The analysis of diffusion-reaction equations with general two-sided fractional derivative characterized by a parameter p∈[0,1] is investigated in this paper. First, we present a Petrov-Galerkin method, derive a proper weak formulation and show the well-posedness of its weak solution. Moreover, on the basis of the two-sided Jacobi polyfractonomials, a priori error analysis of Petrov-Galerkin method

In this paper, we propose approximate solvers for two-phase fluids with general equations of state (EOS) in high dimension. The standard finite volume scheme is used for each fluid away from material interface. The two-phase interfaces are captured by the level set method coupled with a multi-resolution algorithm. For the Riemann solver of two-phase fluids with general equations of state, we construct

In this paper, a second order Gauge Uzawa scheme for the governing equations of an incompressible micropolar fluid flow is introduced and analyzed. The derivation of this scheme is obtained using the second order backward difference approximation. Later, the unconditional stability of the GUM scheme for the micropolar equations will be shown. Then, an a priori error estimate is established to prove

We are interested in finding an approximation for the solution of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst parameter H>12. Based on Taylor expansion we derive a numerical scheme and investigate its convergence. Under some assumptions on drift and diffusion, we show that the introduced method is convergent with strong rate of convergence ΔH, where

We propose an enriched Galerkin-characteristics finite element method for numerical solution of convection-dominated problems. The method uses the modified method of characteristics for the integration of the total derivative in time, combined with the finite element method for the spatial discretization on unstructured grids. The L2-projection method is implemented for the evaluation of numerical

In this paper, we explore the existence and uniqueness theorem for a problem of the fractional Cauchy form, with dependence on the generalized proportional Caputo derivative. Furthermore, a new numerical technique is presented based on a decomposition formula for the generalized proportional Caputo derivative. Convergence analysis of the proposed technique is proved. Finally, numerical results are

In this paper, we revisit two recently published papers on the iterative approximation of fixed points by Kumam et al. [Numer. Funct. Anal. Optim. 40 (2019), 1644–1677] and Maniu [Numer. Funct. Anal. Optim. 41 (2020), 929-949] and reproduce convergence, stability, and data dependency results presented in these papers by removing some strong restrictions imposed on parametric control sequences. We confirm

In this paper, we focus on the fast computation for solving the fourth-order time fractional diffusion wave equations with variable coefficient. A fast difference scheme is derived by applying the FL2−1σ formula to approximate the time Caputo fractional derivative. The resulting FL2−1σ scheme keeps the almost same accuracy with the traditional L2−1σ scheme, but it reduces the computational complexity

In fluid-structure interaction problems, many people use a penalty method for positioning the structure inside the fluid. This is usually performed by considering that the fluid is very stiff or/and very heavy at the place occupied by the structure. These methods are very convenient for the programming point of view but can lead to ill conditioned operators. This is a drawback in the numerical solution

We perform a spectral analysis of matrices arising from isogeometric discretizations based on hyperbolic and trigonometric generalized B-splines. Second-order differential problems with variable coefficients are considered and discretized by means of sequences of both nested and non-nested generalized spline spaces. We prove that an asymptotic spectral distribution always exists when the matrix-size

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. In order to get an optimal approximation in spaces of locally continuous functions equipped with weighted uniform norms, the related Lebesgue constants have to be uniformly bounded. In previous works this has already been proved under different

An inverse problem of identifying an unknown space-dependent source term in a time-space fractional parabolic equation is considered in this paper. Under reasonable boundedness assumptions about the source function, a Hölder-type stability estimate of optimal order is proved. To regularize the inverse source problem, a mollification regularization method is applied. Error estimates of the regularized

The G-transformation is an efficient method for solving the weighted least squares problems. However, the underflows were not considered in the original G-transformation. In order to keep its stability, some authors proposed three specified scaling strategies for guarding against the underflows. Note that these three specific strategies are not easy to be implemented in actual operations, in this paper

In this paper, we propose and analyze a decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on partitioned time-stepping methods, the mixed system with a stabilization term is decoupled, which means that the Navier-Stokes equations and two different Darcy equations

This paper considers the computation of the greatest common divisor (GCD) dt1,t2(x,y) of three bivariate Bernstein polynomials that are defined in a rectangular domain, where t1(t2) is the degree of dt1,t2(x,y) when it is written as a polynomial in x(y) whose coefficients are polynomials in y(x). The Sylvester resultant matrix and its subresultant matrices are used for the computation of the degrees

