Symbol Notation
The following provides a detailed explanation of the operator symbols and function space symbols that appear in the interactive module.
I. Operator Symbols
| Name | Meaning |
|---|
| Caputo time |
Temporal fractional Caputo derivative, denoted $^C D_t^\alpha u$, allowing independent imposition of initial conditions $u(0)=u_0$. |
| Fractional Laplacian |
Integral-type fractional Laplacian $(-\Delta)^s$, defined via a singular integral, characterizing spatial nonlocal jumps. |
| Spectral fractional Laplacian |
$(-\Delta)^s$ defined via the spectral decomposition of Laplacian eigenfunctions, compatible with Dirichlet boundary conditions. |
| Spatio-temporal fractional |
Both temporal and spatial derivatives are of fractional order, commonly used to describe anomalous diffusion coupled with memory and long-range jumps. |
| Convolution nonlocal |
Convolution-type nonlocal operator $K * u$, where the kernel $K$ may contain singularities or rapid decay, generalizing local differentiation. |
| Fractional p-Laplacian |
Nonlinear fractional operator $(-\Delta)_p^s$, combining $p$-Laplacian growth with fractional nonlocality. |
| Pseudodifferential operator |
Strongly elliptic pseudodifferential operator $r^+ OP(p(x,\xi))$, generalizing variable-coefficient fractional operators via Fourier multipliers. |
| Distributed-order time |
Integral form after assigning a weight function $\mu(\alpha)$ to the Caputo derivative: $\int_0^1 \mu(\alpha) \, ^C D_t^\alpha u \, d\alpha$. |
| Variable-order spatio-temporal |
Fractional operator with order varying by spatial position, such as $s(x)$, used to characterize medium heterogeneity. |
II. Function Space Symbols
Spaces for the field $u$
| Symbol | Meaning |
|---|
| $L^\infty(0,T; H_0^1)$ |
Bochner space: essentially bounded on the time interval $[0,T]$, taking values in the zero-boundary Sobolev space $H_0^1$. |
| $H^s(\mathbb{R}^n)$ |
Fractional Sobolev space, Fourier definition: $(1+|\xi|^2)^{s/2}\hat{u} \in L^2$. |
| $C^{2s+\alpha}$ |
Hölder space: the function and its derivatives of order $2s+\alpha$ satisfy $|\partial^k u(x)-\partial^k u(y)| \le C|x-y|^\alpha$. |
| $H^{2s} \cap H_0^1$ |
Intersection of the fractional Sobolev space $H^{2s}$ and the zero-boundary space $H_0^1$, ensuring zero boundary traces. |
| $L^2(0,T; H^s)$ |
Bochner space: square-integrable in time, taking values in the fractional Sobolev space $H^s$. |
| $W^{s,p}$ |
Fractional Sobolev space (Sobolev-Slobodeckij space), fractional differentiability in the $L^p$ framework. |
| $H_q^{a(2a+s)}$ |
Bessel potential space (a special case of Besov spaces), where the parameter $q$ controls summability. |
| $C_*^{a(2a+s)}$ |
Hölder-Zygmund space, characterizing smoothness via differences or Littlewood-Paley decomposition. |
| $H^{s(x)}(\Omega)$ |
Variable-order fractional Sobolev space, with order $s(x)$ varying by spatial position. |
Spaces for the source $f$
| Symbol | Meaning |
|---|
| $L^2(0,T; H^{-1})$ |
Bochner space, taking values in the dual space of $H_0^1$, allowing Dirac-type source terms. |
| $H^{-s}(\Omega)$ |
Negative-order fractional Sobolev space, dual of $H^s$, characterizing low-regularity data. |
| $C^\alpha$ |
Space of Hölder continuous functions of order $\alpha$, requiring $|f(x)-f(y)| \le C|x-y|^\alpha$. |
| $L^2(\Omega)$ |
Lebesgue space: square-integrable functions on the domain $\Omega$. |
| $W^{-s,p'}$ |
Dual space of $W^{s,p}$, where $p'$ is the conjugate exponent of $p$, satisfying $\frac{1}{p}+\frac{1}{p'}=1$. |
| $\bar{H}_q^s$ |
Restricted Bessel potential space, with functions extended by zero outside the domain, suitable for boundary problems. |
| $\bar{C}_*^s$ |
Restricted Hölder-Zygmund space, also requiring functions to vanish outside the domain. |
III. General Notation
| Symbol | Meaning |
|---|
| $P$ |
Differential operator, acting on the field $u$ in $Pu=f$. |
| $u$ |
Unknown field function (field / solution), the target of the equation. |
| $f$ |
Source term (source / right-hand side), known external driving or data. |
| $\Omega$ |
Spatial domain (bounded open domain in $\mathbb{R}^n$). |
| $T$ |
Endpoint of the time interval; typically the time domain is $[0,T]$. |
| $s$ |
Fractional order exponent, $0 < s < 1$ (or more generally $s > 0$), determining the order of the operator. |
| $\alpha$ |
Hölder exponent or fractional order, taking values $0 < \alpha \le 1$. |