Fractional Sobolev Spaces and Variational Problems with Variable-Order Operators on Time Scales

Abstract

We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W^{α(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ through a variable-order Gagliardo-type seminorm and prove completeness and compact embedding properties under standard boundedness assumptions on the order. We then extend the framework to rectangles $\mathcal R=\mathcal I_1\times \mathcal I_2\subset\mathbb T_1\times\mathbb T_2$, introducing the product spaces $W^{(α,β),p}_{\mathrm{rd}}(\mathcal R)$ and establishing completeness, reflexivity, separability, and compact embeddings. To support boundary-value problems, we propose a boundary decomposition of $\partial\mathcal R$ into four sides and a corresponding trace framework (first on $C_{\mathrm{rd}}(\mathcal R)$ and then by density). We also define variable-order Riemann--Liouville and Caputo fractional operators on time scales and derive an Euler--Lagrange equation for variational functionals depending on these operators. The resulting toolkit provides a functional-analytic basis for fractional dynamic equations on mixed time scales and for anisotropic nonlocal models on product time scales.

Figures (7)

• Figure 1: Mixed time-scale illustration (case A) on $\mathbb{T}=[0,10]\cup\{11,12,\dots,20\}$. Left: order profiles $\alpha(t)$. Right: the corresponding solution profiles for the same $\alpha(\cdot)$.
• Figure 2: Mixed product minimizer.
• Figure 3: Mixed product time scale ($\Delta_t$ and fractional operator in $s$): comparison between the non-normalized fractional operator (left) and the spectrally normalized version (Option 2, right) for $\beta=0.6$.
• Figure 4: Benchmark on $f(t)=t^2$ (case A, Caputo). Log--log error curves $\|D_h f-D_{h_{\mathrm{ref}}}f\|_{\infty}$ versus the mesh size $h$ using a fine-grid reference. Variable order yields larger errors at fixed resolution while preserving a stable decay under refinement.
• Figure 5: Benchmark on $f(t)=t e^t$ (case A, Caputo). The mixed-growth test typically produces larger errors than polynomial data at the same $h$, while the refinement trend remains monotone.

Theorems & Definitions (29)

• Proposition 2.6: Ahmadkhanlu2012
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