Phytochemical profile and some biological activities of three Centaurea species from Turkey

Purpose: To characterise the phytochemical profile of whole plants of Centaurea balsamita, C. depressa and C. lycopifolia with LC-ESI-MS/MS, and as well as their antioxidant, anticholinesterase and antimicrobial activities. Methods: Organic and aqueous extracts of the three Centaurea species were evaluated for DPPH free radical, ABTS cation radical scavenging and cupric reducing antioxidant capacity (CUPRAC). Acetyland butyryl-cholinesterase enzyme inhibition abilities of the extracts using petroleum ether, acetone, methanol and water were studied to determine anticholinesterase activity, while antimicrobial activity was determined by disc diffusion method using appropriate antimicrobial standards and organisms. The phytochemical components of the methanol extracts were assessed by LC-MS/MS. Results: The methanol extract of C. balsamita exhibited much higher DPPH free and ABTS cation radicals scavenging activities (with IC50 of 62.65 ± 0.97 and 24.21 ± 0.70 mg/ml, respectively) than the other extracts. The petroleum ether extracts of the plant species exhibited moderate inhibitory activity against butyrylcholinesterase enzymes while the acetone extract of C. balsamita showed good antifungal activity against Candida albicans. Quinic acid (17513 ± 813 μg/g, 63874 ± 3066 μg/g and 108234 ± 5195 μg/g) was the major compound found in the methanol extracts of C. balsamita, C. depressa and C. Lycopifolia, respectively. Conclusion: These results indicate quinic acid is the major compound in the three plant species and that Centaurea balsamita has significant antioxidant, anticholinesterase and antimicrobial properties. Further studies to identify the compounds in the extracts responsible for the activities are required.

An advanced meshless method for time fractional diffusion equation

Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

A preconditioned Lanczos method for space-fractional reaction-diffusion equations on two-dimensional unstructured meshes

We consider a two-dimensional space-fractional reaction diffusion equation with a fractional Laplacian operator and homogeneous Neumann boundary conditions. The finite volume method is used with the matrix transfer technique of Ilić et al. (2006) to discretise in space, yielding a system of equations that requires the action of a matrix function to solve at each timestep. Rather than form this matrix function explicitly, we use Krylov subspace techniques to approximate the action of this matrix function. Specifically, we apply the Lanczos method, after a suitable transformation of the problem to recover symmetry. To improve the convergence of this method, we utilise a preconditioner that deflates the smallest eigenvalues from the spectrum. We demonstrate the efficiency of our approach for a fractional Fisher’s equation on the unit disk.

A finite volume method with linearisation in time for the solution of advection-reaction-diffusion systems

The numerical solution in one space dimension of advection--reaction--diffusion systems with nonlinear source terms may invoke a high computational cost when the presently available methods are used. Numerous examples of finite volume schemes with high order spatial discretisations together with various techniques for the approximation of the advection term can be found in the literature. Almost all such techniques result in a nonlinear system of equations as a consequence of the finite volume discretisation especially when there are nonlinear source terms in the associated partial differential equation models. This work introduces a new technique that avoids having such nonlinear systems of equations generated by the spatial discretisation process when nonlinear source terms in the model equations can be expanded in positive powers of the dependent function of interest. The basis of this method is a new linearisation technique for the temporal integration of the nonlinear source terms as a supplementation of a more typical finite volume method. The resulting linear system of equations is shown to be both accurate and significantly faster than methods that necessitate the use of solvers for nonlinear system of equations.