Efficient and Structure-Preserving Numerical Methods for Time–Space Fractional Diffusion in Heterogeneous Biological Tissues

Abstract

Time–space fractional diffusion equations are widely used to model anomalous transport in heterogeneous biological tissues, where memory effects, spatial nonlocality, and coefficient variability are intrinsically coupled. However, existing numerical approaches typically treat these aspects in isolation, and a fully discrete framework that simultaneously accounts for heterogeneity, long-memory effects, and computational efficiency remains lacking. In this work, a fully discrete numerical method is developed and analyzed. The method integrates heterogeneous diffusion coefficients and memory-efficient temporal discretization within a unified variational framework. It combines a finite element approximation of a spectral fractional elliptic operator with an implicit $L^1$ discretization of the Caputo derivative enhanced by a sum-of-exponentials approximation of the memory kernel. Unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate are established, explicitly separating temporal, spatial, and kernel approximation errors. The proposed approach reduces memory complexity from $\mathcal{O}\left(N\right)$ to $\mathcal{O}(logN)$ without compromising accuracy. Numerical experiments confirm the theoretical convergence rates, demonstrate stable behavior across all tested configurations, and illustrate the impact of heterogeneous coefficients on anomalous transport dynamics.

1. Introduction

1.1. Problem Statement and Main Contribution

Time–space fractional diffusion equations have emerged as a powerful framework for modeling anomalous transport in complex media, particularly in heterogeneous biological tissues. Unlike classical diffusion models, fractional formulations naturally incorporate memory effects and spatial nonlocality, allowing them to capture subdiffusive and superdiffusive dynamics observed in experimental studies. The inclusion of spatially varying diffusion coefficients further enhances their modeling capability by accounting for structural heterogeneity across multiple scales.

Despite these advantages, the numerical approximation of time–space fractional diffusion problems remains challenging due to the simultaneous presence of long-memory effects, nonlocal spatial operators, and heterogeneous coefficients. These features introduce significant analytical and computational difficulties, particularly in the design of stable, accurate, and efficient numerical methods.

A key limitation of existing approaches is that these components are typically treated in a decoupled manner. In particular, memory-efficient temporal discretization techniques, such as sum-of-exponentials (SOE) approximations, are predominantly developed for problems involving homogeneous operators. Conversely, finite element approximations of fractional elliptic operators with heterogeneous coefficients are usually analyzed independently of fast temporal schemes. As a result, there is currently no fully discrete numerical framework that rigorously integrates spatial heterogeneity, long-memory effects, and reduced memory complexity.

The objective of this work is to address this gap by developing and analyzing a fully discrete numerical method that combines heterogeneous spectral fractional operators with memory-efficient temporal discretization within a unified variational framework.

1.2. Biomedical Motivation

Diffusion processes play a central role in numerous physiological and pathological mechanisms, including nutrient transport, metabolite exchange, signaling molecule propagation, and drug delivery within biological tissues. Classical diffusion models, derived from Fick’s laws, rely on assumptions of Brownian motion, spatial homogeneity, and the absence of long-range temporal correlations. Although these assumptions are mathematically convenient, they often fail to capture transport phenomena observed in living tissues [1,2].

Biological tissues are characterized by a highly complex and heterogeneous microstructure, including cellular membranes, organelles, extracellular matrix fibers, vascular networks, and varying pore geometries. These structural features introduce obstacles, trapping effects, transient binding, and spatial heterogeneity, all of which strongly influence particle motion. As a consequence, experimental measurements frequently reveal deviations from the linear mean squared displacement predicted by classical Fickian diffusion [3,4].

Anomalous diffusion has been reported in a wide range of biological systems using experimental techniques, such as fluorescence recovery after photobleaching (FRAP), diffusion-weighted magnetic resonance imaging (DW–MRI), and single-particle tracking. Subdiffusive behavior is commonly observed in crowded intracellular environments and soft biological matter, where viscoelasticity and molecular crowding dominate transport dynamics [5,6]. In contrast, superdiffusive transport may emerge in tissues exhibiting active processes, directed motion, or long-range structural correlations [7].

Fractional-order diffusion models provide a mathematically consistent and physically interpretable framework for describing such non-Fickian transport phenomena. Temporal fractional derivatives naturally incorporate memory effects and long-time correlations, while spatial fractional operators account for nonlocal transport and long-range interactions induced by tissue heterogeneity [8,9]. Importantly, the fractional orders appearing in these models can often be linked to experimentally measurable properties, establishing a meaningful connection between mathematical formulation and biological interpretation [10].