This paper discusses several transform-based methods for solving linear discrete ill-posed problems for third order tensor equations based on a tensor-tensor product defined by an invertible linear transform. Linear transform-based tensor-tensor products were first introduced in Kernfeld et al. (2015) [16]. These tensor-tensor products are applied to derive Tikhonov regularization methods based on

We extend the diffusion-map formalism to data sets that are induced by asymmetric kernels. Analytical convergence results of the resulting expansion are proved, and an algorithm is proposed to perform the dimensional reduction. A coordinate system connected to the tensor product of Fourier basis is used to represent the underlying geometric structure obtained by the diffusion-map, thus reducing the

This paper deals with the discrete maximum principle and energy stability of a new difference scheme for solving Allen-Cahn equations. By combining the second-order central difference approximation in space and the Crank-Nicolson method with Newton linearized technique in time, a two-level linearized difference scheme for Allen-Cahn equations is derived, which can yield accuracy of order two both in

This paper presents a Legendre wavelet spectral method for solving a type of fractional Fredholm integro–differential equations. The fractional derivative is defined in the Caputo–Prabhakar sense. The derivative of Prabhakar consists of an integro–differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes the Riemann–Liouville and Caputo

In this paper we consider a fluid-fluid system coupled with a friction-type interface condition which is described by a jump of the velocities along the tangential interface. Based on domain decomposition, we propose a Schwarz type iterative method to decouple the different physical process in each subdomain occupied by a single fluid, which allows solving in each subdomain only the physical process

In this paper, high-order compact difference method is used to solve the one-dimensional nonlinear advection diffusion reaction equation. The nonlinearity here is mainly reflected in the advection and reaction terms. Firstly, the diffusion term is discretized by using the fourth-order compact difference formula, the nonlinear advection term is approximated by using the fourth-order Padé formula of

In this paper, we consider the magnetohydrodynamics (MHD) equations with variable density, which are a coupled system by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. Such MHD system describes the motions of several conducting incompressible immiscible fluids without surface tension in presence of a magnetic field. A first-order Euler semi-implicit time

A numerical scheme based on the finite volume approach is developed to solve a binary breakage population balance equation (PBE) on nonuniform meshes. The key feature of the new scheme is that it is free of the common requirement of redistributing the particle mass to neighboring pivots, its formulation is simpler compared to other methods such as cell average and fixed pivot techniques. The new scheme

The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical foundation, or equivalently a numerical infrastructure. This foundation should include a collection of suitable elements (usually hypercubes, simplices, or closely related

In this paper, we present a numerical technique for solving the time-fractional Black-Scholes (TFBS) equation describing European options. The time-fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied. Two test problems with the known analytical

Spherulites are growth patterns of average spherical form which may occur in the polycrystallization of binary mixtures due to misoriented angles at low grain boundaries. The dynamic growth of spherulites can be described by a phase field model where the underlying free energy depends on two phase field variables, namely the local degree of crystallinity and the orientation angle. For the solution

This paper provides an effective method for a class of 2-dim nonlinear variable order fractional optimal control problems (2DNVOFOCP). The technique is based on the new class of basis functions namely the generalized shifted Legendre polynomials. The dynamic constraint is described by a nonlinear variable order fractional differential equation where the fractional derivative is in the sense of Caputo

In the last twenty years, the analysis of problems involving dual-phase-lag models has received an increasing attention. In this work, we consider the coupling between one of these models and the microtemperatures effects. In order to overcome the infinite speed paradox, two relaxation parameters are introduced for each evolution equation related to the temperature and the microtemperatures, leading

In this work, a class of inexact block SSOR-like preconditioners are proposed for the non-Hermitian positive definite linear system whose coefficient matrix is of a blockwise form. The novel preconditioners are based on the SSOR-like iteration method proposed by Bai (Numer. Linear Algebra Appl. 23, 37-60, 2016), but each of the diagonal blocks in the SSOR-like method is replaced by an approximation

In this paper, we study the numerical method for solving forward-backward stochastic differential equations driven by G-Brownian motion (G-FBSDEs) which correspond to fully nonlinear partial differential equations (PDEs). First, we give an approximate conditional G-expectation and obtain some feasible methods to calculate the distribution of G-Brownian motion. On this basis, some efficient numerical

