Time-Fractional Porous Medium Equation

• The time-fractional porous medium equation is a class of nonlinear diffusion models replacing the classical first-order time derivative with a fractional derivative, thereby incorporating memory effects.
• Analytical methods such as self-similar reductions and Erdélyi–Kober operators, along with rigorous numerical discretizations, help establish existence, uniqueness, and decay properties.
• Applications span subdiffusive phenomena in porous media, with models derived from deterministic, stochastic, and variational frameworks providing insights into free-boundary behavior and anomalous scaling.

Searching arXiv for recent and foundational papers on the time-fractional porous medium equation. The time-fractional porous medium equation denotes a family of nonlinear diffusion equations in which the classical first-order time derivative is replaced by a fractional derivative, typically of Caputo or Riemann–Liouville type. In its prototypical form, it combines porous-medium degeneracy with temporal nonlocality, so that the instantaneous rate of change depends on the full prior history of the solution. Several distinct but related models appear in the literature, including the local-pressure equation

$\partial_t^\alpha u=\Delta(u^m),$

the divergence-form equation

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$

and nonlocal-pressure variants such as

$$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$

with $0<\alpha<1$ and suitable ranges of $m$ or $\sigma$ (Allen et al., 2015). Across these formulations, the central analytical themes are anomalous subdiffusion, self-similarity, free boundaries, compact support versus heavy tails, weak solvability, and regularity.

1. Model classes and fractional-time operators

A standard starting point is the replacement of the integer-order derivative in the porous medium equation by a Caputo derivative of order $0<\alpha<1$. In one widely studied formulation,

$${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$

with $u(x,0)=0$, where

$${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$

(Płociniczak, 2014). Because $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$0 in that setting, Caputo and Riemann–Liouville derivatives coincide (Płociniczak, 2014).

A second common local-pressure model is written as

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$1

in $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$2, with a Dirac-mass initial trace and conserved total mass (Gómez-Castro et al., 10 Apr 2026). In bounded domains, a more general porous-medium-type problem is

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$3

where $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$4 is a local or nonlocal self-adjoint diffusion operator and zero Dirichlet boundary conditions are imposed (Bonforte et al., 2024).

The literature also contains nonlocal-pressure equations that couple temporal memory with long-range spatial interaction. The model

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$5

uses a Caputo-type time derivative together with the inverse fractional Laplacian

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$6

(Allen et al., 2015). Another nonlocal-pressure formulation is

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$7

with $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$8 (Chung et al., 2024).

The time-fractional porous medium equation is also treated in stochastic and variational settings. On a bounded domain $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$9, the stochastic equation

$$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$0

includes time-fractional porous media equations as special cases when $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$1 and $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$2 is a porous-medium nonlinearity (Liu et al., 2017).

These formulations are linked by a common mechanism: the time-fractional derivative introduces memory, while the porous-medium nonlinearity creates degeneracy or singularity according to whether $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$3 or $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$4. This suggests that the term “time-fractional porous medium equation” is best understood as a class of models rather than a single canonical PDE.

2. Physical derivation and interpretation

One deterministic derivation begins from Darcy’s law $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$5 and mass conservation $$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$6, then replaces the local continuity law by a nonlocal balance

$$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$7

Choosing the kernel

$$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$8

leads to

$$D_t^{\alpha} u - \operatorname{div}\!\bigl(u\,\nabla(-\Delta)^{-\sigma}u\bigr)=f,$$9

and, with $0<\alpha<1$0, to the time-fractional Richards equation and its porous-medium specialization (Płociniczak, 2014). That derivation is presented as dual to the stochastic continuous-time random walk framework and is motivated by waiting-time or trapping effects in porous media (Płociniczak, 2014).

The interpretation of the Caputo derivative as a memory term is explicit in the nonlocal-pressure model of Allen, Caffarelli, and Vasseur, where the local rate at time $0<\alpha<1$1 depends on the full history $0<\alpha<1$2 for $0<\alpha<1$3 (Allen et al., 2015). In that paper, the one-sided integration-by-parts structure of the Caputo derivative is central to the weak formulation and regularity theory (Allen et al., 2015).