From an engineering and clinical perspective, accurate modeling of anomalous diffusion is essential for improving predictive simulations at the tissue scale, optimizing drug delivery strategies, and supporting the design of biomedical devices and therapeutic protocols. These considerations provide a strong motivation for the development of reliable, efficient, and mathematically rigorous computational methods for fractional diffusion equations arising in biological tissue modeling.

1.3. Mathematical Background

Fractional calculus extends classical differential operators by allowing derivatives and integrals of noninteger order. Unlike standard integer-order derivatives, fractional derivatives are inherently nonlocal operators, as they involve integration over an interval in time or space. This nonlocality provides a natural mathematical mechanism to incorporate memory effects and long-range interactions, which are essential features of anomalous transport processes observed in complex media [8,9].

In the temporal domain, fractional derivatives account for history-dependent dynamics. For instance, the Caputo fractional derivative of order $0<\alpha <1$ introduces a convolution kernel with algebraic decay, reflecting long-term memory effects and power-law waiting times between particle movements. Such behavior is a hallmark of subdiffusive transport and cannot be captured by classical first-order time derivatives [3,11]. From a modeling perspective, time-fractional diffusion equations arise naturally as macroscopic limits of continuous-time random walk processes with heavy-tailed waiting-time distributions.

In the spatial domain, fractional operators such as the fractional Laplacian or Riesz derivative generalize the classical second-order Laplace operator by incorporating nonlocal spatial interactions. These operators describe transport processes characterized by long jumps or Lévy flights, which are particularly relevant in heterogeneous or porous media. In biological tissues, spatial fractional derivatives provide a mathematical description of transport influenced by structural irregularities, connectivity across scales, and long-range correlations induced by the underlying microarchitecture [12,13].

Throughout this work, we adopt the spectral definition of the fractional Laplacian ${(-\mathrm{\Delta })}^{\beta /2}$, $\beta \in (1,2]$, defined via the eigenpairs of the classical Laplacian subject to the prescribed boundary conditions. This choice ensures a rigorous variational formulation in fractional Sobolev spaces and is well suited for finite element discretizations on bounded domains.

Time–space fractional diffusion equations combine these two generalizations into a unified mathematical framework, leading to models of the form

$${}_0{}^{C}D_t^{\alpha }u(x,t)={\mathcal{L}}^{\beta }u(x,t)+f(x,t),$$

(1)

where ${}_0{}^{C}D_t^{\alpha }$ denotes a fractional temporal derivative and ${\mathcal{L}}^{\beta }$ represents a fractional spatial operator of order $\beta$. Classical diffusion equations are recovered as special cases when $\alpha =1$ and $\beta =2$, highlighting the role of fractional models as genuine extensions rather than ad hoc modifications of standard theory [14].

From a mathematical standpoint, fractional diffusion equations pose significant analytical and computational challenges. The presence of nonlocal operators complicates the analysis of well-posedness, regularity, and stability, while also increasing the computational cost of numerical approximations. Nevertheless, these models offer remarkable flexibility: the fractional orders $\alpha$ and $\beta$ act as tunable parameters that encode medium heterogeneity, memory strength, and transport modality, enabling precise adaptation to experimentally observed phenomena [10,15].

In the context of biological tissue modeling, the mathematical structure of time–space fractional diffusion equations provides a principled way to bridge microscopic transport mechanisms and macroscopic tissue-scale behavior. This dual interpretability, combined with their strong theoretical foundations, makes fractional diffusion models particularly attractive for the quantitative description of anomalous transport in heterogeneous biological media and motivates the development of robust analytical and numerical tools for their solution.

1.4. Related Work

Fractional diffusion equations have received considerable attention over the last two decades due to their ability to describe anomalous transport processes in complex and heterogeneous media. Their nonlocal nature introduces both analytical and computational challenges, which has motivated the development of a wide range of numerical methods.

Early numerical approaches mainly focused on finite difference schemes for time-fractional diffusion equations, where the Caputo derivative is approximated by classical $L^1$-type formulas combined with standard spatial discretizations. These methods provide a simple and robust framework for the numerical approximation of subdiffusion models, but they often suffer from high computational cost due to the long-memory property of fractional derivatives, and do not address the challenges associated with spatial heterogeneity.

