In this paper, the regularity and finite difference methods for the two-dimensional delay fractional equations are considered. The analytic solution is derived by eigenvalue expansions and Laplace transformation. However, due to the derivative discontinuities resulting from the delay effect, the traditional L1-ADI scheme fails to achieve the optimal convergence order. To overcome this issue and improve the convergence order, a simple and cost-effective decomposition technique is introduced and a fitted L1-ADI scheme is proposed. The numerical tests are conducted to verify the theoretical result.

Higher order finite difference Weighted Essentially Non-Oscillatory (FD-WENO) schemes for conservation laws are extremely popular because, for multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. Such schemes come in two formulations. The very popular classical FD-WENO method (Shu and Osher J Comput Phys 83: 32–78, 1989) relies on two reconstruction steps applied to two split fluxes. However, the method cannot accommodate different types of Riemann solvers and cannot preserve free stream boundary conditions on curvilinear meshes. This limits its utility. The alternative FD-WENO (AFD-WENO) method can overcome these deficiencies, however, much less work has been done on this method. The reasons are three-fold. First, it is difficult for the casual reader to understand the intricate logic that requires higher order derivatives of the fluxes to be evaluated at zone boundaries. The analytical methods for deriving the update equation for AFD-WENO schemes are somewhat recondite. To overcome that difficulty, we provide an easily accessible script that is based on a computer algebra system in Appendix A of this paper. Second, the method relies on interpolation rather than reconstruction, and WENO interpolation formulae have not been documented in the literature as thoroughly as WENO reconstruction formulae. In this paper, we explicitly provide all necessary WENO interpolation formulae that are needed for implementing the AFD-WENO up to the ninth order. The third reason is that the AFD-WENO requires higher order derivatives of the fluxes to be available at zone boundaries. Since those derivatives are usually obtained by finite differencing the zone-centered fluxes, they become susceptible to a Gibbs phenomenon when the solution is non-smooth. The inclusion of those fluxes is also crucially important for preserving the order property when the solution is smooth. This has limited the utility of the AFD-WENO in the past even though the method per se has many desirable features. Some efforts to mitigate the effect of finite differencing of the fluxes have been tried, but so far they have been done on a case by case basis for the PDE being considered. In this paper we find a general-purpose strategy that is based on a different type of the WENO interpolation. This new WENO interpolation takes the first derivatives of the fluxes at zone centers as its inputs and returns the requisite non-linearly hybridized higher order derivatives of flux-like terms at the zone boundaries as its output. With these three advances, we find that the AFD-WENO becomes a robust and general-purpose solution strategy for large classes of conservation laws. It allows any Riemann solver to be used. The AFD-WENO has a computational complexity that is entirely comparable to the classical FD-WENO, because it relies on two interpolation steps which cost the same as the two reconstruction steps in the classical FD-WENO. We apply the method to several stringent test problems drawn from Euler flow, relativistic hydrodynamics (RHD), and ten-moment equations. The method meets its design accuracy for smooth flow and can handle stringent problems in one and multiple dimensions.

In this paper, we present a novel and efficient iterative approach for computing the polar decomposition of rectangular (or square) complex (or real) matrices. The method proposed herein entails four matrix multiplications in each iteration, effectively circumventing the need for matrix inversions. We substantiate that this method exhibits fourth-order convergence. To illustrate its efficacy relative to alternative techniques, we conduct numerical experiments using randomly generated matrices of dimensions $ n\times n $, where n assumes values of 80, 90, 100, 120, 150, 180, and 200. Through two illustrative examples, we provide numerical results. We gauge the performance of different methods by calculating essential metrics based on ten matrices for each dimension. These metrics include the average iteration count, the average total matrix multiplication count, the average precision, and the average execution time. Through meticulous comparison, our newly devised method emerges as a proficient and rapid solution, boasting a reduced computational overhead.

