Numerical Treatments for Crossover Cancer Model of Hybrid Variable-Order Fractional Derivatives

In this paper,two crossover hybrid variable-order derivatives of the cancer model are developed.Grünwald-Letnikov approximation is used to approximate the hybrid fractional and variable-order fractional operators.The existence,uniqueness,and stability of the proposed model are discussed.Adams Bashfourth’s fifth-step method with a hybrid variable-order fractional operator is developed to study the proposed models.Comparative studies with generalized fifth-order Runge-Kutta method are given.Numerical examples and comparative studies to verify the applicability of the used methods and to demonstrate the simplicity of these approximations are presented.We have showcased the efficiency of the proposed method and garnered robust empirical support for our theoretical findings.

A NEW METHOD FOR THE PREPARATION OF PHENOLS AND ITS DERIVATIVES

a-Oxo ketene dithioacetals 2 via 1,2-nucleophilie addition by methallyl magnesius chloride afforded corresponding alcohols (3). Treated with water or methanol and catalyzed by Lewis acid, the alcohols 3 were converted regiospecifical ly to substituted phenols 5' or related phenol methyl ethers 5 respectively. This reaction is a novel approach to the synthesis of phenols and their derivatives starting from non-aromatic precursors.

Determination of Fenofibrate and the Degradation Product Using Simultaneous UV-Derivative Spectrometric Method and HPLC

Two new selective, precise, and accurate methods were developed for the determination of fenofibrate in the presence of its basic degradation product. In the first method fenofibrate was determined using an algorithm bivariate calibration derivative method, in which an optimum pair of wavelengths was chosen for the determination of different binary mixtures. In the second method (HPLC), separation was achieved on RESTEK Pinnacle II phenyl column (5 μm, 250 × 4.6 mm) and Pinnacle II phenyl (5 μm, 10 × 4 mm) guard cartridge using a mobile phase consisting of methanol –0.1% phosphoric acid (60:40, v/v) at a flow rate 2 mL●min–1, and the column oven temperature was set at 50°C. The UV detector was time programmed at 302 nm and 289 nm for the internal standard (I.S.) and fenofibrate, respectively. The proposed methods were successfully applied for the determination of fenofibrate and its degradation product in the laboratory-prepared mixture and in pharmaceutical formulation. The assay results obtained using the bivariate method were statistically compared to those of the HPLC method and good agreement was observed.

DETERMINATION OF STABILITY CONSTANTS OF THE COORDINATION COMPOUNDS OF Au(Ⅲ)-2-MERCAPTOPYRIDINE-1-OXIDE AND METHYL DERIVATIVES——METHOD OF DUAL-WAVELENGTH CORRESPONDING SOLUTIONS

In this paper,we report the deductive formula used for the method of dual-wavelength corresponding solutions under condition of having ligand interference and the stability constants of three new coordination compounds [AuL_2]^+determined with this method.The stability of the three compounds,the necessity of controlling pH in experimental systems and the advantage of this method are discussed in detail.

Design Method of Cultural Derivative Products of Museums from the Perspective of Regional Culture

To perfect innovative design of cultural derivative products of museums,promote the dissemination of museum collection culture,and enhance product connotation and cultural value,based on the research on regional culture,the relationship between regional culture and cultural derivative products of museums is analyzed,and design method of cultural derivative products of museums is summarized.

Constructive Theory of Designing Optimal Eighth-Order Derivative-Free Methods for Solving Nonlinear Equations

This paper stresses the theoretical nature of constructing the optimal derivative-free iterations. We give necessary and sufficient conditions for derivative-free three-point iterations with the eighth-order of convergence. We also establish the connection of derivative-free and derivative presence three-point iterations. The use of the sufficient convergence conditions allows us to design wide class of optimal derivative-free iterations. The proposed family of iterations includes not only existing methods but also new methods with a higher order of convergence.

