AbstractIn this contribution, we investigate an initial-boundary value problem for a fractional diffusion equation with Caputo fractional derivative of space-dependent variable order where the coefficients are dependent on spatial and time variables. We consider a bounded Lipschitz domain and a homogeneous Dirichlet boundary condition. Variable-order fractional differential operators originate in anomalous diffusion modelling. Using the strongly positive definiteness of the governing kernel, we establish the existence of a unique weak solution in $u\in \operatorname{L}^{\infty } ((0,T),\operatorname{H}^{1}_{0}( \Omega ) )$
u
∈
L
∞
(
(
0
,
T
)
,
H
0
1
(
Ω
)
)
to the problem if the initial data belongs to $\operatorname{H}^{1}_{0}(\Omega )$
H
0
1
(
Ω
)
. We show that the solution belongs to $\operatorname{C} ([0,T],{\operatorname{H}^{1}_{0}(\Omega )}^{*} )$
C
(
[
0
,
T
]
,
H
0
1
(
Ω
)
∗
)
in the case of a Caputo fractional derivative of constant order. We generalise a fundamental identity for integro-differential operators of the form $\frac{\mathrm{d}}{\mathrm{d}t} (k\ast v)(t)$
d
d
t
(
k
∗
v
)
(
t
)
to a convolution kernel that is also space-dependent and employ this result when searching for more regular solutions. We also discuss the situation that the domain consists of separated subdomains.
A novel series method for fractional diffusion equation within Caputo fractional derivative
