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We present a new fractional Taylor formula for singular functions whose Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) $L^infty$-estimates and $L^2$-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for $p$ and $hp$ methods for singular problems, and answer some open questions posed in some recent literature.
This work discusses the finite element discretization of an optimal control problem for the linear wave equation with time-dependent controls of bounded variation. The main focus lies on the convergence analysis of the discretization method. The stat
In this paper, we investigate fast algorithms to approximate the Caputo derivative $^C_0D_t^alpha u(t)$ when $alpha$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost
We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corres
The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection-reaction-diffusion equation that exhibits both paraboli
The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematic