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The steady state fractional convection diffusion equation with inhomogeneous Dirichlet boundary is considered. By utilizing standard boundary shifting trick, a homogeneous boundary problem is derived with a singular source term which does not belong to $$L^2$$ anymore. The variational formulation of such problem is established, based on which the finite element approximation scheme is developed. Inf-sup conditions for both continuous case and discrete case are demonstrated thus the corresponding well-posedness is verified. Furthermore, rigorous regularity analysis for the solutions of both original equation and dual problem is carried out, based on which the error estimates for the finite element approximation are derived. Numerical results are presented to illustrate the theoretical analysis.

We consider a finite element method with singularity reconstruction for fractional convection-diffusion equation involving Riemann-Liouville derivative of order $$\alpha \in (1,2)$$ . Jin et al. (ESAIM Math. Model. Numer. Anal. 49(5):1261–1283, 2015) developed a singularity reconstruction strategy for fractional reaction-diffusion equation in which the solution was split into a singular part $$x^{\alpha -1}$$ and a regular part $$u^r$$ with $$u^r(0)=u^r(1)=0$$ . In this paper we transform the original problem into a one-point boundary-value problem whose solution u satisfies the condition $$u(0)=({}_0D_x^{\alpha -1}u)(0)=0$$ . A novel Petrov–Galerkin variational formulation is developed on the domain $${\tilde{H}}_L^{\alpha /2}(\Omega ) \times {\tilde{H}}_R^{\alpha /2}(\Omega )$$ , based on which the finite element approximation scheme is established. The inf-sup conditions for both continuous case and discrete case are analyzed thus the corresponding well-posedness is verified. The $$L^2$$ -error estimate is derived by considering an adjoint variational formulation which is different from the original Petrov–Galerkin weak form. Some numerical results for piecewise linear and quadratic finite elements are presented to verify the theoretical findings.

In this work, we propose a new scheme based on numerical quadrature to calculate the two-parameter Mittag-Leffler function with discrete elliptic operator $$-{\mathcal {L}}_h$$ as input. Except pure mathematical interest from approximation theory, our consideration also arises from solving sub-diffusion equations numerically with time-independent diffusion coefficient. We obtain the scheme by applying Gauss-Legendre quadrature rule for the integral representation of the Mittag-Leffler function. Rigorous error analysis is carried out which shows that the scheme converges exponentially with the increase of quadrature nodes. The computational cost of the algorithm is solving K sparse linear systems with K the number of quadrature nodes. It is worth to point out that the scheme is completely parallel which can save much time if the dimension of $${\mathcal {L}}_h$$ is very large. Some numerical tests are provided to verify the efficiency and robustness of our scheme.

In this paper, we develop and study algorithms for approximately solving linear algebraic systems: ${{\mathcal{A}}}_h^\alpha u_h = f_h$, $ 0< \alpha

In this paper we aim at developing highly accurate and stable method in temporal direction for time-fractional diffusion equations with initial data $$v\in L^2(\varOmega )$$ . To this end we begin with a kind of (time-)fractional ODEs, and a hybrid multi-domain Petrov–Galerkin spectral method is proposed. To match the singularity at the beginning of time, we use geometrically graded mesh together with fractional power Jacobi functions as basis on the first interval. Jacobi polynomials are then chosen to approximate the solution in temporal direction on the intervals hereafter. The algorithm is motivated by the discovery that the solution in temporal direction possesses high regularity in proper weighted Sobolev space on a special mesh in piecewise sense. Combining standard finite element method, we extend it to solve time-fractional diffusion equations with initial data $$v\in L^2(\varOmega )$$ , in which the mesh in temporal direction is determined by spatial mesh size and finial time T. Numerical tests show that the scheme is stable and converges exponentially in time.

We present and analyze a space-time Petrov–Galerkin finite element method for a time-fractional diffusion equation involving a Riemann–Liouville fractional derivative of order α ∈ ( 0 , 1 ) {\alpha\in(0,1)} in time and zero initial data. We derive a proper weak formulation involving different solution and test spaces and show the inf-sup condition for the bilinear form and thus its well-posedness. Further, we develop a novel finite element formulation, show the well-posedness of the discrete problem, and derive error bounds in both energy and L 2 {L^{2}} norms for the finite element solution. In the proof of the discrete inf-sup condition, a certain nonstandard L 2 {L^{2}} stability property of the L 2 {L^{2}} projection operator plays a key role. We provide extensive numerical examples to verify the convergence analysis.

In this paper, we revisit two spectral approximations, including truncated approximation and interpolation for Caputo fractional derivative. The two approaches have been studied to approximate Riemann–Liouville (R–L) fractional derivative by Chen et al. and Zayernouri et al. respectively in their most recent work. For truncated approximation the reconsideration partly arises from the difference between fractional derivative in R–L sense and Caputo sense: Caputo fractional derivative requires higher regularity of the unknown than R–L version. Another reason for the reconsideration is that we distinguish the differential order of the unknown with the index of Jacobi polynomials, which is not presented in the previous work. Also we provide a way to choose the index when facing multi-order problems. By using generalized Hardy's inequality, the gap between the weighted Sobolev space involving Caputo fractional derivative and the classical weighted space is bridged, then the optimal projection error is derived in the non-uniformly Jacobi-weighted Sobolev space and the maximum absolute error is presented as well. For the interpolation, analysis of interpolation error was not given in their work. In this paper we build the interpolation error in non-uniformly Jacobi-weighted Sobolev space by constructing fractional inverse inequality. With combining collocation method, the approximation technique is applied to solve fractional initial-value problems (FIVPs). Numerical examples are also provided to illustrate the effectiveness of this algorithm.

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries.

In this article, we consider a two-dimensional symmetric space-fractional diffusion equation in which the space fractional derivatives are defined in Riesz potential sense. The well-posed feature is guaranteed by energy inequality. To solve the diffusion equation, a fully discrete form is established by employing Crank-Nicolson technique in time and Galerkin finite element method in space. The stability and convergence are proved and the stiffness matrix is given analytically. Three numerical examples are given to confirm our theoretical analysis in which we find that even with the same initial condition, the classical and fractional diffusion equations perform differently but tend to be uniform diffusion at last.