Applications cited in the literature include subdiffusive moisture uptake in building materials such as brick, cement, and zeolite, hydrocarbons in tight rocks, and contaminant transport in heterogeneous soil (Płociniczak, 2014). Experimental support for subdiffusive anomalies is discussed in connection with porous zeolite data showing anomalous scaling $0<\alpha<1$4 with $0<\alpha<1$5, fitted by a three-term approximate profile with parameters

$0<\alpha<1$6

(Płociniczak, 2014).

For nonlocal-pressure variants, the inverse fractional Laplacian implies that the pressure at a point depends on remote density values, so the model incorporates both temporal memory and spatial long-range interaction (Allen et al., 2015). A plausible implication is that such equations interpolate between anomalous transport theory and nonlocal aggregation–diffusion dynamics.

3. Self-similarity, scaling, and profile equations

Self-similar solutions are a principal analytical tool throughout the subject. For the source-type local equation

$0<\alpha<1$7

in $0<\alpha<1$8, one seeks

$0<\alpha<1$9

and mass conservation yields

$m$0

These exponents are positive precisely when

$m$1

which is the optimal range for existence of finite-mass self-similar profiles (Gómez-Castro et al., 10 Apr 2026).

In one dimension for the equation

$m$2

the source-type ansatz

$m$3

combined with mass conservation gives $m$4, and substitution yields

$m$5

(Caballero et al., 25 Jul 2025). The corresponding classical limit as $m$6 is the Barenblatt–Zel’dovich–Kompaneets solution

$m$7

with $m$8 determined by the mass normalization (Caballero et al., 25 Jul 2025).

Boundary-driven self-similar problems on the half-line produce different exponents because the normalization is set by Dirichlet, Neumann, or Robin data rather than by mass conservation. Under constant-flux boundary conditions,

$m$9

Płociniczak obtains

$\sigma$0

for

$\sigma$1

(Płociniczak, 2014). More generally, for half-line free-boundary problems with Dirichlet, Neumann, or Robin data, the self-similar reduction leads to an ordinary integro-differential equation involving the Erdélyi–Kober operator $\sigma$2 (Okrasińska-Płociniczak et al., 2021).

The Erdélyi–Kober operator is a recurring structure in the self-similar reduction of time-fractional porous medium equations. In the one-dimensional half-line setting with $\sigma$3, $\sigma$4, one obtains equations of the form

$\sigma$5

where

$\sigma$6

(López et al., 2023). Equivalent Volterra-type integral equations then follow after integrating twice and using compact support.

For the nonlocal-pressure model, Allen, Caffarelli, and Vasseur prove that if

$\sigma$7

then the rescaled function

$\sigma$8

is again a solution provided

$\sigma$9

(Allen et al., 2015). Their Hölder-regularity argument iterates oscillation decay on nested cylinders at scales synchronized by this relation (Allen et al., 2015).

4. Existence, uniqueness, and weak-solution frameworks

Existence theory depends strongly on the chosen model.

For the nonlocal-pressure equation

$0<\alpha<1$0

with $0<\alpha<1$1, nonnegative data, and exponential decay

$0<\alpha<1$2

Allen, Caffarelli, and Vasseur prove existence of weak solutions (Allen et al., 2015). Their construction regularizes $0<\alpha<1$3 by a smooth kernel $0<\alpha<1$4, adds a viscosity term $0<\alpha<1$5, discretizes time with step $0<\alpha<1$6, and passes to the limit using a priori bounds, discrete Aubin–Lions compactness in time, and Besov/Riesz-potential lifting in space (Allen et al., 2015).

For the one-dimensional equation

$0<\alpha<1$7

with $0<\alpha<1$8, $0<\alpha<1$9, $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$0, and $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$1 for $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$2, existence and uniqueness of the general initial-boundary-value weak solution are proved in (López et al., 2023). The proof uses integration by parts for the Caputo derivative via the right-side fractional integral $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$3 and the Ladyženskaja trick applied to the difference of two solutions (López et al., 2023). In the self-similar setting, the profile equation is recast as

$${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$4

with $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$5, and existence follows from a Leray–Schauder fixed-point argument (López et al., 2023).

