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In order to find the numerical solutions of the two-dimensional (2D) uniform transmission line equation, we first establish a fully second-order finite difference (FD) scheme in matrix-form and analyze the stability and convergence of the FD solutions by matrix analysis. Then we employ a proper orthogonal decomposition (POD) method to establish a reduced-order extrapolated FD (ROEFD) scheme containing very few unknowns for the 2D uniform transmission line equation and also make use of the matrix analysis to analyze the stability and convergence for the ROEFD solutions. Finally, we employ some numerical experiments to validate the effectiveness of the ROEFD scheme.

In this paper we present a stabilizer-free weak Galerkin (SFWG) finite element scheme for the approximation of the non-self adjoint and indefinite elliptic equations. Supercloseness convergence of the SFWG method is derived. After a suitable postprocessing of the SFWG solution, the resulting post-processed solution converges with two order higher than the optimal order in both H 1 semi-norm and L 2 norm on triangular meshes. Numerical experiments are demonstrated to verify the theoretical findings.

The paper introduces a new weak approximation scheme for Kolmogorov type hypoelliptic diffusions based on an operator splitting method and a Malliavin calculus approach. The approximation works even if the test function is not smooth, in other words, irregular functionals of hypoelliptic diffusions appearing in various fields are effectively approximated. A simple numerical algorithm is provided, which is easily implemented by a simulation method with minimum cost on random number generation. The effectiveness of the scheme is confirmed through numerical examples for irregular functionals of hypoelliptic diffusions.

In this paper, a new combined second order fully discrete scheme of an H 1 -Galerkin finite element method ( H 1 -GFEM) with two grid method (TGM) is developed for the bacterial equations. Based on Q 11 / Q 1 , 0 × Q 0 , 1 element pair, the existence and uniqueness of the approximation solution are proved, and the superclose and superconvergent estimates of order O ( h 2 + H 4 + τ 2 ) for the original variables in H 1 -norm and intermediate variables in L 2 -norm are derived, respectively. Some numerical results are provided to show that the proposed method has a very good performance and its computing cost is only about half of H 1 -GFEM for the considered problem. Here, the sizes of coarse mesh, fine mesh and time step are denoted by H , h , τ , respectively.

In this paper, we focus on the numerical approximation of the Cahn-Hilliard-Hele-Shaw system. Firstly, based on the idea of the stabilized method, an unconditionally stable linear scheme with second-order accuracy in time and space is proposed, which is modified from the Crank-Nicolson scheme. Secondly, we derive that the proposed numerical scheme is unconditionally stable, without any restriction for the time step size. After a careful calculation, we get discrete error estimates of the time step size τ and space step size h. Finally, numerical simulations of energy dissipation and spinodal decomposition are presented to demonstrate the stability, accuracy and efficiency of the proposed scheme.

Based on the method of finite differences we propose a numerical blow-up time for a system of nonlinear ordinary differential equations, such system is defined by means of some parameters. Under certain conditions, on the parameters, the system blow-up in finite time. We introduce a numerical time and prove that this converges to the exact blow-up time and an error estimate is also given.

This paper is concerned with numerical solutions of one-dimensional (1D) and two-dimensional (2D) nonlinear coupled Schrodinger-Boussinesq equations (CSBEs) by a type of linearly energy- and mass- preserving finite difference methods (EMP-FDMs) because the existing EMP-FDMs for CSBEs are nonlinear and time-consuming, and corresponding theoretical analyses are not easy to generalize high-dimensional problems. Firstly, a linearized EMP-FDM is created for solving 1D CSBEs. By using the discrete energy analysis method, it is shown that this scheme is uniquely solvable and convergent with an order of O ( Δ t 2 + h x 2 ) in L ∞ -, H 1 - and L 2 -norms, and corresponding numerical energy and mass are conservative. Then, by generalizing this type of EMP-FDM, a linearized EMP-FDM is developed for solving 2D CSBEs. Theoretical findings including the convergence, the discrete conservative laws, and the solvability of this numerical algorithm are strictly derived in detail by using the discrete energy analysis method as well. Finally, numerical results confirm the efficiency of our algorithms and the exactness of the theoretical results.

