This paper presents the new error estimates for a second-order backward-differentiation scheme (BDF2) given in [1] for the evolutionary Stokes-Darcy system. We prove the unconditional optimal error estimates and the uniform in time error estimates with the modest time step (Δt≤C) for the fluid velocity and the hydraulic head in L2 norm. Besides, we also prove the error estimates for the fluid velocity

In this work, we investigate a multidimensional system consisting of a finite (though arbitrary) number of coupled hyperbolic partial differential equations with fractional diffusion, constant damping and inertial times, and nonlinear reaction terms. Under suitable analytical conditions, the model has conserved quantities which are preserved in the absence of damping. We establish rigorously the conservation

In this paper, an energy stable time-stepping method using the finite difference approximation in time and Fourier pseudo-spectral method in space is developed for the symmetric regularized long wave equation with damping mechanism. Based upon a careful treatment of nonlinear term, the suggested numerical scheme is proved to preserve energy dissipation at discrete time levels. The energy stable property

A new collocation method using near-minimal Chebyshev quadrature nodes is proposed on the square [−1,1]2. For a (total) degree of precision 2n−1, the number of nodes of this near-minimal quadrature rule amounts only to n(n+1)2+⌊n2⌋+1, which is one more than the Möller's lower bound, n(n+1)2+⌊n2⌋, i.e., the minimal number of nodes in a quadrature rule of degree 2n−1 in two dimensions. Firstly, a new

Optical tomography is a typical non-invasive medical imaging technique, aiming to reconstruct the optical properties of tissues from boundary measurements by passing near infrared light through the tissues. We are concerned with the forward and inverse problems for a time-dependent diffusion equation model arising in diffuse optical tomography. To solve the forward problem, we develop a boundary integral

In this paper, a hybrid numerical technique is developed to solve the Benjamin-Bona-Mahony-Burgers (BBMB) equation. This technique involves the application of quintic Hermite collocation method (QHCM) and weighted finite difference scheme. The discretization of BBMB equation is done using collocation method with Hermite splines of order five along spatial direction and using weighted finite difference

This paper proposes a discontinuous Galerkin method to solve the generalized Sobolev equation. In this numerical procedure, the temporal variable has been discretized by the Crank–Nicolson idea to get a time-discrete scheme with the second-order accuracy. Then, in the second stage the spatial variable has been discretized by the discontinuous Galerkin finite element method. A prior error estimate has

This paper presents some computer-assisted procedures to prove the invertibility of a second-order linear elliptic operator and to compute a bound for the norm of its inverse. These approaches are based on constructive L2-norm estimates of the Laplacian and improve on previous procedures that use projection and a priori error estimations. Several examples which confirm the actual effectiveness of the

For Laplace's equation in a bounded simply-connected domain Ω in 3D, the method of fundamental solutions (MFS) is studied in this paper. Although some numerical computations can be found in Chen et al. [9], the theoretical analysis is much behind (Li [23] only for unit sphere Ω). Our efforts are devoted to exploring a strict error analysis of the MFS. The error bounds are derived, and the optimal polynomial

In this paper, a non-uniform mesh optimal B-spline collocation method is presented for the numerical solution of a singular two-point boundary value problem describing electrohydrodynamic flow (EF) of a fluid in a circular cylindrical conduit. The EF problem is highly nonlinear and has a singularity at the point r=0. Further, this problem has a boundary layer near right end of the solution domain.

In this paper, we propose the balanced Euler method of a class of stochastic Volterra integro-differential equations with non-globally Lipschitz continuous coefficients. The moment boundedness and strong convergence are shown. Moreover, the theoretical results are illustrated by some numerical examples.

In the paper, a new lowest equal-order stabilized mixed finite element method is proposed for a poroelasticity model in displacement-pressure formulation, which is based on multiphysics approach. The original model is reformulated to reveal the multi-physical process of deformation and diffusion and get a coupled fluid system. Then, a time-stepping algorithm which decouples the reformulated problem

Production-destruction systems (PDS) of ordinary differential equations (ODEs) are used to describe physical and biological reactions in nature. The considered quantities are subject to natural laws. Therefore, they preserve positivity and conservation of mass at the analytical level. In order to maintain these properties at the discrete level, the so-called modified Patankar-Runge-Kutta (MPRK) schemes

