A numerical technique based on B-spline for a class of time-fractional diffusion equation

This paper presents an efficient numerical technique for solving a class of time-fractional diffusion equation. The time-fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B-spline basis function is employed for discretization of space variable. The unconditional stability of the fully-discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)-th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.     

An efficient numerical approach for solving a general class of nonlinear singular boundary value problems

Journal of Mathematical Chemistry - This paper is concerned with the development of a collocation method based on the Bessel polynomials for numerical solution of a general class of nonlinear...     

A Single‐Ion Magnet Based on a Heterometallic CoIII2DyIII Complex

We report a Co^III₂Dy^III complex, which shows single‐ion‐magnet behaviour. AC susceptibility data of this compound reveals the presence of slow relaxation of the magnetization in zero‐field below 15 K. The relaxation barrier is 88 K.     

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

FeIII48-Containing 96-Tungsto-16-Phosphate: Synthesis,Structure, Magnetism and Electrochemistry

The 48-Fe^III-containing 96-tungsto-16-phosphate, [Fe^III₄₈(OH)₇₆(H₂O)₁₆(HP₂W₁₂O₄₈)₈]³⁶⁻ ( Fe₄₈ ), has been synthesized and structurally characterized. This polyanion comprises eight equivalent {Fe^III₆P₂W₁₂} units that are linked in an end-on fashion forming a macrocyclic assembly that contains more iron centers than any other polyoxometalate (POM) known to date. The novel Fe₄₈ was synthesized by a simple one-pot reaction of an {Fe₂₂} coordination complex with the hexalacunary {P₂W₁₂} POM precursor in water. The title polyanion was characterized by single-crystal XRD, FTIR, TGA, magnetic and electrochemical studies.     

Carboxylate-Free Manganese(II) Phosphonate Assemblies: Synthesis, Structure, and Magnetism

The reaction of manganese(II) salts with organophosphonic acid [t-BuPO(3)H(2) or cyclopentyl phosphonic acid (C(5)H(9)PO(3)H(2))] in the presence of ancillary nitrogen ligands [1,10-phenanthroline (phen) or 2,6-bis(pyrazol-3-yl)pyridine (dpzpy)], afforded, depending on the stoichiometry of the reactants and the reaction conditions, dinuclear, trinuclear, and tetranuclear compounds, [Mn(2)(t-BuPO(3)H)(4)(phen)(2)]·2DMF (1), [Mn(3)(C(5)H(9)PO(3))(2)(phen)(6)](ClO(4))(2)·7CH(3)OH (2), [Mn(3)(t-BuPO(3))(2)(dpzpy)(3)](ClO(4))(2)·H(2)O (3), [Mn(4)(t-BuPO(3))(2)(t-BuPO(3)H)(2)(phen)(6)(H(2)O)(2)](ClO(4))(2) (4), and [Mn(4)(C(5)H(9)PO(3))(2)(phen)(8)(H(2)O)(2)](ClO(4))(4) (5). Magnetic studies on 1, 2, and 4 reveal that the phosphonate bridges mediate weak antiferromagnetic interactions between the Mn(II) ions have also been carried out.     

NiII36-Containing 54-Tungsto-6-Silicate: Synthesis,Structure, Magnetic and Electrochemical Studies

The 36-Ni^II-containing 54-tungsto-6-silicate, [Ni₃₆(OH)₁₈(H₂O)₃₆(SiW₉O₃₄)₆]⁶⁻ ( Ni₃₆ ) was synthesized by a simple one-pot reaction of the Ni₂-pivalate complex [Ni₂(μ-OH₂)(O₂CCMe₃)₄(HO₂CCMe₃)₄] with the trilacunary [SiW₉O₃₄]¹⁰⁻ polyanion precursor in water and structurally characterized by a multitude of physicochemical techniques including single-crystal XRD, FTIR, TGA, elemental analysis, magnetic and electrochemical studies. Polyanion Ni₃₆ comprises six equivalent {Ni^II₆SiW₉} units which are linked by Ni−O−W bridges forming a macrocyclic assembly. Magnetic studies demonstrate that the {Ni₆} building blocks in Ni₃₆ remain magnetically intact while forming a hexagonal ring with antiferromagnetic exchange interactions between adjacent {Ni₆} units. Electrochemical studies indicate that the first reduction is reversible and associated with the W^VI/V couple, whereas the second reduction is irreversible attributed to the Ni^II/0 couple.     

Effects of radiation therapy on the human normal brain (white matter) visualized by MR imaging

Sequential MR imaging could provide information related to the pathological changes due to the application of external cytotoxic agents such as radiotherapy on the central nervous system. This paper describes the results of our attempt to demonstrate short-term changes associated with normal brain during and immediately following radiotherapy when the whole brain is irradiated for malignant conditions. No observable changes were found in the normal brain parenchyma in any of the patients (n = 8) in T₁-, T₂-, and proton-density-weighted images in the sequential scans in the first and second weeks and immediately following radiotherapy. Also, no changes were observed in the normal brain appearance at 2 mo (n = 6), up to 6 mo (n = 1), and up to 15 mo (n = 1) after completion of radiotherapy.