In recent years, significant progress has been made in the development of more accurate and stable temporal discretization schemes. Several studies have proposed corrected $L^1$ methods, transformed $L^1$ schemes, and other improved time-stepping techniques designed to capture the weak singular behavior of solutions near the initial time. For example, corrected $L^1$ schemes and related approaches have been analyzed for time-fractional diffusion problems with improved convergence properties [16]. Similarly, transformed $L^1$ methods have been proposed for multiterm fractional diffusion equations, providing enhanced numerical stability and accuracy [17].

Considerable research has also been devoted to the spatial discretization of fractional diffusion operators. Several numerical approaches have been developed, including finite difference methods based on Grünwald–Letnikov approximations, spectral methods, matrix-transfer techniques, and finite element formulations. Alternative analytical and numerical approaches have also been proposed for fractional transport equations. In particular, Lie symmetry techniques have been used to construct numerical procedures for space–fractional advection–diffusion equations with source terms (see, for example, [18]). Finite element methods are particularly attractive for bounded domains and heterogeneous media because they provide a natural variational framework and can accommodate complex geometries and variable material coefficients. Recent work has explored finite element discretizations for fractional diffusion problems with heterogeneous coefficients, highlighting the importance of spatial variability in realistic applications [19].

Another important research direction concerns the reduction in the computational complexity associated with the history dependence of fractional derivatives. Direct implementations of convolution-based schemes typically require $\mathcal{O}\left(N\right)$ memory and $\mathcal{O}\left(N^2\right)$ computational work over N time steps. To overcome these limitations, several fast algorithms have been proposed, including methods based on SOE approximations of the fractional kernel. Such techniques allow efficient evaluation of the Caputo derivative and significantly reduce memory and computational costs while preserving numerical accuracy [20,21]. However, their analysis is typically restricted to problems with homogeneous operators, limiting their applicability in heterogeneous settings.

Recent advances have also addressed the numerical approximation of fractional elliptic operators using finite element techniques. For instance, variational formulations based on spectral fractional powers of elliptic operators have been analyzed in several works, providing a rigorous mathematical framework for fractional diffusion problems posed on bounded domains. Finite element approximations of the fractional Laplacian and related operators have been investigated in detail in [19,22,23], where stability and convergence properties were established within appropriate fractional Sobolev spaces. These developments have significantly improved the numerical treatment of nonlocal operators arising in fractional PDE models. However, these approaches are usually developed independently of memory-efficient temporal discretizations, and, therefore, do not address the interaction between spatial heterogeneity and long-memory effects.

Another important line of research concerns the development of efficient time-stepping algorithms for fractional evolution equations. Classical convolution quadrature techniques introduced by Lubich [24] provide a general framework for the numerical approximation of fractional derivatives. More recently, fast time-stepping algorithms based on kernel compression and sum-of-exponentials approximations have been proposed to alleviate the computational burden associated with the long-memory structure of fractional derivatives. For example, fast $L^1$-type schemes and related algorithms have been developed to significantly reduce the storage and computational complexity of time-fractional diffusion simulations [20,21]. However, most existing analyses treat either the temporal discretization or the spatial operator in isolation. Existing analyses of $L^1$-type schemes for time-fractional diffusion equations (see, e.g., [25,26]) typically assume homogeneous spatial operators. On the other hand, finite element discretizations of fractional elliptic operators with heterogeneous coefficients have been investigated in [19,22], but without considering temporal fractional dynamics.

Fast convolution techniques based on SOE approximations have been studied for temporal fractional operators (e.g., [20]), where they provide efficient evaluation of the Caputo derivative and significantly reduce memory requirements. However, these approaches are typically developed under the assumption of homogeneous operators, and their extension to heterogeneous fractional elliptic problems remains largely unexplored.

Consequently, existing numerical frameworks still rely on a partial separation between temporal discretization, spatial fractional operators, and fast convolution techniques. In particular, there is currently no rigorously analyzed fully discrete method that simultaneously accounts for spatial heterogeneity, long-memory effects, and logarithmic memory complexity.

1.5. Contributions and Novelty

To address this gap, we develop and analyze a fully discrete numerical framework that directly couples heterogeneous fractional operators with memory-efficient temporal discretization, thereby eliminating the separation between spatial and temporal components present in existing approaches.

This coupling is nontrivial. In heterogeneous media, the spectral fractional operator depends explicitly on the spatial variability of $\kappa \left(x\right)$, which directly affects the eigenstructure of the discrete operator. At the same time, SOE-based compression techniques rely on a temporal convolution structure that is usually analyzed independently of spatial variability. Combining these two aspects at the fully discrete level precludes standard decoupling strategies and introduces analytical and computational challenges that are not addressed in existing frameworks.