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta

In this paper, we consider a prototypical convex optimization problem with multi-block variables and separable structures. By adding the Logarithmic Quadratic Proximal (LQP) regularizer with suitable proximal parameter to each of the first grouped subproblems, we develop a partial LQP-based Alternating Direction Method of Multipliers (ADMM-LQP). The dual variable is updated twice with relatively larger

It has been reported that convergence stalls on Bakhvalov-Shishkin mesh in the case of N−1≤ε, where ε is the singular perturbation parameter and N is the number of mesh intervals. To analyze these phenomena, we present uniform convergence analysis of finite element methods on Bakhvalov-type meshes, one popular kind of graded meshes closely related to Bakhvalov-Shishkin mesh, when N−1≤ε. An optimal

The error of approximation of a function using Poly-Sinc approximation was shown to have a convergence rate of exponential order over a global partition. However, it was assumed that the function values at the interpolation points were known. In this paper, we extend our work on poly-Sinc-based discontinuous Galerkin approximation, in which the function values at the interpolation points are replaced

We establish the uniform convergence of the control polygons generated by repeated degree elevation of q-Bézier curves (i.e., polynomial curves represented in the q-Bernstein bases of increasing degrees) on [0,1], q>1, to a piecewise linear curve with vertices on the original curve. A similar result is proved for q<1, but surprisingly the limit vertices are not on the original curve, but on the q−1-Bézier

In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank–Nicolson scheme in time. A third-order scheme for the tempered Black–Scholes equation is also proposed

In this paper, a derivative-free RMIL conjugate gradient projection method for solving large-scale nonlinear monotone equations with convex constraints is proposed. The proposed method is a modification of an RMIL conjugate gradient method combined with the projection techniques. We establish its global convergence under appropriate conditions. The method is then compared with other existing methods

In this article, the non-singular variable-order fractional derivative in the Heydari-Hosseininia concept is used to formulate the variable-order fractional form of the coupled nonlinear Ginzburg-Landau equations. To solve this system, a numerical scheme is constructed based upon the shifted Vieta-Lucas polynomials. In this method, with the help of classical and fractional derivative matrices of the

In this paper we study the numerical approximation of the solution of a Cauchy problem for a first-order-in-time differential equation involving a fractional power of a self-adjoint positive operator. One popular approach for the approximation of fractional powers of such operators is based on rational approximations. The purpose of this work is to construct special approximations in time so that the

In this paper we propose and analyze some a posteriori error estimator for the lowest-order triangular Raviart–Thomas mixed finite element method. This error estimator is a simple modification of the ones proposed in Vohralík (2007) [34] and Ainsworth (2008) [1], and relies on the use of a superconvergent vector approximation. It is shown that the new error estimator not only guarantees an upper bound

In this paper, we propose the stabilized penalty-projection finite element method for solving the Navier-Stokes-Cahn-Hilliard-Oono system, which is used to simulate the movement of thrombus in the blood vessels. The proposed algorithms are based on mixed finite element method in spatial direction, combined with a backward-Euler scheme and penalty-projection scheme for temporal discretization. It is

In this work, a class of fractional delay differential equations with four-point boundary condition and p-Laplacian operator is discussed. Based on the Avery-Peterson theorem, the existence of at least triple positive solutions is derived. A simple example is given to show the validity of the conditions of our main theorem.

In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar

We are concerned with robust iterative solution methods for solving the Stokes optimal control problems. Two efficient preconditioners are proposed for the discretized saddle point linear systems arising from the velocity tracking of the Stokes control problem. The proposed preconditioners are similar in structure to and can be viewed as modifications of the preconditioner in Axelsson et al. (2017)

This paper considers an explicit continuation method and the trust-region updating strategy for the unconstrained optimization problem. Moreover, in order to improve its computational efficiency and robustness, the new method uses the switching preconditioning technique. In the well-conditioned phase, the new method uses the L-BFGS method as the preconditioning technique in order to improve its computational

This paper deals with the construction of a computational approach based on B-spline functions for solving a nonlinear boundary value problem describing the electro-hydrodynamic flow (EHF) of a fluid in a circular cylindrical conduit. The radial dependence of the velocity field emerging in the EHF is computed. We study the effects of two relevant parameters, namely the Hartmann electric number H and