APPLICATION OF PARAMETRIC DERIVATION METHOD TO THE CALCULATION OF PEIERLS ENERGY AND PEIERLS STRESS IN LATTICE THEORY

Applying the parametric derivation method, Peierls energy and Peierls stress are calculated with a non-sinusoidal force law in the lattice theory, while the results obtained by the power-series expansion according to sinusoidal law can be deduced as a limiting case of non- sinusoidal law. The simplified expressions of Peierls energy and Peierls stress are obtained for the limit of wide and narrow. Peierls energy and Peierls stress decrease monotonically with the factor of modification of force law. Present results can be used expediently for prediction of the correct order of magnitude of Peierls stress for materials.

A FAMILY OF MODAL METHODS FOR COMPUTING EIGENVECTOR DERIVATIVES WITH REPEATED ROOTS

A family of modal methods for computing eigenvector derivatives with repeated roots are directly derived from the constraint generalized inverse technique which is originally formulated by Wang and Hu. Extensions are made to Akgun's method to allow treatment of eigensensitivity with repeated roots for general nondefective systems, and Bernard and Bronowicki's modal expansion approach is expanded to a family of modal methods.

A Novel BEM for Modeling and Simulation of 3T Nonlinear Generalized Anisotropic Micropolar-Thermoelasticity Theory with Memory Dependent Derivative

The main aim of this paper is to propose a new memory dependent derivative(MDD)theory which called threetemperature nonlinear generalized anisotropic micropolar-thermoelasticity.The system of governing equations of the problems associated with the proposed theory is extremely difficult or impossible to solve analytically due to nonlinearity,MDD diffusion,multi-variable nature,multi-stage processing and anisotropic properties of the considered material.Therefore,we propose a novel boundary element method(BEM)formulation for modeling and simulation of such system.The computational performance of the proposed technique has been investigated.The numerical results illustrate the effects of time delays and kernel functions on the nonlinear three-temperature and nonlinear displacement components.The numerical results also demonstrate the validity,efficiency and accuracy of the proposed methodology.The findings and solutions of this study contribute to the further development of industrial applications and devices typically include micropolar-thermoelastic materials.

Synthesis and Anti-Cancer Activities of Resveratrol Derivatives

A novel series of resveratrol derivatives were synthesized according to Wittig-Horner reaction with 3,5-dihydroxybenzyl alcohol or 3,5-dimethoxybenzyl alcohol or 4-hydroxybenzyl alcohol as raw material and the inhibitory activities on breast carcinoma (MDA-MB-231) and gastric carcinoma cell lines (SGC-7901) in vitro were evaluated by the standard methyl thiazole tetrazolium (MTT) method. The result of biological test shows that some of resveratrol derivatives possess stronger anti-cancer activities than 5-FU. Compound 5c shows the strongest activity against breast carcinoma (MDA-MB-231) and gastric carcinoma cell lines (SGC-7901) with IC50 value of 50.19 ± 1.02 μM, 122.68.27 ± 2.04 μM, compared to that IC50 value of 5-FU is 98.59±3.61 μM,156.74±6.16 μM, respectively.

The Efficient Finite Element Methods for Time-Fractional Oldroyd-B Fluid Model Involving Two Caputo Derivatives

In this paper,we consider the numerical schemes for a timefractionalOldroyd-B fluidmodel involving the Caputo derivative.We propose two efficient finite element methods by applying the convolution quadrature in time generated by the backward Euler and the second-order backward difference methods.Error estimates in terms of data regularity are established for both the semidiscrete and fully discrete schemes.Numerical examples for two-dimensional problems further confirmthe robustness of the schemes with first-and second-order accurate in time.

An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid

In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes＇ first problem for a heated generalized second grade fluid.