This leads to a fully discrete scheme in which the spectral fractional operator with variable coefficients is consistently coupled with a memory-efficient approximation of the Caputo derivative, enabling the simultaneous treatment of spatial heterogeneity and long-memory effects.

The main contributions of this work are the following:

• Fully discrete formulation with heterogeneous coefficients. We develop a finite element discretization of the fractional elliptic operator and couple it with an implicit $L^1$ approximation of the Caputo derivative, resulting in a scheme applicable to strongly varying diffusion coefficients and not covered by existing analyses.
• Energy-structure preservation. We prove that the numerical method preserves a coercive discrete energy structure, ensuring physically consistent dissipative behavior.
• Unconditional stability. An $L^2\left(\mathsf{\Omega }\right)$ stability estimate is established that is independent of the mesh size h, time step $\mathrm{\Delta }t$, and fractional order $\alpha$.
• Error estimate with kernel compression. We derive a fully discrete error estimate that explicitly separates temporal, spatial, and kernel approximation contributions, showing that the use of SOE-based compression does not degrade the convergence order:

  $$\parallel u\left(t_n\right)-u_h^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\left(\mathrm{\Delta }{t}^{min(1,2-\alpha )}+h^s+{\epsilon }_{\mathrm{SOE}}\right).$$

  (2)
• Computational validation. Numerical experiments confirm stability, convergence, and the expected reduction in memory requirements in heterogeneous configurations.

The resulting scheme achieves unconditional stability, preserves a discrete energy structure, and reduces the memory requirement from $\mathcal{O}\left(MN\right)$ to $\mathcal{O}(MlogN)$ without loss of accuracy.

To the best of our knowledge, this is the first rigorously analyzed fully discrete framework that eliminates the separation between temporal discretization, spatial heterogeneity, and memory-efficient convolution, while simultaneously achieving logarithmic memory complexity.

1.6. Relation with Existing Numerical Approaches

The numerical components employed in this work, including the $L^1$ time discretization, finite element approximation of fractional operators, and SOE-based memory compression, have each been studied extensively in the literature.

However, these components are typically analyzed in isolation and often under simplifying assumptions such as homogeneous diffusion coefficients. In contrast, the present work investigates their interaction within a single fully discrete framework, incorporating heterogeneous coefficients, spectral fractional operators, and memory-efficient temporal discretization.

This unified approach is particularly relevant for applications involving heterogeneous biological tissues, where spatial variability and long-memory effects must be treated simultaneously.

1.7. Organization of the Paper

This paper is organized as follows. Section 2 introduces the mathematical model and operator framework. Section 3 establishes the functional setting and well-posedness. Section 4 presents the fully discrete numerical method, including temporal and spatial discretization, and provides stability and error analysis. Section 5 reports numerical experiments validating the theoretical results.

2. Mathematical Model of Anomalous Diffusion in Biological Tissues

This section establishes the mathematical formulation of the time–space fractional diffusion model studied in this work. Our objective is to define the governing equation in a rigorous operator-theoretic framework that accommodates variable diffusion coefficients and fractional-order dynamics. We begin by introducing the evolution problem on a bounded domain together with precise definitions of the temporal and spatial operators involved. The resulting formulation provides the analytical foundation required for the functional setting and numerical discretization developed in the subsequent sections.

2.1. Governing Equation

Let $\mathsf{\Omega }\subset {\mathbb{R}}^d$, with $d\ge 1$, be a bounded Lipschitz domain representing a biological tissue region, and let $T>0$ denote the final observation time. We consider the following time–space fractional diffusion problem:

$${}_0{}^{C}D_t^{\alpha }u(x,t)+A^{\beta /2}u(x,t)=f(x,t),(x,t)\in \mathsf{\Omega }\times (0,T],$$

(3)

where $u(x,t)$ denotes the concentration of a diffusing quantity within the tissue, such as a solute or tracer, and $f(x,t)$ represents a source or sink term.

2.1.1. Caputo Time Derivative

The operator ${}_0{}^{C}D_t^{\alpha }$ denotes the Caputo fractional derivative of order $0<\alpha <1$, defined by

$${}_0{}^{C}D_t^{\alpha }u(x,t)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{\displaystyle \frac{\partial u(x,s)}{\partial s}}ds,$$

(4)

which incorporates memory effects through a convolution kernel with algebraic decay [8,9]. The parameter $\alpha$ controls the strength of temporal nonlocality, with subdiffusive behavior arising when $\alpha <1$.

