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A time-fractional partial differential equation with a hidden-memory space-time dependent variable order is studied. The well-posedness and high-order regularity estimates of the problem are proved via the resolvent estimates. Accordingly, an error estimate of the L-1 discretization without any artificial regularity assumption of the true solution is analyzed. Numerical experiments are performed to substantiate the theoretical results and to study the behavior of the variable-order fractional partial differential equations.

We develop an indirect collocation method for a variable-order fractional wave equation. Fractional differential equations are well known to exhibit initial weak singularity, which makes it unreali...

We analyze finite element approximations to variable-coefficient two-sided space-fractional advection-reaction-diffusion equations in three space dimensions. It is known that the corresponding bilinear form may lose coercivity due to the variable diffusivity coefficient. We prove the weak coercivity of the bilinear form via the Garding's inequality and thus derive the optimal-order error estimate of the finite element method. The developed method is further extended to analyze the time-dependent problems. Numerical experiments are given to validate the theoretical findings.

We proved the unique determination of the variable order in a two-scale mobile–immobile variable-order time-fractional partial differential equation with a variable diffusivity tensor imposed on a ...

Fractional diffusion equations (FDEs) were shown to provide very competitive descriptions of challenging phenomena of anomalously diffusive transport or long-range interactions. However, FDEs intro...

A fully discrete two-grid modified method of characteristics (MMOC) scheme is proposed for nonlinear variable-order time-fractional advection–diffusion equations in two space dimensions. The MMOC is used to handle the advection-dominated transport and the two-grid method is designed for efficiently solving the resulting nonlinear system. Optimal L 2 error estimates are derived for both the MMOC scheme and the corresponding two-grid MMOC scheme. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed method.

We study a fully discretized finite element approximation to variable-order Caputo and Riemann–Liouville time-fractional diffusion equations (tFDEs) in multiple space dimensions, which model solute transport in heterogeneous porous media and related applications. We prove error estimates for the proposed methods, which are discretized on an equidistant or graded temporal partition predetermined by the behavior of the variable order at the initial time, only under the regularity assumptions of the variable order, coefficients and the source term but without any regularity assumption of the true solutions. Roughly, we prove that the finite element approximations to variable-order Caputo tFDEs have optimal-order convergence rates on a uniform temporal partition. In contrast the finite element approximations to variable-order Riemann–Liouville tFDEs discretized on a uniform temporal partition achieve an optimal-order convergence rate if $\alpha (0)=\alpha ^{\prime}(0) = 0$ but a suboptimal-order convergence rate if $\alpha (0)>0$. In the latter case, optimal-order convergence rate can be proved by employing the graded temporal partition. We conduct numerical experiments to investigate the performance of the numerical methods and to verify the mathematical analysis.

We prove wellposedness of a variable-order linear space-time fractional diffusion equation in multiple space dimensions. In addition we prove that the regularity of its solutions depends on the behavior of the variable order (and its derivatives) at time [Formula: see text], in addition to the usual smoothness assumptions. More precisely, we prove that its solutions have full regularity like its integer-order analogue if the variable order has an integer limit at [Formula: see text] or have certain singularity at [Formula: see text] like its constant-order fractional analogue if the variable order has a non-integer value at time [Formula: see text].

Variable-order space-time fractional diffusion equations, in which the variation of the fractional orders determined by the fractal dimension of the media via the Hurst index characterizes the stru...

In this article, we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post-processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Numerical experiments are presented to test the performance of the proposed method.

We study an indirect finite element approximation for two-sided space-fractional diffusion equations in one space dimension. By the representation formula of the solutions u(x) to the proposed variable coefficient models in terms of v(x), the solutions to the constant coefficient analogues, we apply finite element methods for the constant coefficient fractional diffusion equations to solve for the approximations $$v_h(x)$$ to v(x) and then obtain the approximations $$u_h(x)$$ of u(x) by plugging $$v_h(x)$$ into the representation of u(x). Optimal-order convergence estimates of $$u(x)-u_h(x)$$ are proved in both $$L^2$$ and $$H^{\alpha /2}$$ norms. Several numerical experiments are presented to demonstrate the sharpness of the derived error estimates.

We prove the wellposedness of a nonlinear variable-order fractional differential equation and the regularity of its solutions. The regularity of the solutions is determined solely by the values of the variable order and its high-order derivatives at time t = 0 (in addition to the usual regularity assumptions on the variable order and the coefficients). If the variable-order reduces to an integer order at t = 0, then the solution has full regularity as the solution to a first-order ordinary differential equation. In this case, we prove that the corresponding finite difference scheme discretized on a uniform mesh has an optimal-order convergence rate. However, if the variable order does not reduce to an integer order at t = 0, then the solution has a singularity at time t = 0, as Stynes et al. proved in [15] for the constant-order time-fractional diffusion equations. The corresponding finite difference scheme discretized on a uniform mesh has only a suboptimal-order convergence rate. Instead, we prove that the finite difference scheme discretized on a graded mesh determined by the value of the variable order at time t = 0 has an optimal-order convergence rate in terms of the number of the time steps. Numerical experiments substantiate these theoretical results.

We analyze the well-posedness and smoothing properties of a variable-order time-fractional wave partial differential equation in multiple space dimensions. Accordingly, we prove the unique determination of the variable order in this model with the observations of the unknown solutions on an arbitrarily small spatial domain over a sufficiently small time interval. The proved theorem provides a guidance where the measurements should be taken and ensure the unique identification of the variable order.

We present a logarithmic transformation reducing the (variable-order) Caputo–Hadamard fractional problems to their Caputo analogues. Then the analysis of the former may be directly obtained from the existing results for the latter by the inverse transformation. By employing this transformation method, we connect these two kinds of problems and obtain new analysis results for variable-order Caputo–Hadamard fractional operators and models.

We study an initial-boundary value problem of variable-order time-fractional diffusion equations in one space dimension. Based on the wellposedness of the proposed model and the smoothing properties of its solutions, which are shown to be determined by the behavior of the variable order at the initial time, a uniqueness result for an important inverse problem of determination of the variable order in the time-fractional derivative contained in the proposed model from observations of its solutions is obtained.

We prove the existence and uniqueness of the solution to a variable-order fractional stochastic differential equation driven by a multiplicative white noise, which describes the random phenomena with nonlocal effect. We further develop a Euler-Maruyama scheme and prove the strong convergence of the scheme. Numerical experiments are presented to substantiate the mathematical analysis.

We prove the wellposedness of a variable-order fractional stochastic differential equation driven by a multiplicative white noise, which models random phenomena with memory effects, and the regularity of its solutions.

We prove the wellposedness and smoothing properties of the initial–boundary value problem of a variably distributed-order time-fractional diffusion partial differential equation in multiple space dimensions, which models the subdiffusive transport of solutes traveling through heterogeneous porous media. We accordingly derive a finite element approximation to the problem and prove its optimal-order error estimate, only under the regularity assumptions of the variably distributed order, coefficients and the source term, but without any regularity assumption on the true solution of the problem. Numerical experiments are presented to verify the mathematical and numerical analyses.