A related half-line Dirichlet problem with $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$6,

$${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$7

together with a moving free boundary and no-flux condition, is reduced to a nonlinear Volterra integral equation in $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$8 (Płociniczak et al., 2018). A shooting parameter $${}^{C}\partial_t^\alpha u(x,t)=\partial_x\!\bigl(u^m(x,t)\,\partial_xu(x,t)\bigr),$$9 is introduced; Schauder’s fixed-point theorem yields a compactly supported candidate for each $u(x,0)=0$0, and an intermediate-value argument identifies exactly one $u(x,0)=0$1 producing the no-flux condition (Płociniczak et al., 2018).

For broader operator classes on bounded domains, the equation

$u(x,0)=0$2

admits a unique nonnegative $u(x,0)=0$3-solution for every $u(x,0)=0$4, assuming the inverse kernel $u(x,0)=0$5 satisfies

$u(x,0)=0$6

and the weak Kato inequality holds (Bonforte et al., 2024). The comparison principle is established in the same general setting (Bonforte et al., 2024).

In the stochastic and variational framework of Gelfand triples, Liu, Röckner, and da Silva prove existence and uniqueness for

$u(x,0)=0$7

under assumptions $u(x,0)=0$8–$u(x,0)=0$9 on the monotone nonlinearity $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$0, with the deterministic time-fractional porous-medium equation recovered when the noise vanishes (Liu et al., 2017). The proof combines pseudomonotonicity of the fractional derivative operator with Yosida approximation and Brezis-type surjectivity (Liu et al., 2017).

A point of contrast is that uniqueness is open for the fully nonlocal time-and-pressure model of (Allen et al., 2015), whereas uniqueness is obtained for several local-pressure one-dimensional and bounded-domain problems (López et al., 2023).

5. Regularity, decay, and qualitative behavior

Regularity theory for time-fractional porous medium equations must address both degeneracy in $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$1 and one-sided temporal nonlocality.

For the equation with fractional potential pressure,

$${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$2

Allen, Caffarelli, and Vasseur prove that every nonnegative weak solution constructed under the exponential-decay assumptions is locally Hölder continuous in $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$3 (Allen et al., 2015). The proof adapts the De Giorgi–Nash–Moser method to a nonlocal-in-time, nonlocal-in-space, degenerate setting. It uses “pull-up” and “pull-down” lemmas, one-sided integration by parts for the Caputo derivative, mixed space-time energy inequalities, Sobolev embeddings in time and space, and a nonlinear $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$4 iteration (Allen et al., 2015).

For the nonlocal time porous medium equation

$${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$5

with Marchaud time derivative and $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$6, existence and uniqueness of weak solutions are established under

$${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$7

and bounded weak solutions are shown to be locally Hölder continuous for positive time by a De Giorgi–Nash–Moser argument (Djida et al., 2018).

In bounded domains, sharp decay and regularization results are available for

$${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$8

Under the assumptions of (Bonforte et al., 2024), there is an $${}^{C}\partial_t^\alpha u(x,t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}\,\partial_su(x,s)\,ds$$9 smoothing estimate

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$00

for $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$01 and suitable $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$02 depending on $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$03 (Bonforte et al., 2024). The same paper proves Benilan–Crandall-type monotonicity:

• if $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$04, then $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$05 is nonincreasing;
• if $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$06, then $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$07 is nondecreasing (Bonforte et al., 2024).

The long-time behavior in that bounded-domain framework is particularly notable. For every $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$08, there is no finite-time extinction, and every $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$09-norm decays like

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$10

(Bonforte et al., 2024). The paper emphasizes that the decay exponent $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$11 is independent of $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$12, and that temporal memory mitigates the distinction between slow and fast diffusion regimes (Bonforte et al., 2024).

For one-dimensional Barenblatt profiles of

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$13

the profile $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$14 is strictly increasing on $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$15, real-analytic on $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$16, globally Hölder continuous with exponent $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$17, and satisfies explicit upper and lower bounds together with free-boundary asymptotics

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$18

(Caballero et al., 25 Jul 2025).

These results indicate that the regularity theory bifurcates according to geometry and operator structure: bounded domains favor comparison-based decay estimates, while whole-space self-similar and nonlocal-pressure problems emphasize Hölder continuity and free-boundary structure.

