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In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical solution of Volterra integro‐differential equations with weakly singular kernels. A smoothing transformation is first used to convert the original equation to a new equation with better regularity. A collocation method based on barycentric rational interpolation is introduced. The convergence and supercovergence of the numerical solution are studied in detail. Some numerical results are illustrated to confirm the theoretical prediction of the rate of convergence and superconvergence. [ABSTRACT FROM AUTHOR]

In this paper, we propose an hp -version fractional collocation method for solving second kind Volterra integro-differential equations with weakly singular kernels. We derive hp -version error estimates in a weighted H 1 -norm for the method on arbitrary meshes. The results show that for any given mesh partition, exponential rates of convergence can be achieved for certain weakly singular solutions by linearly increasing the degrees of piecewise fractional polynomials. The results also imply that in the case of uniform mesh, the method has no ( h -version) order barrier for weakly singular solutions, which is different from classical polynomial collocation methods. The method is easy to implement and has the same computational complexity as polynomial collocation methods. Numerical experiments are presented to demonstrate the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

A generalized quadrature method is studied for Volterra integral equations with highly oscillatory kernels. According to the kernel, a two-point quadrature rule is constructed by Lagrange's identity at first. The error of the quadrature formula is presented as well. Then, it is employed to discretize the highly oscillatory equation without the need to compute moment. For the convergence, the asymptotic order as well as the classical order of the quadrature method for equation is analyzed. It is shown that the method has asymptotic order two and converges with order two as step length decreases. Some numerical examples are conducted to test its efficiency. [ABSTRACT FROM AUTHOR]

In this paper we first establish the existence, uniqueness and Hölder continuity of the solution to stochastic Volterra integral equations (SVIEs) with weakly singular kernels, with singularities |$\alpha \in (0, 1)$| for the drift term and |$\beta \in (0, 1/2)$| for the stochastic term. Subsequently, we propose a |$\theta $| -Euler–Maruyama scheme and a Milstein scheme to solve the equations numerically and obtain strong rates of convergence for both schemes in |$L^{p}$| norm for any |$p\geqslant 1$|⁠. For the |$\theta $| -Euler–Maruyama scheme the rate is |$\min \big\{1-\alpha ,\frac{1}{2}-\beta \big\}~ $| and for the Milstein scheme is |$\min \{1-\alpha ,1-2\beta \}$|⁠. These results on the rates of convergence are significantly different from those it is similar schemes for the SVIEs with regular kernels. The source of the difficulty is the lack of Itô formula for the equations. To get around this difficulty we use the Taylor formula subsequently carrying out a sophisticated analysis of the equation. [ABSTRACT FROM AUTHOR]

In this paper, collocation methods for linear Volterra integral equations with highly oscillatory trigonometric kernels are considered. Different from the cases with linear oscillator, we need a further action to obtain a practical scheme for general oscillators. To deal with it, the moment free Filon type method is employed in the exact collocation equation resulting to a fully discretized one. In order to conduct the convergence analysis, we firstly study the highly oscillatory integrals occurring in the error equations. Both the theoretical and numerical results show that our method not only converges with respect to step length but also has an asymptotic order which could be two. The asymptotic order indicates that the numerical results will become more accurate as the increase of frequency. [ABSTRACT FROM AUTHOR]

This work is to analyze a Legendre collocation approximation for third-kind Volterra integro-differential equations. The rigorous error analysis in the L ∞ and L ω 0 , 0 2 -norms is provided for the proposed method. In fact when converting the original equation to an equivalent second kind one, the integral operator of the obtained equation contains two singularities and may become non-compact under certain conditions. In addition, in order to avoid the low-order accuracy caused by the singularity of the solution at the initial point, we adopted the idea of smooth transformation at the beginning to convert the original equation into a new equation with a more regular solution. Finally, the validity and applicability of the method are verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we propose a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations with a cubic nonlinear term. The employed time discretization is a weighted combination of the L2 − 1σ formula introduced recently by Lyu and Vong (Numer. Algorithms 78(2):485–511, 2018), Galerkin finite element method is used for the spatial discretization, and the cubic nonlinear term is handled explicitly. Using mathematical induction, we prove that the numerical solution is bounded and the fully discrete scheme is convergent with second-order accuracy in time. In numerical experiments, some problems with both smooth and non-smooth exact solutions are considered. [ABSTRACT FROM AUTHOR]