2.1.2. Elliptic Operator with Variable Coefficients

Consider the second-order elliptic operator defined in Equation = (3)

$$Au:=-\nabla ·\left(\kappa \right(x)\nabla u),$$

(5)

posed on $\mathsf{\Omega }$ with homogeneous Dirichlet boundary conditions. The diffusion coefficient $\kappa \left(x\right)$ is assumed to satisfy

$$0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max}<\infty \mathrm{for}\mathrm{almost}\mathrm{every}x\in \mathsf{\Omega },$$

(6)

ensuring uniform ellipticity.

Under these assumptions, we further assume that

$$\kappa \in L^{\infty }\left(\mathsf{\Omega }\right),0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max}<\infty .$$

(7)

These conditions guarantee that the operator A is uniformly elliptic and defines a self-adjoint positive operator in $L^2\left(\mathsf{\Omega }\right)$ with compact inverse. Consequently, there exists an orthonormal basis of eigenfunctions ${\left\{{\varphi }_m\right\}}_{m=1}^{\infty }\subset H_0^1\left(\mathsf{\Omega }\right)$ and a sequence of eigenvalues ${\left\{{\lambda }_m\right\}}_{m=1}^{\infty }$ satisfying

$$A{\varphi }_m={\lambda }_m{\varphi }_m,0<{\lambda }_1\le {\lambda }_2\le \cdots ,{\lambda }_m\to \infty .$$

(8)

2.1.3. Spectral Fractional Power

For $\beta \in (1,2]$, the fractional power of A is defined spectrally by

$$A^{\beta /2}u=\sum _{m=1}^{\infty }{\lambda }_m^{\beta /2}(u,{\varphi }_m){\varphi }_m,$$

(9)

for all u such that

$$\sum _{m=1}^{\infty }{\lambda }_m^{\beta }{\left|(u,{\varphi }_m)\right|}^2<\infty .$$

(10)

The domain of the fractional operator is, therefore,

$$D\left(A^{\beta /2}\right)=\left\{u\in L^2\left(\mathsf{\Omega }\right):\sum _{m=1}^{\infty }{\lambda }_m^{\beta }{\left|(u,{\varphi }_m)\right|}^2<\infty \right\},$$

(11)

which coincides with an appropriate fractional Sobolev space.

This spectral construction guarantees that $A^{\beta /2}$ remains self-adjoint and coercive in $L^2\left(\mathsf{\Omega }\right)$, and is particularly suitable for variable diffusion coefficients. Throughout this work, the spatial fractional diffusion operator is interpreted exclusively in this spectral sense, thereby avoiding ambiguity with Riesz-type integral definitions.

2.1.4. Connection with Classical Diffusion

Equation (3) generalizes the classical diffusion equation. In the limiting case $\alpha =1$ and $\beta =2$, one recovers

$${\partial }_{t}u-\nabla ·(\kappa \left(x\right)\nabla u)=f,$$

(12)

which corresponds to the standard Fickian diffusion model. The fractional formulation, therefore, constitutes a mathematically consistent extension of classical diffusion theory rather than a purely phenomenological modification [14].

2.2. Biological Interpretation

The fractional temporal order $\alpha$ reflects the strength of memory effects in the transport process. Values of $\alpha <1$ correspond to subdiffusive behavior, which is commonly observed in biological tissues due to molecular crowding, binding interactions, and viscoelasticity of the intracellular and extracellular environments [5,6]. In this context, the Caputo derivative provides a causal representation of memory, making it suitable for modeling physiological processes with well-defined initial states.

The spatial fractional order $\beta$ characterizes the degree of nonlocal spatial transport. Values of $\beta <2$ indicate the presence of long-range particle displacements or Lévy-type transport mechanisms, which may arise from heterogeneous tissue architecture, porous structures, or multiscale connectivity within the biological medium [4,7]. The combination of time- and space-fractional operators allows the model to capture a wide spectrum of anomalous diffusion behaviors observed experimentally.

The spatially varying diffusivity $\kappa \left(x\right)$ further enhances the biological realism of the model by accounting for tissue heterogeneity, such as variations in cellular density, extracellular matrix composition, or perfusion. Together, the parameters $(\alpha ,\beta ,\kappa )$ form a compact yet expressive representation of tissue-scale transport dynamics [10,15].