In this paper, we give a stabilized second order scheme for the time fractional Allen-Cahn equation. The scheme uses the fractional backward difference formula (FBDF) for the time fractional derivative and the Legendre spectral method for the space approximation. The nonlinear terms are treated implicitly with a second order stabilized term. Based on the fractional Grönwall inequality, we strictly

In this paper, we study the Hamiltonian structure and develop a novel energy-preserving scheme for the two-dimensional fractional nonlinear Schrödinger equation. First, we present the variational derivative of the functional with fractional Laplacian to derive the Hamiltonian formula of the equation and obtain an equivalent system by defining a scalar variable. An energy-preserving scheme is then presented

In the paper, the high-order conservative schemes are presented for space fractional nonlinear Schrödinger equation. First, we give two class high-order difference schemes for fractional Risze derivative by compact difference method and extrapolating method, and show the convergence analysis of the two methods. Then, we apply high-order conservative difference schemes in space direction, and Crank-Nicolson

The present work introduces a numerical scheme that preserves the dissipation of energy of the Higgs boson equation in the de Sitter space-time. More precisely, the model considered in this work is a mathematical generalization of Higgs' model which includes a general time-dependent diffusion coefficient and a generalized potential. The mathematical system is dissipative, and we propose an implicit

We know that the exact solutions for most of stochastic age-structured capital systems are difficult to find. The numerical approximation method becomes an important tool to study properties for stochastic age-structured capital models. In the paper, we study the numerical solution for stochastic age-dependent capital system based on explicit-implicit algorithm and discuss the convergence of numerical

A BDDC preconditioner with adaptive coarse space for advection-diffusion problems discretized by stabilized finite element method is proposed. Since the bilinear form of the corresponding variational form is nonsymmetric and positive definite (NSPD), the adaptive BDDC preconditioner, which is always used for solving the symmetric and positive definite (SPD) problems, is extended to solve the nonsymmetric

In this paper, we focus on a numerical method for computing the ground state of the fractional stationary Schrödinger equation (FSSE). We first construct a continuous normalized fractional gradient flow (CNFGF) and prove its L2− norm conservation and energy diminishing properties. Then we introduce the fractional gradient flow with discrete normalization (FGFDN) to solve the CNFGF, and give the fully

The purpose of this paper is to investigate a high order numerical method for solving time-fractional Black-Scholes equation in which the fractional operator is defined by the Caputo fractional derivative. The proposed space-time spectral method employs the Jacobi polynomials for the temporal discretisation and Fourier-like basis functions for the spatial discretisation. The stability and convergence

In this study, we provide a novel third-order weighted essentially non-oscillatory (WENO) method to solve Hamilton-Jacobi equations. The key idea is to incorporate exponential polynomials to construct numerical fluxes and smoothness indicators. First, the new smoothness indicators are designed by using the finite difference operator annihilating exponential polynomials such that singular regions can

In this paper, a parallel, non-spatial iterative, and rotational pressure projection method for the coupled Navier-Stokes equations is proposed and developed. As for each Navier-Stokes equation, one elliptic equation and one Possion equation at each time step determine the values of the velocity and pressure, respectively. Moreover, we only need to solve four simple linear equations, and the computational

In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in H−1 and Hαs (s=1,2,3) spaces independent of time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes. The uniform Hα3 bound is obtained on basis of the uniform H1

A family of novel time-stepping methods for the fractional calculus operators is presented with a shifted parameter. The truncation error with second-order accuracy is proved under the framework of the shifted convolution quadrature. To improve the efficiency, two aspects are considered, that i) a fast algorithm is developed to reduce the computation complexity from O(Nt2) to O(Ntlog⁡Nt) and the memory

We analyze the penalty and Nitsche's methods, in the continuous and discrete senses, for the Stokes–Darcy system with a curved interface. In the continuous sense, we prove the stability and optimal convergence for the penalty approach. In discretization, the curved interface is approximated by polygonal surface. We propose the discontinuous Galerkin (DG) penalty/Nitsche schemes, and establish the stability

In this paper, we first study a relaxed inertial projection and contraction method for variational inequalities to obtain weak convergence results under standard assumptions in Hilbert spaces. Next, we propose another inertial projection and contraction method for which the inertial factor θ is chosen in [0,1] with θ=1 possible. Weak and linear convergence are obtained for this second proposed method