In this paper, we first construct a linearized Galerkin‐Legendre spectral method for the one‐dimensional nonlinear fractional Ginzburg‐Landau equation, where a three‐level linearized Crank‐Nicolson scheme is used for time discretization. The unique solvability and boundedness properties of the fully discrete scheme are analyzed. It is shown that the method is unconditionally convergent in the maximum norm with second‐order accuracy in time and spectral accuracy in space. Then, two‐dimensional problems are considered and a split‐step alternating direction implicit Galerkin‐Legendre spectral method is introduced without theoretical analysis. Finally, some numerical examples are presented to illustrate the effectiveness of the two proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multistep Runge–Kutta methods for Volterra integro-differential equations. First, we derive order conditions for methods of order p and stage order q = p − 1 for p ⩽ 4. Then, stability properties of these methods with respect to the basic and convolution test equations are analyzed. V 0 -stable methods of order 2 and 3 are obtained. Finally, some numerical experiments are given to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical solutions of the coupled fractional Klein-Gordon-Schrödinger equation. The numerical schemes are constructed by combining the Crank-Nicolson/leap-frog difference methods for the temporal discretization and the Galerkin finite element methods for the spatial discretization. We give a detailed analysis of the conservation properties in the senses of discrete mass and energy. Then the numerical solutions are shown to be unconditionally bounded in L2 −norm, H α 2 − semi-norm and L ∞ − norm, respectively. Based on the well-known Brouwer fixed-point theorem and the mathematical induction, the unique solvability of the discrete solutions is proved. Moreover, the schemes are proved to be unconditionally convergent with the optimal order O τ 2 + h r + 1 , where τ is the temporal step, h is the spatial grid size, and r is the order of the selected finite element space. Furthermore, by using the proposed decoupling and iterative algorithms, several numerical examples are included to support theoretical results and show the effectiveness of the schemes. Finally, the fast Krylov subspace solver with suitable circulant preconditioner is designed to effectively solve the Toeplitz-like linear systems. In each iterative step, this method can effectively reduce the memory requirement of above each finite element scheme from O M 2 to O(M), and the computational complexity from O M 3 to O (M log M) , where M is the number of grid nodes. Numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization/Cholesky decomposition methods, in terms of memory requirement and computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Galerkin–Legendre spectral method for solving the two-dimensional distributed-order time fractional fourth-order partial differential equation. By utilizing the composite Simpson formula to discretize the distributed-order integral, we transform the considered equation into a multi-term time fractional sub-diffusion equation. Then the L 2 - 1 σ formula is used to approximate the multi-term Caputo fractional derivatives and the Legendre spectral method is employed for the spatial discretization. The scheme is proved to be unconditionally stable and convergent in both L 2 - and L ∞ -norms with fourth-order accuracy in distributed order, second-order accuracy in time and spectral accuracy in space. Finally, some numerical tests are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper gives a rigorous error analysis of the multisymplectic Fourier pseudospectral method for the nonlinear fractional Schrödinger equation. The method preserves some intrinsic structure properties including the generalized multisymplectic conservation law. By rewriting it in a matrix form similar to that in the finite difference method, the method is shown to be convergent in the discrete L2 norm with the second‐order accuracy in time and spectral accuracy in space. The key techniques in the analysis include the discrete energy method, cutoff of the nonlinearity, and a posterior bound of numerical solutions by using the inverse inequality. In a similar line, the convergence result for the symplectic Fourier pseudospectral method can also be established. Moreover, the errors in the local and global energy conservation laws of discrete systems are also investigated. Numerical tests are performed to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability properties of Runge-Kutta methods for Volterra integro-differential equations. Both the basic test equation and a convolution test equation are considered. Some fixed order recurrence relations and the corresponding stability conditions are derived for general methods. The concept of V 0 -stability is introduced for the convolution test equation and some V 0 -stable one-stage methods are found. The A 0 -stability and V 0 -stability of the fully implicit discretized collocation methods with one or two stages are investigated in details. Finally, some numerical experiments are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of problems have a weak singularity at the initial time t = 0 . A rigorous error analysis of our method shows that, with an appropriate choice of the fractional-order polynomials and a suitably graded mesh, one can attain optimal orders of convergence to the exact solution and its derivative, and certain superconvergence results are also derived. In particular, our analysis shows that on a uniform mesh our method attains a higher order of convergence than standard piecewise polynomial collocation. Numerical examples are presented to demonstrate the sharpness of our theoretical results. [ABSTRACT FROM AUTHOR]