Local Discontinuous Galerkin Method for the Time-Fractional KdV Equation with the Caputo-Fabrizio Fractional Derivative

This paper studies the time-fractional Korteweg-de Vries (KdV) equations with Caputo-Fabrizio fractional derivatives. The scheme is presented by using a finite difference method in temporal variable and a local discontinuous Galerkin method (LDG) in space. Stability and convergence are demonstrated by a specific choice of numerical fluxes. Finally, the efficiency and accuracy of the scheme are verified by numerical experiments.

A NEW GENERAL OPTIMAL PRINCIPLE OF DESIGNING EXPLICIT FINITE DIFFERENCE METHOD FOR VALUING DERIVATIVE SECURITIES

A new general optimal principle of designing explicit finite difference method was obtained. Several applied cases were put forward to explain the uses of the principle. The validity of the principal was tested by a numeric example.

ON FICTITIOUS DOMAIN METHOD FOR THE NUMERICAL SOLUTION TO HEAT CONDUCTION EQUATION WITH DERIVATIVE BOUNDARY CONDITIONS

This paper investigates the stability and convergence of some knowndifference schemes for the numerical solution to heat conduction equation withderivative boundary conditions by the fictitious domain method.The discrete vari-ables at the false mesh points are firstly eliminated from the difference schemes andthe local truncation errors are then analyzed in detail.The stability and convergenceof the schemes are proved by energy method.An improvement is proposed to obtainbetter schemes over the original ones.Several numerical examples and comparisonswith other schemes are presented.

Study on State of Terbium Ion in Liposome by One order Derivative Fluorescence Quenching Method

The state of Tb3+ is investigated in liposome. When the concentration of PC is below CMC (critical micell concentration), most of Tb3+ is associated with PC, the binding constant is about 3.35×103 L/mol. When the concentration of PC is beyond CMC, most of Tb3+ is dimerized, the dimerization constant is about 3.92×104L/mol. In PC?CH?H2O system, the binding constant of Tb3+?CH complex 2.93×104L/mol is obtained.

Rederivation of AB Scattering Cross Section by the Method of Path Integration

Taking advantage of the calculation based on the original fundamental ideas of Aharonov and Bohm(AB)on the one hand,and making some necessary revisions on the other hand,this paper rederives the famous AB scattering cross section from the known propagator by the path integral method.

Film-Forming Properties of Fullerene Derivatives in Electrospray Deposition Method

Thin films of three types of fullerene derivatives were prepared through the electrospray deposition (ESD) method. The optimized conditions for the fabrication of the thin films were investigated for different types of fullerene derivatives: [6,6]-phenyl-C61-butyric acid methyl ester, [6,6]-phenyl-C71-butyric acid methyl ester, and indene-C60-monoadduct. The spray diameter during the ESD process was observed as a function of the supply rate achieved by changing the applied voltage. In all cases, the spray diameter increased with increasing applied voltage, reaching the maximum diameter (Dmax) in the voltage range 4 to 6 kV. It was clear that Dmax was influenced by the dipole moments of the fullerene derivatives (as calculated by density functional theory methods). Scanning electron microscopy observation of the?fabricated thin films showed that imbricated structures were formed through the stacking of the fullerene-derivative sheets. Atomic force microscopy images revealed that the density of the imbricated structure was dependent on the spray diameter during the ESD process, and the root-mean-square roughness of the film surface decreased with increasing applied voltage. These findings suggest that the ESD method will be effective for the preparation of fullerene-derivative thin films for the production of organic devices.

Formal Derivation of the Combinatorics Problems with PAR Method

Partition-and-Recur (PAR) method is a simple and useful formal method. It can be used to design and testify algo-rithmic programs. In this paper, we propose that PAR method is an effective formal method on solving combinatorics problems. Furthermore, we formally derive combinatorics problems by PAR method, which cannot only simplify the process of algorithmic program's designing, but also improve its automatization, standardization and correctness. We develop algorithms for two typical combinatorics problems, the number of string scheme and the number of error per-mutation scheme. Lastly, we obtain accurate C++ programs which are transformed by automatic transforming system of PAR platform.