In this paper, the difference method with intrinsic parallelism is studied in order to meet the needs of quickly solving the variable-coefficient compound KdV-Burgers equation. The classical Crank-Nicolson (C-N) scheme is segmented on the basis of the alternating segment difference technique. And four different types of Saul'yev asymmetric schemes are alternately used at the segment points. Then we obtain the alternating segment Crank-Nicolson (ASC-N) difference scheme for the variable-coefficient compound KdV-Burgers equation. Theoretical analyses prove the existence and uniqueness of solution to ASC-N scheme, the linear absolute stability and the second-order convergence in time and space. Numerical experiments show that when the computational accuracy of the ASC-N scheme is similar to that of the C-N scheme, the computational efficiency of the ASC-N scheme is significantly higher than that of the C-N scheme. It shows that the difference method with intrinsic parallelism in this paper can effectively solve the variable-coefficient compound KdV-Burgers equation.

A fully discrete ADI scheme is proposed for solving the two-dimensional time-fractional diffusion equation with weakly singular solutions, where L1 scheme on graded mesh is adopted to tackle the initial singularity. An improved discrete fractional Gronwall inequality is employed to give an α-robust H 1 -norm convergence analysis of the fully discrete ADI scheme, where the error bound does not blow up when the order of fractional derivative α → 1 − . Numerical results show that the theoretical analysis is sharp.

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 the convergence. Finally, we present some numerical simulations that confirm the theoretical results.

In this paper, we study a regularized Lie-Trotter splitting spectral method for a regularized space-fractional logarithmic Schrodinger equation by introducing a small regularized parameter 0 e ≪ 1 . The regularized method can be used to avoid numerical blow-up in the space-fractional logarithmic Schrodinger equation. The regularized space-fractional logarithmic Schrodinger equation is proved to approximate the space-fractional logarithmic Schrodinger equation with linear convergence rate O ( e ) . The proposed numerical scheme can preserve the discrete mass and step-energy for regularized space-fractional logarithmic Schrodinger equation. The first order convergence in time for the numerical method is rigorous proved for the regularized space-fractional logarithmic Schrodinger equation. Due to the appearance of fractional Laplace operator with order α, we can adjust the parameters of order to make the equation show much more dynamic characteristics than the classical logarithmic Schrodinger equation. Numerical simulations for 1D case based on the Fourier spectral approximation in space are presented to validate the theoretical analysis.

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 time and space. Under appropriate conditions, the scheme is proved to be uniquely solvable and able to preserve the maximum principle and energy stability of the equations in the discrete sense. With some numerical experiments, the theoretical results and computational effectiveness of the scheme are further illustrated.

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 be distinguished from smooth regions more efficiently. Moreover, to construct numerical flux, we employ an interpolation method based on exponential polynomials which yields improved results around steep gradients. The proposed scheme retains the optimal order of accuracy (i.e., three) in smooth areas, even near the critical points. To illustrate the ability of the new scheme, some numerical results are provided along with comparisons with other WENO schemes.

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 shifted Vieta-Lucas polynomials (which are extracted in this study), solving the studied problem is transformed into solving a system of nonlinear algebraic equations. The convergence analysis and the truncation error of the shifted Vieta-Lucas polynomials in two dimensions are investigated. Numerical problems are demonstrated to confirm the convergence rate of the presented algorithm.

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 and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.

In this paper, a novel numerical scheme is proposed to numerically solve the fractional activator-inhibitor system, which is a coupled nonlinear model. In the temporal direction, we employ two families of novel fractional θ-methods, the FBT-θ and FBN-θ methods, in spatial direction, the Legendre spectral method is used. Based on some positivity properties of the coefficients of both methods, the stability and convergence of the numerical scheme are proved. We derive an optimal convergence rate in space and second order accuracy in time. Finally, some numerical results are given to confirm the theoretical analysis.

This paper focuses on developing the reduced-order extrapolated model based on the splitting implicit finite difference (SIFD) scheme and the proper orthogonal decomposition (POD) for the two-dimensional (2D) fourth-order nonlinear Rosenau equation. For this purpose, we first construct the SIFD scheme and analyze the stability and convergence. And then, we develop a reduced-order extrapolated SIFD (ROESIFD) scheme for the Rosenau equation by POD technique and analyze the ability and convergence for the ROESIFD solutions. Finally, we enumerate an example to illustrate the efficacy and feasibility of the ROESIFD scheme.