This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

• TFS coupling experimental and numerical technology is proposed. • A novel friction reduction structure is proposed for blade cooling. • The friction reduction mechanism of the cross bridge is revealed. • The action mechanism of cooling performance on solid strain is revealed. • The heat transfer uniformity and overall strain are improved. To supply the higher safety design demands of the new generation of gas turbines, it is urgent to solve the difficulties in mid-chord region of high-temperature turbine blades, including excessive friction loss of traditional two-pass channel, lack of experimental data, lack of stress-strain characteristics in the solid domain and so on. Therefore, this research aims to propose an integrated method of the high-efficiency and low-friction cooling design and Thermo-Fluid-Solid (TFS) coupling analysis. Firstly, a turbine blade rotating cooling experimental system is designed and built independently. Then, a novel friction reduction method with cross bridge structure is proposed. The friction reduction mechanism and the effect mechanism of the cooling performance on the solid strain are revealed through experimental test and TFS coupling numerical study. The results show that the cross brige has an excellent friction reduction effect on the channel. The influence of 'peak cutting and valley filling' effectively enhances the heat transfer uniformity as well as successfully reduces the overall strain. The average Nu/Nu 0 of the leading edge surface decreases by 14.7% (stationary condition) and 16.4% (rotating condition), respectively. And the Nu/Nu 0 of the trailing edge surface achieves a maximum improvement effect of 3.3%. The f/f 0 of each channel decreases initially and followed by an increase as Ro increasing. Among them, the channel with two and three cross bridges can achieve friction reduction effects of 24.7–37.3% and 32.2–40.9%, respectively. The strain is related to the Nu/Nu 0 of each region, and the value of the rotating trailing edge surface is generally higher than that of the rotating leading edge surface. [ABSTRACT FROM AUTHOR]

In this article, an efficient difference scheme for the coupled fractional Ginzburg–Landau equations with the fractional Laplacian is studied. We construct the discrete scheme based on the implicit midpoint method in time and a weighted and shifted Grünwald difference method in space. Then, we prove that the scheme is uniquely solvable, and the numerical solutions are bounded and unconditionally convergent in the lh2 norm. Finally, numerical tests are given to confirm the theoretical results and show the effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the delay dependent asymptotic mean square stability of the stochastic split-step θ method for a scalar linear stochastic delay differential equation with real coefficients is studied. The full stability region of this method is given by using root locus technique. The necessary and sufficient condition with respect to the equation coefficients, time stepsize and method parameter θ is derived. It is proved that the stochastic split-step backward Euler can preserve the asymptotic mean square stability of the underlying system completely. In addition, the numerical stability regions of the stochastic split-step θ method and the stochastic θ method are compared with each other. At last, we validate our conclusions by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a high-order implicit-explicit difference scheme for the nonlinear complex fractional Ginzburg-Landau equation involving the Riesz fractional derivative. For the time discretization, the second-order backward differentiation formula combined with the explicit second-order Gear’s extrapolation is adopted. While for the space discretization, a fourth-order fractional quasi-compact method is used to approximate the Riesz fractional derivative. The scheme is efficient in the sense that, at each time step, only a linear system with a coefficient matrix independent of the time level needs to be solved. Despite of the explicit treatment of the nonlinear term, the scheme is shown to be unconditionally convergent in the lh2 norm, semi-Hhα/2 norm and lh∞ norm at the order of O(τ2+h4) with τ time step and h mesh size. Numerical tests are provided to confirm the accuracy and efficiency of the scheme. [ABSTRACT FROM AUTHOR]