Fractional differential equations have become an important modeling technique in describing various natural phenomena. A variety of numerical methods for solving fractional differential equations has been developed over the last decades. Among them, finite difference methods are most popular owing to relative easiness for implementation. In this paper, we show that the finite difference method with

A Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev–Gauss–Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equation. The proposed method enables us to derive optimal error estimates

The truncated Euler-Maruyama (EM) method is proposed to approximate a class of non-autonomous stochastic differential equations (SDEs) with the Hölder continuity in the temporal variable and the super-linear growth in the state variable. The strong convergence with the convergence rate is proved. Moreover, the strong convergence of the truncated EM method for a class of highly non-linear time-changed

In this paper, the backward heat conduction problem is solved numerically using a fourth-order compact difference scheme. For the regularization of this ill-posed problem, fourth-order compact filtering which is a spatial filtering technique is proposed. The second derivative approximation in compact difference scheme has been compared with the fourth-order standard finite difference scheme via Fourier

Meshfree methods (MMs) enjoy advantages in discretizing problem domains over mesh-based methods. Extensive progress has been made in the development of the MMs in the last three decades. The commonly used MMs, such as the reproducing kernel particle methods (RKP), the moving least-square methods (MLS), and the meshless Petrov-Galerkin methods, have main difficulties in numerical integration and in

This paper deals with the efficient numerical solution of the two-dimensional partial integro-differential complementarity problem (PIDCP) that holds for the value of American-style options under the two-asset Merton jump-diffusion model. We consider the adaptation of various operator splitting schemes of both the implicit-explicit (IMEX) and the alternating direction implicit (ADI) kind that have

The design of efficient numerical methods, which produce an accurate approximation of the solutions, for solving time-dependent Schrödinger equation in the semiclassical regime, where the Planck constant ε is small, is a formidable mathematical challenge. In this paper a new method is shown to construct exponential splitting schemes for linear time-dependent Schrödinger equation with a linear potential

SimRank is popularly used for evaluating similarities between nodes in a graph. Though some recent work has transformed the SimRank into a linear system which can be solved by using conjugate gradient (CG) algorithm. However, the diagonal preconditioner used in the algorithm still left some room for improvement. There is no work showing what should be an optimal preconditioner for the linear system

This paper develops an accurate computational method based on the shifted Chebyshev cardinal functions (CCFs) for a new class of systems of nonlinear variable-order fractional quadratic integral equations (QIEs). In this way, a new operational matrix (OM) of variable-order fractional integration is obtained for these cardinal functions. In the proposed method, the unknown functions of a system of nonlinear

Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schrödinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive

In this work, the Discontinuous Galerkin (DG) schemes were studied for the solution of convection-diffusion-reaction equations which arise from mathematical modeling of fluidized bed spray granulation process (FBSG). The discontinuous Galerkin method for space discretization is employed to treat the dominated convection behavior in the governing equations. The implicit Euler method for the temporal

We intend to numerically solve the famous two-dimensional Bratu's problem through developing a new iterative finite difference algorithm. The proposed scheme is capable of accurately approximating both branches of the solution of the considered problem. We first introduce a new transformation of Bratu's problem conserving the solution bifurcated behavior. Then, using Newton-Raphson-Kantorovich approximation

We derived two adaptive choices for the nonnegative parameter of the Hager-Zhang conjugate gradient method. The first was achieved by minimizing the Frobenius condition number of the search direction matrix and the other by minimizing the difference between the smallest and the largest singular value. Global convergence is based on an appropriate assumption. Preliminary numerical results demonstrate

In this article a two-sided variable coefficient fractional diffusion equation (FDE) is investigated, where the variable coefficient occurs outside of the fractional integral operator. Under a suitable transformation the variable coefficient equation is transformed to a constant coefficient equation. Then, using the spectral decomposition approach with Jacobi polynomials, we proved the wellposedness

In the current research paper, the generalized moving least squares technique is considered to approximate the spatial variables of two time-dependent phase field partial differential equations on the spheres in Cartesian coordinate. This is known as a direct approximation (it is the standard technique for generalized finite difference scheme [69], [77]), and it can be applied for scattered points

We consider the problem of quantifying uncertainty for converging beam triple LIDAR when the input uncertainty follows a uniform distribution. We determine expressions for the range (i.e. set of reachable points) for the reconstructed velocity vector as a function of any particular setting of the nominal input parameters and determine an explicit lower (and upper) bound on the (averaged) volume (with