This paper deals with the qualitatively correct numerical solution of differential equations with known invariants. We use general linear methods with standard projection technique to preserve the invariants numerically by projecting the solution onto the desired manifold. The projection scheme admits the integration without any drift from the desired manifold and is computationally cost effective. The numerical experiments presented in this paper show the advantages of the proposed schemes.

With the help of the generalized viscosity implicit method and hybrid steepest-descent method, we introduce an iterative scheme for approximating the solution of a variational inequality over the set of fixed points of an asymptotically nonexpansive mapping in the intermediate sense. Some strong convergence results for our proposed iterative scheme are established in the framework of Banach spaces. Applicability of our proposed method is shown in variational inclusion problem and convex minimization problem. We discuss some examples to demonstrate the numerical implementation and efficiency of our main results in comparison of other related results. Our results improve, extend and unify previously known results given in literature.

In this article, we present an accurate finite difference scheme for solving the KdV equation with time-periodic forcing at boundary. We first employ a variable transformation to change the time-periodic boundary to a homogeneous boundary. We then develop a three-level fourth-order accurate finite difference scheme for solving the transformed KdV problem. The stability, existence and uniqueness of the numerical solution are analyzed. Numerical examples are provided to confirm the accuracy of the scheme, and the effectiveness for time-periodic simulation.

In this paper, a number of energy stable numerical schemes are proposed for the fractional Cahn–Hilliard equation. We prove mass conservation, unique solvability and energy stability for three time semi-discretized schemes based on the first-order semi-implicit scheme, the Crank–Nicolson scheme and the BDF2 scheme respectively. Then we present error analysis for these numerical schemes with the Fourier spectral approximation in space. Some numerical experiments are finally carried out to confirm accuracy and effectiveness of these proposed methods.

In the past years several unconditionally positivity preserving numerical methods for production-destruction equations arising in a wide variety of real life applications were developed. The BBKS schemes suggested by Bruggemann et al. (2007) [4] and Broekhuizen et al. (2008) [3] modifications of Heun's method and conserve all linear invariants while still maintaining positivity independent of the time step size used. In this paper, we present a comprehensive generalization of these schemes as modifications of arbitrary first and second order Runge-Kutta methods with nonnegative parameters. This is achieved by the introduction of novel Patankar weights and sufficient and necessary conditions w.r.t. the Patankar weight denominators for first as well as second order consistency are proven. Furthermore, a convergence proof for the incorporated Newton scheme is given such that the efficiency compared to the original BBKS schemes, which make use of the bisection method is significantly improved. Numerical simulations are presented confirming the theoretical results concerning the validity, efficiency and accuracy of the new class of generalized BBKS schemes.

In this paper, the virtual element method for the approximation of Laplacian eigenvalue problem in mixed form is studied. We show that the discrete form satisfies the hypotheses required by the Brezzi-Babǔska theory. Under some assumptions on polygonal meshes, we employ the spectral theory of compact operators to prove the spectral approximation and the optimal order for the eigenvalues. Finally, some numerical results show that numerical eigenvalues obtained by the proposed numerical scheme can achieve the optimal convergence order.

This paper is concerned with the numerical solutions of nonlinear Schrodinger equation (NLSE) in one-dimensional unbounded domain. The unbounded problem is truncated by the third-order absorbing boundary conditions (ABCs), and the corresponding initial-boundary value problem is solved by a three-level linearized difference scheme. By introducing auxiliary functions to simplify the difference scheme, the proposed difference scheme for NLSE with third-order ABCs is theoretically analyzed. It is strictly proved that the difference scheme is uniquely solvable and unconditionally stable, and has second-order accuracy both in time and space. Finally, several numerical examples are given to verify the theoretical results.

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 been proposed for the semi-discrete scheme based on the spatial discretization. By applying the Crank–Nicolson idea a full-discrete scheme is driven. Furthermore, an error estimate has been proved to get the convergence order of the developed scheme. Finally, some numerical examples have been presented to show the efficiency and theoretical results of the new numerical procedure.

In this paper, we develop an implicit difference method for solving the nonlinear time-space fractional Schrodinger equation. The scheme is constructed by using the L2- 1 σ formula to approximate the Caputo fractional derivative, while the weighted and shifted Grunwald formula is adopted for the spatial discretization. The stability and unique solvability of the difference scheme are analyzed in detail. Moreover, we prove that the numerical solution is convergent with second-order accuracy in both temporal and spatial directions. Finally, a linearized iterative algorithm is provided and some numerical tests are presented to validate our theoretical results.

