分数阶Whitham-Boroer-Kaup方程,得到了Whitham-Boroer-Kaup方程新的精确解,验证了该方法的有效性和可行性.     

一类变号(k,n-k)共轭边值问题正解的存在性

通过构造一个特殊的锥,利用范数形式的锥拉伸锥压缩不动点定理,在允许非线性项变号无下界且没有任何单调性假设的条件下,得出了一类高阶(k,n-k)共轭两点边值问题方程组正解的存在性结论.     

Prediction of 195Pt NMR chemical shifts of dissolution products of H2[Pt(OH)6] in nitric acid solutions by DFT methods: how important are the counter‐ion effects?

¹⁹⁵Pt NMR chemical shifts of octahedral Pt(IV) complexes with general formula [Pt(NO₃)_n(OH)_{6 ? n}]^2?, [Pt(NO₃)_n(OH₂)_{6 ? n}]^{4 ? n} (n = 1–6), and [Pt(NO₃)_{6 ? n ? m}(OH)_m(OH₂)_n]^{?2 + n ? m} formed by dissolution of platinic acid, H₂[Pt(OH)₆], in aqueous nitric acid solutions are calculated employing density functional theory methods. Particularly, the gauge‐including atomic orbitals (GIAO)‐PBE0/segmented all‐electron relativistically contracted–zeroth‐order regular approximation (SARC–ZORA)(Pt) ∪ 6–31G(d,p)(E)/Polarizable Continuum Model computational protocol performs the best. Excellent second‐order polynomial plots of δ_calcd(¹⁹⁵Pt) versus δ_exptl(¹⁹⁵Pt) chemical shifts and δ_calcd(¹⁹⁵Pt) versus the natural atomic charge Q_Pt are obtained. Despite of neglecting relativistic and spin orbit effects the good agreement of the calculated δ ¹⁹⁵Pt chemical shifts with experimental values is probably because of the fact that the contribution of relativistic and spin orbit effects to computed σ^{iso 195}Pt magnetic shielding of Pt(IV) coordination compounds is effectively cancelled in the computed δ ¹⁹⁵Pt chemical shifts, because the relativistic corrections are expected to be similar in the complexes and the proper reference standard used. To probe the counter‐ion effects on the ¹⁹⁵Pt NMR chemical shifts of the anionic [Pt(NO₃)_n(OH)_{6 ? n}]^2? and cationic [Pt(NO₃)_n(OH₂)_{6 ? n}]^{4 ? n} (n = 0–3) complexes we calculated the ¹⁹⁵Pt NMR chemical shifts of the neutral (PyH)₂[Pt(NO₃)_n(OH)_{6 ? n}] (n = 1–6; PyH = pyridinium cation, C₅H₅NH⁺) and [Pt(NO₃)_n(H₂O)_{6 ? n}](NO₃)_{4 ? n} (n = 0–3) complexes. Counter‐anion effects are very important for the accurate prediction of the ¹⁹⁵Pt NMR chemical shifts of the cationic [Pt(NO₃)_n(OH₂)_{6 ? n}]^{4 ? n} complexes, while counter‐cation effects are less important for the anionic [Pt(NO₃)_n(OH)_{6 ? n}]^2? complexes. The simple computational protocol is easily implemented even by synthetic chemists in platinum coordination chemistry that dispose limited software availability, or locally existing routines and knowhow. Copyright © 2016 John Wiley & Sons, Ltd.     

Effect of chain architecture on the phase transition of star and cyclic poly(N‐isopropylacrylamide) in water

The heat‐induced phase transition of aqueous solutions of Poly(N‐isopropylacrylamide) (PNIPAM) in water is examined for a four‐arm PNIPAM star (s‐PNIPAM), a cyclic PNIPAM (c‐PNIPAM), and their linear counterparts (l‐PNIPAM) in the case of polymers (1.0 g L^?1) of 12,700 g mol^?1 < M_n < 14,700 g mol^?1. Investigations by turbidity, high‐sensitivity differential scanning calorimetry (HS‐DSC), and light scattering (LS) indicate that the polymer architecture has a strong effect on the cloud point (T_c: decrease for s‐PNIPAM; increase for c‐PNIPAM), the phase transition enthalpy change (ΔH decrease for s‐PNIPAM and c‐PNIPAM), and the hydrodynamic radius of the aggregates formed above T_c (R_H: c‐PNIPAM < s‐PNIPAM < l‐PNIPAM). The properties of s‐PNIPAM are compared with those of previously reported PNIPAM star polymers (3 to 52 arms). The overall observations are described in terms of the arm molecular weight and the local chain density in the vicinity of the core of the star, by analogy with the model developed for PNIPAM brushes on nanoparticles or planar surfaces. © 2016 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2016 , 54, 2059–2068.