In order to approximate the Caputo fractional derivative of order α, 0<α<1, we construct here a new class of formulae on the basis of B-spline interpolation. These new formulae called S1, S2 and S3 have 2−α, 3−α and 4−α order of convergence, respectively. The proposed formulae are as simple as the well-known L1 formula and the main advantage of them lies in the fact that their accuracy is fixed in

The compact finite difference scheme is designed to numerically solve the nonlinear Schrödinger equation and coupled nonlinear Schrödinger equations on an unbounded domain in this paper. The original problem on an unbounded domain is reduced to an initial boundary value problem defined on a bounded computational domain by applying the artificial boundary method. Then, the reduced problem on the bounded

Here we consider the second order Inhomogeneous Linear Initial Value Problems with constant coefficients. Two–step hybrid methods (i.e. of Numerov–type) are considered for addressing this problem. A special set of order conditions is given and solved for derivation a low cost new method. Actually, we manage to save one stage (function evaluation) per step for this type of problems in comparison with

In this paper, we first present the nonlinear stability and convergence of extended Runge-Kutta-Nyström (ERKN) integrators for nonlinear multi-frequency highly oscillatory second-order ODEs with a takanami number and dominant frequency matrix. We then rigorously analyse the global errors of the blend of the ERKN time integrators and the Fourier pseudospectral spatial discretisation (ERKN-FP) when applied

We present and analyze an adaptive finite element method for a semilinear parabolic interface problem subject to nonzero flux jump in a two-dimensional bounded convex polygonal domain. The residual-based a posteriori error estimates are derived using energy argument. Our strategy is to avoid solving the nonlinear system by considering a linearized fully discrete scheme. An adaptive algorithm is constructed

In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrödinger equation. The scheme is constructed by using the L2-1σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Grünwald formula is adopted for the spatial discretization. The stability and unique solvability of the difference scheme are analyzed in detail

In this article, a parameter-uniform implicit scheme is constructed for a class of parabolic singularly perturbed reaction-diffusion initial-boundary value problems with large delay in the spatial direction. In general, the solution of these problems exhibits twin boundary layers and an interior layer (due to the presence of the delay in the reaction term). Crank-Nicolson difference formula (on a uniform

In this paper, a three-level compact alternating direction implicit (ADI) scheme is formulated for the two-dimensional coupled nonlinear Schrödinger system. By using an important new lemma and a mathematical induction method, it is proved that the compact ADI scheme is unconditionally convergent in L∞ norm with the convergence order of two in time and four in space. Numerical experiments have been

In this research, we present an iterative spectral method for the approximate solution of a class of Lane–Emden equations. In this procedure, we initially extend the Legendre wavelet which is appropriate for any time interval. Thereafter, the Guass-Legendre collection points of the Legendre wavelet are acquired. Employing this new approach, the iterative spectral technique converts the differential

A step-2 backward differential formula (BDF) temporal discretization scheme is constructed for nonlinear reaction-diffusion equation and superconvergence results are studied by mixed finite element method (FEM) with the elements Q11 and Q01×Q10 unconditionally. In particular, we apply an artificial regularization term to guarantee the energy stability of the step-2 BDF scheme. Splitting technique is

We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. This class of equations is numerically challenging to solve - the computational complexity of “straightforward” algorithms grows exponentially with respect to the number of internal coordinates describing particle properties. Here, we develop algorithms which reduce the storage and

The multiblock mortar expanded mixed method is explored to solve the second order linear elliptic problem. In this method, the domain is expressed as a union of smaller blocks (subdomains) separated by interface. The original problem is posed on each block and discretized locally. A scalar unknown is introduced on the shared boundaries between the blocks which is treated as Dirichlet boundary condition

We consider extension of quasilinearization technique and method of lower and upper solutions for nonlinear weakly singular Volterra integral equations. We obtain linear weakly singular integral equations and discuss on their existence, uniqueness and regularity properties of a solution. On the other hand, we show that the solutions of these linear equations are quadratically convergent to the solution

In this paper, high-order Runge-Kutta discontinuous Galerkin (RKDG) methods with multi-resolution weighted essentially non-oscillatory (WENO) limiters are designed for solving hyperbolic conservation laws on triangular meshes. These multi-resolution WENO limiters are new extensions of the associated multi-resolution WENO finite volume schemes [49], [50] which serve as limiters for RKDG methods from