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 mesh) is used in time to semi-discretize the given PDE, and then the standard finite difference scheme (on a piecewise-uniform mesh) is used for the system of ordinary differential equations obtained in the semi-discretization. The convergence analysis shows that the method is e-uniformly convergent of order two in the temporal direction and almost first-order in the spatial direction. Two test examples are encountered to show the efficiency of the method, validate the computational results, and to confirm the predicted theory.

A stable and convergent Nystrom scheme is proposed to solve Fredholm integral equations (FIEs). Our approximation is based on the barycentric rational interpolants. By introducing barycentric quadratures to the integral operator that appears in the FIE and modifying the standard Nystrom scheme, we demonstrate that the new Nystrom scheme is a viable option for the numerical solution of FIEs. Convergence rates of the method are proved taking into account the effect of grading the domain. The final convergence result shows clearly that one can choose an optimal domain grading. Numerical examples and comparisons with competitive methods of tunable accuracy are provided to support the theoretical analysis and illustrate the efficiency of the proposed numerical scheme.

In the article Martinez et al. (2019) [7] , the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.

This work is devoted to obtain optimal temporal error estimates for a first-order time-splitting scheme for solving the incompressible Navier-Stokes equations with variable density. This scheme was firstly proposed by Li, Mei et al. (2013) [9] . First-order convergence rates have been numerically observed, but never hitherto proved. In this work, we will give a rigorous temporal error analysis for this scheme under some regularity assumptions.

In this research, the presented method combines advantages of the finite element and three-level finite difference techniques to obtain the solution of the two-dimensional general Rosenau-RLW equation. An important point is that nevertheless the two-dimensional general Rosenau-RLW is a non-linear equation, but with the proposed scheme we are able to linearize this system and efficiently solve it due to the implicit nature of the system of equations. In addition, the existence and uniqueness of the approximate solution are approved. The convergence and stability of the approximate solution are also examined. The general formulation of the procedure is given and numerical results are carried out to confirm the accuracy of our theoretical results and the efficiency of the scheme.

A fully discrete L 2 -stability analysis is carried out for linear advection-diffusion problems in one space dimension, discretized in space by discontinuous Galerkin methods based on the ( σ , μ ) -family of diffusion schemes and implicit-explicit Runge-Kutta schemes in time. Advection terms are discretized explicitly while diffusion terms are solved implicitly. Conditions on the parameters σ and μ are derived which guarantee L 2 -stability for time steps Δ t = O ( d / a 2 ) , where a and d denote the advection and diffusion coefficient, respectively, i.e. the allowable time step size does not decrease under grid refinement. These conditions are fulfilled in particular by the BR2 scheme as well as the recent ( 1 4 , 9 4 ) -recovery scheme. However, the BR1 scheme and the Baumann-Oden method do not possess a similar stability property which is also confirmed by numerical experiments.

In this paper, we develop a Crank-Nicolson Legendre spectral method for solving the two-dimensional nonlinear time fractional diffusion-wave equation in bounded rectangular domains. In terms of the error splitting argument technique, an optimal error estimate of the numerical scheme is obtained without any time-step size conditions, while the usual analysis for high dimensional nonlinear fractional problems always required certain time-step restrictions dependent on the spatial mesh size. Some numerical results are given to justify the theoretical analysis.

In this paper, a cascadic multigrid method is proposed to solve nonsymmetric eigenvalue problems. Based on the multilevel correction method, the proposed method transforms a nonsymmetric eigenvalue problem solving on the finest finite element space to linear smoothing steps on a sequence of multilevel finite element spaces and some nonsymmetric eigenvalue problems solving on a very low dimensional space. Choosing the sequence of finite element spaces and the number of smoothing steps appropriately, we obtain the optimal convergence rate with the optimal computing complexity. Some numerical examples are provided to validate the theoretical results and the efficiency of this proposed scheme.

