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High-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant preconditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(N log N ) operation cost and O(N ) memory requirement. Illustrative numerical examples are also presented.

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.Keywords Two-dimensional fractional differential equation · High order difference scheme · Discrete energy method · Preconditioned GMRES method Mathematics Subject Classification (2010) 35R11 · 65M06 · 65M12 · 65M15

For the first time, we have studied the crystallization behavior of FeCl 3 in an aqueous alkaline pseudocapacitor system, where FeCl 3 can be effectively transformed into Fe 2 O 3 •H 2 O colloids by an in situ crystallization process in KOH electrolyte under an electric field. In alkaline aqueous solution, the Fe 3+ cations firstly react with OH − and crystallize into goethite Fe 2 O 3 •H 2 O colloids. During the electrochemical tests, the crystallization of Fe 3+ was interrupted by an external electric field due to a Faradaic redox reaction, resulting in fine and highly electroactive Fe 2 O 3 •H 2 O colloids. For the currently designed FeCl 3 pseudocapacitor system, a very high capacitance of 977 F g −1 at the current density of 3 A g −1 and a high energy density of 23.6 Wh kg −1 at the power density of 3400 W kg −1 can be obtained.Rate performance tests show that 86% of the capacitance can be maintained after the galvanostatic current density increases from 3 to 15 A g −1 . Our current pseudocapacitor system can provide a versatile method for the construction of high-performance inorganic pseudocapacitors by in situ crystallization via chemical/electrochemical reactions in room temperature aqueous solution.

In recent years, considerable literature has proposed the more general class of exponential Lévy processes as the underlying model for prices of financial quantities, which thus better explain many important empirical facts of financial markets. Finite moment log stable (FMLS), CGMY and KoBoL models are chosen from those above-mentioned models as the dynamics of underlying equity prices in this paper. With such models pricing barrier options, one kind of financial derivatives, is transformed to solve specific fractional partial differential equations (FPDEs). This study focuses on numerically solving these FPDEs via the fully implicit scheme, with the shifted Grünwald approximation. The circulant preconditioned generalized minimal residual method which converges very fast with theoretical proof is incorporated for solving resultant linear systems. Numerical examples are given to demonstrate the effectiveness of the proposed preconditioner and show the accuracy of our method compared with that done by the Fourier cosine expansion method as a benchmark.