This paper considers the long-time integration of the nonlinear fractional Schrödinger equation involving the fractional Laplacian from the point of view of symplectic geometry. By virtue of a variational principle with the fractional Laplacian, the equation is first reformulated as a Hamiltonian system with a symplectic structure. Then, by introducing a pair of intermediate variables with a fractional operator, the equation is reformulated in another form for which more conservation laws are found. When reducing to the case of integer order, they correspond to multi-symplectic conservation law and local energy conservation law for the classic Schrödinger equation. After that, structure-preserving algorithms with the Fourier pseudospectral approximation to the spatial fractional operator are constructed. It is proved that the semi-discrete and fully discrete systems satisfy the corresponding symplectic or other conservation laws in the discrete sense. Numerical tests are performed to validate the efficiency of the methods by showing their remarkable conservation properties in the long-time simulation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a split-step quasi-compact finite difference method to solve the nonlinear fractional Ginzburg–Landau equations both in one and two dimensions. The original equations are split into linear and nonlinear subproblems. The Riesz space fractional derivative is approximated by a fourth-order fractional quasi-compact method. Furthermore, an alternating direction implicit scheme is constructed for the two dimensional linear subproblem. The unconditional stability and convergence of the schemes are proved rigorously in the linear case. Numerical experiments are performed to confirm our theoretical findings and the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conservative linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Schrödinger equation. We construct5 a time-discrete system, to which, the mass conservation, semi-discrete error estimates and the suitable regularity of the numerical solution are obtained. With the spatial direction discreted by FEMs, the fully discrete conservative linearized finite element scheme is presented. Moreover, by a new error splitting technique, an unconditional-norm error estimates are derived by the boundedness of the fully-discrete numerical solution in-norm. Finally, some numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical solution of the nonlinear fractional Ginzburg–Landau equation. Galerkin finite element method is used for the spatial discretization, and an implicit midpoint difference method is employed for the temporal discretization. The boundedness, existence and uniqueness of the numerical solution, and the unconditional error estimates in the L 2 -norm are investigated in details. To numerically solve the nonlinear system, linearized iterative algorithms are also considered. Finally, some numerical examples are presented to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive Filon-type method to approximate oscillatory integrals. By using some S-shaped functions connecting Gauss-Legendre nodes and Filon-type method, the adaptive schemes behave well for small as well as for large frequencies. Adding frequency-dependent nodes can improve the asymptotic order at the same time. Moreover, there exist some complex nodes leading to a higher asymptotic order method approximating to the integral, and the special nodes work in other methods as well. The error can be further decreased by taking extra Chebyshev nodes. The efficiency and accuracy are tested by some experiments. [ABSTRACT FROM AUTHOR]

In this paper, we study Galerkin finite element methods for a class of higher dimensional multi-term fractional diffusion equations. The finite difference approximation of Caputo derivative on non-uniform meshes is used in temporal direction and Galerkin finite element method is used in spatial direction. We prove that semi-discrete and fully discrete finite element schemes are unconditionally stable. Meanwhile,-norm convergence properties of the two schemes are proved rigorously. To confirm our theoretical analysis, we give some numerical examples in both two-dimensional (2D) and three-dimensional (3D) spaces. Finally, a moving local refinement technique in temporal direction is used to improve the accuracy of numerical solution. [ABSTRACT FROM AUTHOR]

In this paper the existence, uniqueness, regularity properties, and in particular, the local representation of solutions for general Volterra functional integral equations with non-vanishing delays, are investigated. Based on the solution representation, we detailedly analyze the attainable (global and local) convergence order of (iterated) collocation solutions on θ -invariant meshes. It turns out that collocation at the m Gauss (-Legendre) points neither leads to the optimal global convergence order m + 1 , nor yields the local convergence order 2 m on the whole interval, which is in sharp contrast to the case of the classical Volterra delay integral equations. However, if the collocation is based on the m Radau II points, the local superconvergence order 2 m − 1 will exhibit at all mesh points. Finally, some numerical experiments are performed to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

