The artificial boundary method of Li and Wu (2021) reduces the nonlinear time-fractional Burgers equation on the real line to an initial-boundary value problem on a finite interval, using a [1/1] Padé approximation of the Laplace symbol sα\sqrt{s^{\alpha}} about a free expansion point s0s_0 to localize the transparent boundary condition, together with a one-directional linearization and operator-splitting treatment of the nonlinear advection. Two parts of the method's justification were left incomplete: the uniform boundedness theorem for the fully discrete scheme assumes only that the time step τ\tau is 'small enough', with no explicit admissible range in terms of the fractional order α\alpha, viscosity μ\mu, expansion point s0s_0, or mesh size hh; and no rigorous bound exists for the boundary modeling error introduced by the Padé approximation and the exterior linearization, nor any principled rule for choosing s0s_0 or the rational approximant order. This problem asks for the missing quantitative theory: an explicit parameter-dependent sufficient condition on τ\tau (or a proof of unconditional boundedness) for the scheme's stability theorem, and a rigorous estimate of the artificial-boundary-condition modeling error — its dependence on s0s_0, the approximant order, and the solution's frequency content — together with an adaptive or optimized rule for selecting s0s_0.

Background

The viscous Burgers equation with a Caputo derivative of order α∈(0,1)\alpha\in(0,1) in time models nonlinear advection superposed on anomalous (memory-laden) diffusion:

0cDtαu=μuxx−u ux+f,(x,t)∈R×[0,T],{}_0^{c}D_t^{\alpha}u=\mu u_{xx}-u\,u_x+f,\quad (x,t)\in\mathbb{R}\times[0,T],

where the Caputo derivative is defined by 0cDtαu(t)=1Γ(1−α)∫0t(t−ξ)−α∂u∂ξ(ξ) dξ{}_0^{c}D_t^{\alpha}u(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t(t-\xi)^{-\alpha}\frac{\partial u}{\partial \xi}(\xi)\,d\xi, μ>0\mu>0 is the viscosity, and the initial datum and source are compactly supported with u→0u\to 0 at spatial infinity. Because the spatial domain is unbounded, direct finite-difference discretization is impossible: truncating the domain at artificial boundaries xl<xrx_l<x_r produces spurious reflections unless the boundary conditions reproduce the outgoing behavior of the solution on the exterior half-lines, so that the finite-domain solution coincides with the restriction of the whole-line solution. Such conditions are called artificial boundary conditions (ABCs); the exact condition is often called a transparent boundary condition. For the linear part 0cDtαu=μuxx{}_0^{c}D_t^{\alpha}u=\mu u_{xx}, a Laplace transform in time (transform variable ss, transform u^\hat u) yields on the right exterior the exact boundary relation ∂xu^+sα/μ u^=0\partial_x\hat u+\sqrt{s^{\alpha}/\mu}\,\hat u=0 at x=xrx=x_r, which is nonlocal in time after inversion. Li and Wu (2021) approximate sα\sqrt{s^{\alpha}} by its [1/1] Padé approximant about a free expansion point s0>0s_0>0 — a Padé approximant is a ratio of polynomials matching the Taylor expansion to highest order at the chosen point; here s0α/2(s0α+3sα)/(3s0α+sα)s_0^{\alpha/2}(s_0^{\alpha}+3s^{\alpha})/(3s_0^{\alpha}+s^{\alpha}), which is the standard [1/1] approximant of z1/2z^{1/2} at z=1z=1 composed with z=(s/s0)αz=(s/s_0)^{\alpha} — converting the transparent condition into a local-in-time nonlinear ABC; the directionality of the exterior operator is derived for the linear diffusion part only, the nonlinear term −u ux-u\,u_x being reinserted through an operator-splitting argument. For the resulting reduced initial-boundary value problem on [xl,xr][x_l,x_r] they build an implicit finite-difference scheme whose Caputo derivative is discretized by the L1 formula (a piecewise-linear-in-time quadrature with weights al=(l+1)1−α−l1−αa_l=(l+1)^{1-\alpha}-l^{1-\alpha}), and prove uniform boundedness of the discrete solution (Theorem 1) and convergence at rate O(τ+h2)O(\tau+h^2) (Theorem 2), where τ\tau and hh are the time and space steps. The boundedness proof, however, requires that τ\tau be 'small enough' for a combination of discrete weights and energy-estimate constants to yield a constant γ>12\gamma>\tfrac12, without quantifying the admissible range of τ\tau; and the modeling error committed at the boundary — by the Padé replacement of sα\sqrt{s^{\alpha}}, by the choice of s0s_0, and by the one-directional linearization of the exterior problem — is never bounded. The same Padé-based construction recurs across the fractional-ABC literature: exact convolutional ABCs for subdiffusion (Gao, Sun and Zhang 2012), high-order local ABCs for fractional diffusion (Zhang, Li, Wu and Zhang 2017, who prove stability and convergence for the reduced problem but not a boundary modeling-error bound), nonlinear time-fractional parabolic problems on unbounded domains (Li and Zhang 2016), two-dimensional subdiffusion (Li, Wu and Zhang 2017), diffusion-wave equations (Brunner, Han and Yin 2014), and recent phase-lag heat-conduction extensions (Bu et al. 2026; Bu, Xie and Wang 2025) — in all of which the expansion point is chosen heuristically and the boundary approximation error is measured only empirically, if at all.

Problem Statement

Complete the quantitative justification of the Li–Wu (2021) artificial boundary scheme for the nonlinear time-fractional Burgers equation 0cDtαu=μuxx−u ux+f{}_0^{c}D_t^{\alpha}u=\mu u_{xx}-u\,u_x+f on the real line (Caputo order α∈(0,1)\alpha\in(0,1), viscosity μ>0\mu>0, compactly supported data), by resolving the two limitations that paper left open for its own construction — the reduced initial-boundary value problem with the Padé artificial boundary conditions and its L1-in-time, central-difference-in-space fully discrete scheme:

(1) Quantitative stability. Determine an explicit, parameter-dependent sufficient condition on the time step τ\tau — stated as a computable function of the fractional order α\alpha, the viscosity μ\mu, the expansion point s0s_0, the mesh size hh, and any norms of the initial and source data on which the constants of the energy estimate depend, with those constants tracked explicitly — under which the uniform boundedness result (Theorem 1) holds; or prove that the scheme is unconditionally bounded, thereby removing the 'small enough τ\tau' assumption altogether. The condition must be uniform in the time level, so that a user can select τ\tau a priori.

(2) Boundary-condition modeling error. Derive a rigorous bound on the error introduced by (a) replacing the exact Laplace symbol sα\sqrt{s^{\alpha}} in the transparent boundary condition by its rational (Padé) approximant about s0s_0 and (b) the one-directional linearization/operator-splitting treatment of the exterior nonlinear term — quantifying how this error depends on s0s_0, on the order of the rational approximant, and on the frequency content or regularity of the solution, measured against the solution of the original whole-line problem. On this basis, derive an adaptive or optimized rule for choosing s0s_0 (or a higher-order rational approximant) that provably controls the total boundary modeling error.

References

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