Optimal long-time decay rate of solutions of complete monotonicity-preserving schemes for nonlinear time-fractional evolutionary equations
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The solution of the nonlinear initial-value problem 𝒟_t^αy(t)=-λ y(t)^γ for t>0 with y(0)>0, where 𝒟_t^α is a Caputo derivative of order α∈ (0,1) and λ, γ are positive parameters, is known to exhibit O(t^α/γ) decay as t→∞. No corresponding result for any discretisation of this problem has previously been proved. In the present paper it is shown that for the class of complete monotonicity-preserving (𝒞ℳ-preserving) schemes (which includes the L1 and Grünwald-Letnikov schemes) on uniform meshes {t_n:=nh}_n=0^∞, the discrete solution also has O(t_n^-α/γ) decay as t_n→∞. This result is then extended to 𝒞ℳ-preserving discretisations of certain time-fractional nonlinear subdiffusion problems such as the time-fractional porous media and p-Laplace equations. For the L1 scheme, the O(t_n^-α/γ) decay result is shown to remain valid on a very general class of nonuniform meshes. Our analysis uses a discrete comparison principle with discrete subsolutions and supersolutions that are carefully constructed to give tight bounds on the discrete solution. Numerical experiments are provided to confirm our theoretical analysis.
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