Numerical solution of time-dependent differential equations with quasi-interpolation usually uses derivatives of a quasi-interpolant directly to approximate corresponding spatial derivatives (called the direct technique) in equations. This in turn requires that the quasi-interpolant should possess high-order derivatives for solving high-order differential equations. In addition, the resulting numerical solution usually gives a lower approximation accuracy. To circumvent these limitations, the paper proposes a new scheme for getting high-order numerical solutions of time-dependent differential equations based on quasi-interpolation. The scheme uses an iterated technique instead of the direct quasi-interpolation technique for approximating spatial derivatives. Moreover, it requires only computing the first-order derivative of the involved quasi-interpolant. Numerical examples of solving several benchmark equations using the proposed scheme are provided at the end of the paper to demonstrate these features vividly.

This paper presents a second-order semi-implicit Crank-Nicolson scheme for a hybrid magnetohydrodynamics system coupled by the nonstationary Navier-Stokes equations and the stationary Maxwell equations. Main features of proposed scheme are two-fold. One is that it is a decoupled scheme and the magnetic field and velocity field can be solved independently at the same time discrete level. Another is that two subproblems both are linear and are easy to be solved numerically. A rigorous temporal-spatial error analysis is done and we prove that the proposed scheme is of second-order convergence rate O ( ( Δ t ) 2 ) for approximations of the magnetic field, the velocity field and the pressure under the condition Δ t = O ( h 1 / 2 ) . Finally, a numerical result is displayed to illustrate the theoretical results.

Spectral collocation methods are developed for weakly singular Volterra integro-differential equations (VIDEs). Convergence analysis results show that global convergence order depends on the regularities of the kernels functions and solutions to VIDEs, and the number of collocation points is independent of regularities of given functions and the solution to VIDEs. Numerical experiments are carried out to confirm these theoretical results. In numerical experiments, we develop a numerical scheme for nonlinear VIDEs, and investigate the local convergence on collocation points.

The nonlinear Degasperis-Procesi (DP) equation allows shock wave solutions and possesses bi-Hamiltonian properties. Considering both of the two properties, we introduce a numerical scheme which involves three key features. First, a stable moving knots strategy is introduced to simulate the shock waves with less knots. Second, the moving knots equation together with the DP equation are simulated using the meshless multi-quadric (MQ) quasi-interpolation method, which allows scattered data approximation. Third, the proposed moving knots meshless scheme preserves the conservation laws. Both theoretical analysis and numerical examples demonstrate that the algorithm saves the computational time, improves the accuracy, and possesses a long-time tracking capability.

In this paper, we propose a linear, unconditionally energy stable and second-order (in time) numerical scheme based on a convex splitting scheme and the semi-implicit spectral deferred correction (SISDC) method for the phase field crystal equation. The convex splitting scheme we use is linear, uniquely solvable and unconditionally energy stable but is of first-order, so we take the SISDC method to improve the rate of convergence. The resulted scheme inherits the advantages of the convex splitting scheme and thus leads to linear equations at each time step, which is easy to implement. We also prove that the scheme is unconditionally weak energy stable and of second-order accuracy in time. Numerical experiments are presented to validate the accuracy, efficiency and energy stability of the proposed numerical strategy.

A convergence analysis is given for the Grunwald–Letnikov discretisation of a Riemann–Liouville fractional initial-value problem on a uniform mesh t m = m τ with m = 0 , 1 , … , M . For given smooth data, the unknown solution of the problem will usually have a weak singularity at the initial time t = 0 . Our analysis is the first to prove a convergence result for this method while assuming such non-smooth behaviour in the unknown solution. In part our study imitates previous analyses of the L1 discretisation of such problems, but the introduction of some additional ideas enables exact formulas for the stability multipliers in the Grunwald–Letnikov analysis to be obtained (the earlier L1 analyses yielded only estimates of their stability multipliers). Armed with this information, it is shown that the solution computed by the Grunwald–Letnikov scheme is O ( τ t m α − 1 ) at each mesh point t m ; hence the scheme is globally only O ( τ α ) accurate, but it is O ( τ ) accurate for mesh points t m that are bounded away from t = 0 . Numerical results for a test example show that these theoretical results are sharp.

In this paper, for biharmonic eigenvalue problems with clamped boundary condition in R n which include plate vibration problem and plate buckling problem, we primarily study the two-grid discretization based on the shifted-inverse iteration of Ciarlet–Raviart mixed method. With our scheme, the solution of a biharmonic eigenvalue problem on a fine mesh π h can be reduced to the solution of an eigenvalue problem on a coarser mesh π H and the solution of a linear algebraic system on the fine mesh π h . With a new argument which is not covered by existing work, we prove that the resulting solution still maintains an asymptotically optimal accuracy when H > h ≥ O ( H 2 ) . The surprising numerical results show the efficiency of our scheme.