[Display omitted] • Coordination environments of Ni precursor complexes were precisely controlled. • Bifunctional Ni@S-1 catalysts developed from the evolution of hydroxylated Ni phyllosilicate. • Superb CO 2 -CH 4 reforming behavior benefited from the synergy of tailored metal–acid sites. • A benign balance between formation and elimination of "process carbon" species was established. Ni-based catalysts using siliceous materials as the supports have suffered from two challenging issues of sintering and carbon deposition for dry reforming of methane (DRM), resulted from a weak metal-support interaction. Given these, a novel metal–acid bifunctional Ni@S-1 zeolite catalyst is tailored from the evolution of hydroxylated nickel phyllosilicates that are formed by subtly controlling the coordination environments of Ni precursor complexes in the silicalite-1 (S-1) zeolite. The developed Ni 0.44 @S-1 catalyst renders almost equivalent CH 4 and CO 2 conversions (85.1 % and 88.7 %), high H 2 /CO molar ratio (about 1) and superb long-term stability during 100 h with no activity loss at 750 °C with a WHSV of 100 L·g cat −1·h−1, outperforming other control catalysts under investigation. Such superior catalytic behavior benefits from the synergy between highly uniform Ni nanoparticles (3–5 nm) confined in the micropores of zeolite and substantially formed Lewis acid sites due to coordinatively unsaturated Ni2+ sites located at the reduced 2:1 nickel phyllosilicate, as revealed by detailed characterizations. Meanwhile, it further discloses that a benign balance between formation and elimination of "process carbon" species is responsible for improving the carbon resistance of catalyst during the DRM reaction. The present design broadens a new insight for the development of new Ni-based catalysts with simultaneous coking- and sintering-resistant for the DRM application. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal (global and local) convergence orders of the (iterated) collocation solutions for second-kind Volterra integral equations with vanishing delays on quasi-geometric meshes. It turns out that the classical global convergence results still hold under certain regularity conditions of the given functions. In particular, it is shown that the optimal local superconvergence order p = 2 m can be attained if collocation is at the m Gauss(–Legendre) points, which contrasts with collocations both on uniform meshes and on geometric meshes. Numerical experiments are performed to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a new conservative difference scheme is proposed for solving the nonlinear space-fractional Schrödinger equation. The Riesz space-fractional derivative is approximated by the second-order accurate weighted and shifted Grünwald difference operator. Building on a careful analysis of this operator, the conservation properties including the mass and energy are analyzed and some norm equivalences are established. Then the numerical solution is shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme in the l 2 norm, semi- H α / 2 norm and l ∞ norm are derived. The uniform bounds on the numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial mesh size needs to be assumed. Numerical tests are performed to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider generalized fractional derivatives, which are characterized by a scale function and a weight function. It is proposed to replace the time variable with a new variable, associated with the scale and weight functions, which allows us to reduce the problems for equations with generalized fractional derivatives to problems for equations with the usual fractional derivative. The equations obtained after the above change of variables are quite well studied, so that one can apply well-known effective numerical methods and use the reverse substitution to find solutions to the original problems. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an efficient difference scheme for the nonlinear complex Ginzburg–Landau equation involving fractional Laplacian. The scheme is based on the implicit midpoint rule for the temporal discretization and a weighted and shifted Grünwald difference operator for the spatial fractional Laplacian. By virtue of a careful analysis of the difference operator, some useful inequalities with respect to suitable fractional Sobolev norms are established. Then the numerical solution is shown to be bounded, and convergent in the l h 2 norm with the optimal order O ( τ 2 + h 2 ) with time step τ and mesh size h . The a priori bound as well as the convergence order holds unconditionally, in the sense that no restriction on the time step τ in terms of the mesh size h needs to be assumed. Numerical tests are performed to validate the theoretical results and effectiveness of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conservative and effective difference scheme for solving the two dimensional nonlinear space-fractional Schrödinger equation with the Riesz fractional derivative. The scheme is constructed by combining the split-step method for handling the nonlinearity with the alternating direction implicit (ADI) method for resolving the multi-dimensions difficulty. The Riesz space-fractional derivative is approximated by the second order accurate fractional centered difference. Based on matrix analysis, we show that in the discrete sense the scheme conserves the mass and energy for linear problems and conserves the mass for nonlinear problems. The unconditional convergence is proved rigorously in the linear case. Numerical tests are performed to support our theoretical results and show the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