In this paper, we investigate the global dynamics of a predator-prey model with a general growth rate function and carrying capacity. We prove that the origin is unstable using the blow-up method. Also, by constructing a new Lyapunov function and using LaSalle’s invariance principle, we obtain the global stability of the positive equilibrium state of the system. In addition, the system undergoes the Hopf bifurcation at the positive equilibrium point when the predator birth rate $ \delta $ is used as the bifurcation parameter. Finally, two examples are given to verify the feasibility of the theoretical results. One example is given to reconsider the global stability of the positive equilibrium of a Leslie-Gower predator-prey model with prey cannibalism, and the obtained results confirm the conjecture proposed by Lin et al. (Adv Differ Equ 2020, 153, 2020). The other example is given to verify the occurrence of the Hopf bifurcation of a Leslie-Gower predator-prey model with a square root response function, and obtain the Hopf bifurcation diagram by the numerical simulation.

We introduce tensor generalized bilateral inverses (TGBIs) under the Einstein tensor product as an extension of generalized bilateral inverses (GBIs) in the matrix environment. Moreover, the TBGI class includes so far considered composite generalized inverses (CGIs) for matrices and tensors. Applications of TBGIs for solving multilinear systems are presented. The characterizations and representations of the TGBI were studied and verified using a specific algebraic approach. Further, a few characterizations of known CGIs (such as the CMP, DMP, MPD, MPCEP, and CEPMP) are derived. The main properties of the TGBIs were exploited and verified through numerical examples.

Higher order finite difference Weighted Essentially Non-oscillatory (WENO) schemes for conservation laws represent a technology that has been reasonably consolidated. They are extremely popular because, when applied to multidimensional problems, they offer high order accuracy at a fraction of the cost of finite volume WENO or DG schemes. They come in two flavors. There is the classical finite difference WENO (FD-WENO) method (Shu and Osher in J. Comput. Phys. 83: 32–78, 1989). However, in recent years there is also an alternative finite difference WENO (AFD-WENO) method which has recently been formalized into a very useful general-purpose algorithm for conservation laws (Balsara et al. in Efficient alternative finite difference WENO schemes for hyperbolic conservation laws, submitted to CAMC, 2023). However, the FD-WENO algorithm has only very recently been formulated for hyperbolic systems with non-conservative products (Balsara et al. in Efficient finite difference WENO scheme for hyperbolic systems with non-conservative products, to appear CAMC, 2023). In this paper, we show that there are substantial advantages in obtaining an AFD-WENO algorithm for hyperbolic systems with non-conservative products. Such an algorithm is documented in this paper. We present an AFD-WENO formulation in a fluctuation form that is carefully engineered to retrieve the flux form when that is warranted and nevertheless extends to non-conservative products. The method is flexible because it allows any Riemann solver to be used. The formulation we arrive at is such that when non-conservative products are absent it reverts exactly to the formulation in the second citation above which is in the exact flux conservation form. The ability to transition to a precise conservation form when non-conservative products are absent ensures, via the Lax-Wendroff theorem, that shock locations will be exactly captured by the method. We present two formulations of AFD-WENO that can be used with hyperbolic systems with non-conservative products and stiff source terms with slightly differing computational complexities. The speeds of our new AFD-WENO schemes are compared to the speed of the classical FD-WENO algorithm from the first of the above-cited papers. At all orders, AFD-WENO outperforms FD-WENO. We also show a very desirable result that higher order variants of AFD-WENO schemes do not cost that much more than their lower order variants. This is because the larger number of floating point operations associated with larger stencils is almost very efficiently amortized by the CPU when the AFD-WENO code is designed to be cache friendly. This should have great, and very beneficial, implications for the role of our AFD-WENO schemes in the Peta- and Exascale computing. We apply the method to several stringent test problems drawn from the Baer-Nunziato system, two-layer shallow water equations, and the multicomponent debris flow. The method meets its design accuracy for the smooth flow and can handle stringent problems in one and multiple dimensions. Because of the pointwise nature of its update, AFD-WENO for hyperbolic systems with non-conservative products is also shown to be a very efficient performer on problems with stiff source terms.