This paper provides a novel inversion free variant of the fixed point iteration strategy to obtain a maximal positive definite solution for the nonlinear matrix equation X+A⁎X−1A=Q. The conditions for the convergence of the method and for the existence of solution of this nonlinear matrix equation are derived. At the end, numerical examples are also presented to show the behavior of the proposed technique

The high velocity, variety and volume of data generation by today's systems have necessitated Big Data (BD) analytic techniques. This has penetrated a wide range of industries; BD as a notion has various types and characteristics, and therefore a variety of analytic techniques would be required. The traditional analysis methods are typically unable to analyse spatial-temporal BD. Interpolation is required

The discretizations of the left and the right fractional derivatives based on the shifted finite-difference formulas of the Grünwald-Letnikov type can result in Toeplitz matrices T and T⁎. Combining with the generating function of Toeplitz matrix T, we analyse the dominant property of the Hermitian part of T relative to its skew-Hermitian part. Then we construct a dominant Hermitian splitting iteration

We introduce a modified QHSS (MQHSS) iteration method for solving a class of complex symmetric linear systems, and prove that this MQHSS iteration method is convergent under certain conditions. We also consider acceleration of the MQHSS iteration by Krylov subspace methods. Numerical experiments show the good performance of the MQHSS iteration method.

In this paper, numerical solutions of space-fractional advection-diffusion equations, involving the Riemann-Liouville derivative with a nonlinear source term, are presented. We propose a procedure that combines the fractional Lie symmetries analysis, to reduce the original fractional partial differential equations into fractional ordinary differential equations, with a numerical method. By adopting

We present a new two-layer closed form model for the dynamics of desert dunes under the effect of a horizontal wind blowing in an arbitrary direction. This model is an extension of a very simplified model previously introduced by Hadeler and Kuttler [12]. Our extension, inspired by the sandpile dynamics approach, includes the effects of gravity on both sides (upwind and downwind) of the dune, and allows

We generalize the nonstandard Euler and Heun schemes in order to provide explicit geometric numerical integrators for biochemical systems, here denoted as GeCo schemes, that preserve both positivity of the solutions and linear invariants. We relax the request on the order convergence of the denominator function for the first-order approximation and we let it depend on the step size also throughout

The Dynamic Programming approach allows to compute a feedback control for nonlinear problems, but suffers from the curse of dimensionality. The computation of the control relies on the resolution of a nonlinear PDE, the Hamilton-Jacobi-Bellman equation, with the same dimension of the original problem. Recently, a new numerical method to compute the value function on a tree structure has been introduced

A continuation of previous authors' work on adaptive parameter estimation for linear dynamical systems having irrational transfer function is presented in this work. An original modification of the gradient algorithm, inspired by the variable structure control techniques and additionally featuring a time-varying adaptation gain, is presented and analyzed using Lyapunov techniques. The exposition is

In this paper we validate the implementation of the numerical scheme proposed in [3]. The validation is made by comparison with an explicit solution here obtained, and the solutions of Riemann problems for several networks. We then perform some simulations in order to qualitatively validate the model under consideration. Such results represent also a first step for the validation of the finite volumes

We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show

In this work we show that numerical integration during sliding motion, for piecewise-smooth systems, can be performed without the need to project the approximate solutions on the constraints' surface. We give a general algorithm, an order of approximation result, and show the effectiveness of the technique on several examples. Comparison between projected and non-projected methods and the standard

Multi-derivative one-step methods based upon Euler–Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any multi-derivative Runge–Kutta methods, we show that the Euler–MacLaurin method of order p is conjugate-symplectic up to order p+2. This feature entitles them to play a role

The dynamic behaviour of rigid blocks subjected to support excitation is represented by discontinuous differential equations with state jumps. In the numerical simulation of these systems, the jump times corresponding to the numerical trajectory do not coincide with the ones of the given problem. When multiple state jumps occur, this approximation may affect the accuracy of the solution and even cause

We extend the nonconforming Trefftz virtual element method introduced in [30] to the case of the fluid-fluid interface problem, that is, a Helmholtz problem with piecewise constant wave number. With respect to the original approach, we address two additional issues: firstly, we define the coupling of local approximation spaces with piecewise constant wave numbers; secondly, we enrich such local spaces

We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep

This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss–Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and