[Display omitted] • HZSM-5 zeolites cross-linked with ultrathin siliceous layer are fabricated. • Catalytic cracking performance of n-butane on the silanization zeolites are largely enhanced. • Diffusion properties of n-butane on the zeolites are quantified by in situ FTIR spectroscopy. • Correlation between n-butane conversion rate per Brønsted acid site and diffusivity is negative. • HZSM-5 with siliceous monolayer intensifies catalytic performance and diffusivity of n-butane. A series of HZSM-5 zeolites are fabricated to make it cross-link with an external ultrathin siliceous layer by chemical liquid deposition and the corresponding thickness is measured by TEM and XPS analysis. In situ FTIR experiments disclose that the external surface acidic sites of HZSM-5 zeolite can be covered even by siliceous monolayer (0.7 ∼ 1.3 nm, HZ5-S1). In comparison to parent HZSM-5, the catalytic cracking performance of n-butane over the modified zeolites are evidently enhanced. In particular, the n-butane conversion rate per Brønsted acid site over the HZ5-S2 (16.02 min−1) at 550 °C is 2.82 times than that over the HZSM-5 (5.68 min−1). Meanwhile, the diffusion properties of n-butane over the zeolites are quantified by in situ FTIR spectroscopic analysis. Negative correlation between steady-state diffusivity and n-butane conversion rate per Brønsted acid site can be found, which unravels the origin of the progressive decrease in the catalytic stability. Under comparable conditions, more advantages can be greatly achieved by HZ5-S1 catalyst in the catalytic activity and the yields to light olefins due to the benign diffusivity and the elimination of external surface acidic sites. This study can provide a guidance for the understanding of diffusion intensification and the rational design of superior catalyst for the catalytic cracking reactions of alkanes. [ABSTRACT FROM AUTHOR]

In this paper, an energy conservative Crank–Nicolson difference scheme for nonlinear Riesz space-fractional Schrödinger equations is studied. We give a rigorous analysis of the conservation properties, including mass conservation and energy conservation in the discrete sense. Based on Brouwer fixed point theorem, the existence of the difference solution is proved. By virtue of the energy method, the difference solution is shown to be unique and convergent at the order of O ( τ 2 + h 2 ) in the l 2 -norm with time step τ and mesh size h . Finally a linearized iterative algorithm is presented and numerical experiments are given to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we propose a conservative linearized difference scheme for the nonlinear fractional Schrödinger equation. The scheme efficiently avoids the time consuming iteration procedure necessary for the nonlinear scheme and thus is time saving relatively. It is rigorously proved that the scheme is mass conservative and uniquely solvable. Then employing mathematical induction, we further show that the proposed scheme is convergent at the order of O( τ + h) in the l norm with time step τ and mesh size h. Moreover, an extension to coupled nonlinear fractional Schrödinger systems is presented. Finally, numerical tests are carried out to corroborate the theoretical results and investigate the impact of the fractional order α on the collision of two solitons. [ABSTRACT FROM AUTHOR]

We investigate the strong convergence of a full discretization for the stochastic elastic equation driven by multiplicative noise. Finite element method is used for the spatial approximation and an exponential integrator scheme is proposed for the temporal discretization. We show that the fully discrete scheme strongly converges to the mild solution of the continuous problem, with precise convergence rates given. As shown in our result, the exponential integrator scheme achieves a higher strong convergence rate than the backward Euler method does in the literature. [ABSTRACT FROM AUTHOR]

Limestone hills are an unusual habitat for primates, prompting them to evolve specific behavioral adaptations to the component karst habitat. From September 2012 to August 2013, we collected data on the diet of one group of Assamese macaques living in limestone forests at Nonggang National Nature Reserve, Guangxi Province, China, using instantaneous scan sampling. Assamese macaques were primarily folivorous, young leaves accounting for 75.5% and mature leaves an additional 1.8% of their diet. In contrast, fruit accounted for only 20.1%. The young leaves of Bonia saxatilis, a shrubby, karst-endemic bamboo that is superabundant in limestone hills, comprised the bulk of the average monthly diet. Moreover, macaques consumed significantly more bamboo leaves during the season when the availability of fruit declined, suggesting that bamboo leaves are an important fallback food for Assamese macaques in limestone forests. In addition, diet composition varied seasonally. The monkeys consumed significantly more fruit and fewer young leaves in the fruit-rich season than in the fruit-lean season. Fruit consumption was positively correlated with fruit availability, indicating that fruit is a preferred food for Assamese macaques. Of seventy-eight food species, only nine contributed >0.5% of the annual diet, and together these nine foods accounted for 90.7% of the annual diet. Our results suggest that bamboo consumption represents a key factor in the Assamese macaque's dietary adaptation to limestone habitat. Am. J. Primatol. 77:171-185, 2015. © 2014 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper, we first prove the strong convergence of the split-step theta methods for non-autonomous stochastic differential equations under a linear growth condition on the diffusion coefficient and a one-sided Lipschitz condition on the drift coefficient. Then, if the drift coefficient satisfies a polynomial growth condition, we further get the rate of convergence. Finally, the obtained results are supported by numerical experiments. [ABSTRACT FROM PUBLISHER]