In this article, numerical algorithms are derived for ultra-slow (or superslow) diffusion equations in one and two space dimensions, where the ultra-slow diffusion is characterized by the Caputo-Hadamard fractional derivative of order $ \alpha \in (0,1) $. To describe the non-locality in spatial interaction, the Riesz fractional derivative and the fractional Laplacian are used in one and two space dimensions, respectively. The Caputo-Hadamard derivative is discretized by two typical approximate formulae, i.e., $ \textrm{L2-1}_{\sigma } $ and L1-2 ones. The spatial fractional derivatives are discretized by the second order finite difference methods. When the $ \textrm{L2-1}_{\sigma } $ discretization is used, the derived numerical schemes are unconditionally stable, with both theoretical and numerical convergence order $ \mathcal {O}(\tau ^{2}+h^{2}) $ for all $ \alpha \in (0, 1) $, in which $ \tau $ and h are temporal and spatial stepsizes, respectively. When the L1-2 discretization is used, the derived numerical schemes are proved to be stable with the error estimate $ \mathcal {O}(\tau ^{2}+h^{2}) $ for $ \alpha \in (0, 0.373\,8) $, and numerically exhibit the stability for all $ \alpha \in (0, 1) $ with the numerical error being $ \mathcal {O}(\tau ^{3-\alpha }+h^2) $. The illustrative examples displayed are in line with the theoretical analysis.

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation (SE) both analytically and numerically, where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations (PDEs) that can be derived using the standard rules of standard derivatives. However, for the space fractional Schrödinger equation (FSE), the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations, because not only the problem becomes non-local, but also the rules for the standard derivatives generally do not hold for the fractional derivatives. In this work, we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) form for the space FSE based on the quantum Riesz fractional operators. We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local, the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE, and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order. We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes. Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.

We propose a dual Markov chain model to accommodate probabilities as well as perturbation, error bounds, or variances, in the Markov chain process. This motivates us to extend the Perron-Frobenius theory to dual number matrices with primitive and irreducible nonnegative standard parts. It is shown that such a dual number matrix always has a positive dual number eigenvalue with a positive dual number eigenvector. The standard part of this positive dual number eigenvalue is larger than or equal to the modulus of the standard part of any other eigenvalue of this dual number matrix. An explicit formula to compute the dual part of this positive dual number eigenvalue is presented. The Collatz minimax theorem also holds here. The results are nontrivial as even a positive dual number matrix may have no eigenvalue at all. An algorithm based upon the Collatz minimax theorem is constructed. The convergence of the algorithm is studied. An upper bound on the distance of stationary states between the dual Markov chain and the perturbed Markov chain is given. Numerical results on both synthetic examples and the dual Markov chain including some real world examples are reported.

The present article is devoted to developing new finite difference schemes with a higher order of the convergence for the generalized time-fractional diffusion equations (GTFDEs) that are characterized by a weight function w(t). Three different discrete analogs with different orders of approximations are designed for the generalized Caputo derivative. The major contribution of this paper is the development of an L2 type difference scheme that results in the $ (3-\alpha ) $ order of convergence in time. The spatial direction is discretized using a second-order difference operator. Fundamental properties of the coefficients of the L2 difference operator are examined and proved theoretically. The stability and convergence analysis of the developed L2 scheme are established theoretically using the energy method. An efficient algorithm is developed and implemented on numerical test problems to prove the numerical accuracy of the scheme.

The existing numerical approximation formulae for two kinds of typical fractional derivatives—the exponential Caputo and Caputo-Hadamard derivatives both of order $ \alpha \in (1,2) $ include L2, $ \hbox {L2}_1 $, H2N2, but their convergence orders are all less than 2. To obtain a higher accuracy convergence order, we construct H3N3 approximation formulae based on the H2N2 formulae of these two kinds of derivatives and the $ \hbox {H3N3-2}_\sigma $ formula of the Caputo derivative, determine their truncation errors, and show the coefficients’ properties. Simultaneously, we display the numerical examples which support the theoretical analysis.