In this paper, the Bernstein polynomials method is proposed for the numerical solution of a class of multipoint problems with linear Volterra–Fredholm integro-differential equations of the neutral type. The main characteristic behind this method lie in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of Bernstein polynomials method for solving these equations are high, which will be shown by preparing some theorems. Also, the existence and uniqueness of solution have been proved. Finally, the numerical experiments are presented to show the excellent behavior and high accuracy of proposed scheme in comparison with some other well-known methods.

The numerical solution of distributed order time fractional partial differential equations based on the midpoint quadrature rule and linear B-spline interpolation is studied. The proposed discretization algorithm follows the Du Fort–Frankel method. The unconditional stability and the convergence of the scheme are analyzed. Simulations illustrate the application of the new algorithm.

In this paper, we construct a kind of product integration scheme for the nonlinear fractional ordinary differential equation. We design the scheme by considering the equivalent Volterra integral equation and using the idea of local Fourier expansion. We prove the high accuracy property of the new scheme and provide the stability analysis. Numerical experiments demonstrate the effectiveness of the scheme.

A family of orthogonal systems of fractional functions is introduced. The proposed orthogonal systems are based on Jacobi polynomials through a fractional coordinate transform. This family of orthogonal systems offers great flexibility to match a wide range of fractional differential models. Approximation errors by the basic orthogonal projection are established. Three new kinds of fractional Jacobi–Gauss-type interpolations are introduced. As an example of application, an efficient approximation based on the proposed fractional functions to a fractional variational problem is presented and implemented. This approximation takes into account the potential irregularity of the solution, and so we are able to obtain a result on optimal order of convergence without the need to impose inconvenient smoothness conditions on the solution. Implementation details are provided for the scheme, together with a series of numerical examples to show the efficiency of the proposed method.

In this paper we present and analyze a polynomial spectral-Galerkin method for nonlinear elliptic eigenvalue problems of the form − div ( A ∇ u ) + V u + f ( u 2 ) u = λ u , ‖ u ‖ L 2 = 1 . We estimate errors of numerical eigenvalues and eigenfunctions. Spectral accuracy is proved under rectangular meshes and certain conditions of f. In addition, we establish optimal error estimation of eigenvalues in some hypothetical conditions. Then we propose a simple iteration scheme to solve the underlying an eigenvalue problem. Finally, we provide some numerical experiments to show the validity of the algorithm and the correctness of the theoretical results.

In this paper, we focus on a decoupled, linearized semi-implicit Galerkin FEM scheme for a MHD system coupled by the time-dependent Navier–Stokes equation with the steady Maxwell's equations in three-dimensional convex domain. First, additional regularities of the solution to the coupled MHD system are derived. By using H 1 -conforming finite element to approximate the magnetic field, it is shown that the proposed semi-linearized scheme is of the first-order convergence order of the velocity field, the magnetic field and the pressure under the time step condition Δ t = O ( h ) .

A numerical method for fractional order differential equations (FDEs) and constant or time-varying delayed fractional differential equations (FDDEs) is constructed. This method is of BDF-type which is based on the interval approximation of the true solution by truncated shifted Chebyshev series. This approach can be reformulated in an equivalent way as a RungeKutta method and its Butcher tableau is given. A detailed local and global truncating errors analysis is deduced for the numerical solutions of FDEs and FDDEs. Illustrative examples are included to demonstrate the validity and applicability of the proposed approach.

This paper deals with nonlinear Fredholm integral equations of the second kind. We study the case of a weakly singular kernel and we set the problem in the space L 1 ( [ a , b ] , C ) . As numerical method, we extend the product integration scheme from C 0 ( [ a , b ] , C ) to L 1 ( [ a , b ] , C ) .

In this article, a class of second order parabolic initial-boundary value problems in the framework of primal hybrid principle is discussed. The interelement continuity requirement for standard finite element method has been alleviated by using primal hybrid method. Finite elements are constructed and used in spatial direction, and backward Euler scheme is used in temporal direction for solving fully discrete scheme. Optimal order estimates for both the semidiscrete and fully discrete method are derived with the help of modified projection operator. Numerical results are obtained in order to verify the theoretical analysis.