In this paper, a two-parameter Milstein method for stochastic Volterra integral equations is introduced. First, the method is proved to be strongly convergent with order one in L p norm (p ≥ 1). Then, we investigate the mean square stability of the exact and numerical solutions of a stochastic convolution test equation. Stability conditions are derived. Based on these conditions, analytical and numerical stability regions are plotted and compared with each other. The results show that additional implicitness offers benefits for numerical stability. Finally, some numerical experiments are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping principle. Secondly, we develop an Euler–Maruyama (EM) method and obtain its strong convergence rate. Moreover, we propose a fast EM method via the exponential-sum-approximation technique to reduce the EM method's computational cost. More specifically, if one disregards the Monte Carlo sampling error, then the fast EM method reduces the computational cost from O (N 2) to O (N log 2 N) and the storage from O(N) to O (log 2 N) , where N is the total number of time steps. Moreover, if the sampling error is taken into account, we employ the multilevel Monte Carlo method based on the fast EM method to reduce computational costs further. Significantly, the computational costs of the EM method and the fast EM method to achieve an accuracy of O(ε) (ε < 1) are reduced from O (ε - 2 - 2 α ~ ) and O (ε - 2 - 1 α ~ log 2 (ε)) , respectively, to O (ε - 1 α ~ (log (ε - 1)) 3) , where α ~ = min { 1 - α * , 1 2 - β * } is related to the exponents of the singular kernel in the equations. Finally, numerical examples are provided to illustrate our theoretical results and demonstrate the superiority of the fast EM method. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients. This condition admits that the diffusion coefficient can be highly nonlinear, i.e., it does not necessarily satisfy a linear growth or global Lipschitz condition. It is proved that, for all positive stepsizes, the classical stochastic theta method with is asymptotically mean square stable and the split-step theta method with is exponentially mean square stable. Conditional stability results for the methods with are also obtained under a stronger assumption. Finally, we further investigate the mean square dissipativity of the split-step theta method with and prove that the method possesses a bounded absorbing set in mean square independent of initial data. [Copyright &y& Elsevier]

Abstract: In this paper, we propose and analyze a spectral Jacobi-collocation method for the numerical solution of general linear fractional integro-differential equations. The fractional derivatives are described in the Caputo sense. First, we use some function and variable transformations to change the equation into a Volterra integral equation defined on the standard interval . Then the Jacobi–Gauss points are used as collocation nodes and the Jacobi–Gauss quadrature formula is used to approximate the integral equation. Later, the convergence order of the proposed method is investigated in the infinity norm. Finally, some numerical results are given to demonstrate the effectiveness of the proposed method. [Copyright &y& Elsevier]

We collected fecal samples of white-headed langurs from 3 of the 4 remaining habitat fragments (Fa, Fb and CZ) located in southwestern Guangxi, China in Nov 2005, and used 5 microsatellite loci and the SRY gene to assess the relatedness between 46 langurs within and between groups. We observed 2 forms of group structure: one-male/multi-female groups (OMGs) and all-male groups (AMGs). One AMG in Fa was composed of 2 generations, included a father, 2 sons and 1 unrelated male, and all OMGs in all 3 habitats included 1 resident male, several adult females and offspring. Of the 21 identified father-offspring cases, the resident male fathered 20 (95%) and the non-resident male sired 1 (5%), suggesting that adult males had overwhelming priority of access to females as the resident male in an OMG, while the non-resident male may also have the opportunity to adopt surreptitious mating strategies. [ABSTRACT FROM AUTHOR]