6. Barenblatt solutions, compact support, and fast-diffusion profiles

The analogue of the classical Barenblatt solution is central to the subject. For the one-dimensional equation

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$19

a 2025 construction proves existence of a compactly supported, symmetric mass-one self-similar profile $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$20 with support $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$21, giving

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$22

inside the support and zero outside (Caballero et al., 25 Jul 2025). The profile is obtained from a nonlinear Volterra integral equation with an Erdélyi–Kober-type kernel expressed באמצעות incomplete Beta functions (Caballero et al., 25 Jul 2025). The half-mass condition

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$23

selects a unique $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$24 (Caballero et al., 25 Jul 2025).

A broader whole-space classification is given for

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$25

in all dimensions $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$26. There exist self-similar finite-mass solutions for all

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$27

and this range is optimal (Gómez-Castro et al., 10 Apr 2026). Two regimes are identified:

• for $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$28, there is a unique, nonincreasing, compactly supported profile with support $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$29;
• for $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$30, there is a family of positive, strictly decreasing profiles with heavy tails (Gómez-Castro et al., 10 Apr 2026).

In the slow-diffusion range $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$31, the profile vanishes at the free boundary according to

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$32

with an explicit coefficient involving Gamma functions (Gómez-Castro et al., 10 Apr 2026). In the mildly fast-diffusion range $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$33, the asymptotics are

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$34

with $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$35 defined implicitly by an integral equation involving $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$36 (Gómez-Castro et al., 10 Apr 2026).

The behavior near the similarity origin also depends on dimension. As $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$37,

• in $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$38, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$39 with power-law blow-up;
• in $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$40, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$41 grows like a logarithm;
• in $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$42, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$43 tends to a constant (Gómez-Castro et al., 10 Apr 2026). The authors describe this as reflecting a “Newtonian-potential-type” memory at the origin (Gómez-Castro et al., 10 Apr 2026).

These results clarify a frequent misconception inherited from the classical theory. In the time-fractional setting, compact support is not universal: it holds in the slow-diffusion regime $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$44, but for $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$45 the finite-mass self-similar solutions have heavy tails rather than compact support (Gómez-Castro et al., 10 Apr 2026).

7. Symmetry analysis and numerical methodologies

Lie-symmetry methods provide explicit reductions for the one-dimensional fractional porous medium equation

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$46

when the time derivative is taken in the Riemann–Liouville sense (Yang et al., 2019). The admitted Lie algebra is generated by

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$47

with $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$48 (Yang et al., 2019). An optimal one-dimensional subalgebra system is computed, similarity reductions are derived, and several invariant solutions are exhibited, including

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$49

for the translation-invariant case $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$50 (Yang et al., 2019).

Numerically, much of the literature exploits the self-similar reduction to convert the PDE into a Volterra integral equation or an EK-based integral equation. For the half-line equation with Dirichlet data, the nonlinear Abel-type Volterra equation

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$51

admits a family of explicit discretizations, including second-order trapezoidal schemes (Okrasińska-Płociniczak et al., 2021). The main convergence theorem states that if

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$52

then the scheme converges with order

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$53

(Okrasińska-Płociniczak et al., 2021).

For the EK operator itself, rectangle and trapezoid discretizations satisfy error bounds

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$54

with optimal truncations

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$55

respectively (López et al., 2023). In the same work, the rectangle backward-shooting scheme is explicit and yields an $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$56 algorithm (López et al., 2023).

For the Barenblatt integral equation in one dimension, product-integration on a uniform mesh yields a discrete fixed-point system

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$57

with the initial step selected from the free-boundary asymptotics (Caballero et al., 25 Jul 2025). If the quadrature error is $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$58, then the global error is

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$59

(Caballero et al., 25 Jul 2025).

A distinct numerical and variational direction is the JKO formulation for the nonlocal-pressure equation

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$60

where the discrete dynamics minimize

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$61

with modified Wasserstein distance $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$62 and energy

$$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$63

(Chung et al., 2024). Under $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$64, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$65, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$66, $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$67, and $$\partial_t^\alpha u=\partial_x\!\bigl(u^m\,\partial_x u\bigr),$$68, the scheme converges to a global weak solution (Chung et al., 2024).

The numerical literature therefore reflects two dominant strategies. One reduces to self-similar integral equations and exploits their one-dimensional weakly singular structure; the other builds variational time discretizations directly at the PDE level. This suggests that the computational treatment of time-fractional porous medium equations is shaped as much by their memory structure as by their nonlinear degeneracy.