Three zeolites ZSM-5, ZSM-35 and ZSM-22 are applied in n -butane cracking to reveal the influence of regional differences in Brønsted acid sites on reaction pathways. The Brønsted acid sites located in the channels of ZSM-5 exhibit a higher catalytic activity and yields to light olefins (especially the ethylene), which can be attributed to more relative contribution of bimolecular pathway. In comparison, more Brønsted acid sites are situated on the external surface or micropore mouths of ZSM-35 and ZSM-22, which are beneficial to propylene production. Note-worthily, almost no propane production on the two zeolites implies the predomination of monomolecular pathway. In situ FTIR technique further discloses that the difference in the reaction pathways is tailored by adsorption configurations of n -butane at two types of Brønsted acid sites. Under comparable conditions, the ZSM-5 performs a relatively good catalytic stability through bimolecular pathway. Furthermore, the correlation between deactivation rate and light olefins reveals that the enhancement of propylene selectivity accelerates the deactivation of ZSM-35 and ZSM-22. While, promotion on the ethylene selectivity is a significant inducement for the deactivation of ZSM-5. Reaction pathways of n-butane cracking tailored by the influence of regional differences in Brønsted acid sites. [Display omitted] • Three typical 10-member ring zeolites ZSM-5, ZSM-35 and ZSM-22 are prepared. • Reaction pathways of n -butane cracking over the zeolites are investigated. • Regional differences in Brønsted acid sites tailor the selectivities to propylene and ethylene. • ZSM-5 zeolite exhibits high catalytic activity and stability compared to ZSM-35 and ZSM-22. • Catalyst deactivation mechanisms are revealed on different reaction pathways. [ABSTRACT FROM AUTHOR]

A highly stable bimetallic-sites catalyst of Ni-Zn anchored in the silicalite-1 zeolite framework is facilely fabricated by ligand-protected strategy, synergistically boosting propane dehydrogenation to propylene. [Display omitted] • Bimetallic Ni-Zn site anchored in the silicalite-1 zeolitic framework are fabricated. • Superior catalytic performance on the 0.2Ni0.06Zn-S-1 for PDH is achieved. • High stability and activity of catalyst are remained after regeneration for 4 cycles. • Synergistic effect between Ni-Zn sites favors propylene desorption to reduce coke deposition. Propane dehydrogenation (PDH) is a prospective benign process for the production of propylene, but generally depends on high-price Pt and toxic Cr(VI) catalysts that often suffer from the rapid deactivation derived from metal sintering and severe coke deposition. To overcome these problems, the breakthroughs in an alternative catalyst design with efficient, low-cost and eco-friendly play a crucial role in successfully developing this process. Here, we facilely fabricate the bimetallic-site catalyst of Ni-Zn anchored in the zeolitic framework of silicalite-1 (depicted as NiZn-S-1) by employing ligand-protected strategy, which performs high temperature stability. Under comparable conditions, the NiZn-S-1 catalyst with high Ni/Zn molar ratio of 3:1 (named 0.2Ni0.06Zn-S-1) exhibits superior catalytic activity (initial propane conversion of 32.5 % and actual TOF propane of 28.5 min−1) with low deactivation rate (0.0778 h−1) and great propylene selectivity (>90%) after 30 h on stream. In particular, the catalytic performance on this catalyst almost keeps same as that of fresh catalyst after 4 successive oxidation–reduction cycles. Multiple characterizations further demonstrate that the enhancement of PDH performance ought to be attributed to the synergistic effect between bimetallic Ni-Zn sites, which favors the selective activation of C–H bond and propylene desorption to reduce the formation of coke species with high graphitization degree, boosting the improvement of catalyst stability. The design of novel bimetallic-sites catalyst can give a new idea on the development of alternative PDH catalysts with economical- and environment-friendly in the future. [ABSTRACT FROM AUTHOR]

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3 − α) th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems. [ABSTRACT FROM AUTHOR]

Abstract: Fractional integro-differential equations have been recently solved by many methods, such as Adomian decomposition method, differential transform method, collocation method and Taylor expansion approach. In this paper a hybrid collocation method is used which combines a non-polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval. A theoretical analysis for the convergence order of the method is presented. Some numerical examples are given which confirm the theoretical results. [Copyright &y& Elsevier]