2.3. Initial and Boundary Conditions

To complete the mathematical model, Equation (3) is supplemented with an initial condition

$$u(x,0)=u_0\left(x\right),x\in \mathsf{\Omega },$$

(13)

where $u_0\left(x\right)$ represents the initial distribution of the diffusing quantity, assumed to be nonnegative and sufficiently regular.

Boundary conditions are chosen to reflect physiological constraints and depend on the specific biomedical application. Common choices include homogeneous Dirichlet conditions,

$$u(x,t)=0,x\in \partial \mathsf{\Omega },t>0,$$

(14)

which may model complete absorption or clearance at tissue boundaries, and homogeneous Neumann-type conditions,

$$\left(\kappa \left(x\right){\nabla }^{\beta }u(x,t)\right)·n\left(x\right)=0,x\in \partial \mathsf{\Omega },t>0,$$

(15)

representing no-flux or impermeable boundaries [13]. We remark that, for nonlocal fractional operators, classical Neumann boundary conditions require careful interpretation. In the present work, boundary conditions are understood in the variational sense associated with the chosen spectral definition of the operator.

More general boundary conditions, including mixed or nonlocal formulations, can also be considered to describe exchange with surrounding tissues or vascular compartments. The well-posedness and numerical treatment of such conditions in the fractional setting require particular care due to the nonlocal nature of the spatial operator [19,27]. These modeling choices are discussed in greater detail in the numerical experiments section.

3. Functional Setting and Analytical Properties

3.1. Fractional Sobolev Spaces

The analysis of time–space fractional diffusion equations requires an appropriate functional framework that accounts for the nonlocal nature of the involved operators. Let $\mathsf{\Omega }\subset {\mathbb{R}}^d$ be a bounded Lipschitz domain. For $s\in (0,1)$ and $1\le p<\infty$, the fractional Sobolev space $W^{s,p}\left(\mathsf{\Omega }\right)$ is defined as

$$W^{s,p}\left(\mathsf{\Omega }\right)=\left\{v\in L^p\left(\mathsf{\Omega }\right):{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{{|v\left(x\right)-v\left(y\right)|}^p}{{|x-y|}^{d+sp}}}dxdy<\infty \right\},$$

(16)

endowed with the norm

$${\parallel v\parallel }_{W^{s,p}\left(\mathsf{\Omega }\right)}={\left({\parallel v\parallel }_{L^p\left(\mathsf{\Omega }\right)}^p+{\int }_{\mathsf{\Omega }}{\int }_{\mathsf{\Omega }}{\displaystyle \frac{{|v\left(x\right)-v\left(y\right)|}^p}{{|x-y|}^{d+sp}}}dxdy\right)}^{1/p}.$$

(17)

In the Hilbertian case $p=2$, we write $H^s\left(\mathsf{\Omega }\right):=W^{s,2}\left(\mathsf{\Omega }\right)$.

Fractional Sobolev spaces provide the natural variational setting for space-fractional operators, such as the fractional Laplacian. In particular, for $\beta \in (1,2]$, the operator $A^{\beta /2}$ is typically associated with the space $H^{\beta /2}\left(\mathsf{\Omega }\right)$, where weak formulations and energy estimates can be rigorously established [19,28]. These spaces generalize classical Sobolev spaces and retain key properties such as completeness, reflexivity, and compact embeddings under suitable conditions.

In the temporal dimension, the fractional Caputo derivative naturally leads to solution spaces involving $L^2(0, T; H^{\beta /2}\left(\mathsf{\Omega }\right))$ together with weighted Sobolev spaces in time that reflect the history dependence of the solution. Detailed discussions of fractional Sobolev regularity in time can be found in [11,29].

3.2. Well-Posedness

We now address the well-posedness of the time–space fractional diffusion problem introduced in Section 2. Let $u_0\in L^2\left(\mathsf{\Omega }\right)$, $f\in L^2(0, T; L^2\left(\mathsf{\Omega }\right))$, and assume that the diffusion coefficient $\kappa \left(x\right)$ satisfies

$$0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max}<\infty \mathrm{for}\mathrm{almost}\mathrm{every}x\in \mathsf{\Omega }.$$

(18)

Definition 1.

A function $u\in L^2(0, T; H^{\beta /2}\left(\mathsf{\Omega }\right))$ is called a weak solution of the time–space fractional diffusion problem if $u(·, 0)=u_0$ and, for all test functions $v\in H^{\beta /2}\left(\mathsf{\Omega }\right)$ and almost every $t\in (0, T)$,

$$\left({}_0{}^{C}D_t^{\alpha }u\left(t\right),v\right)+a(u\left(t\right),v)=(f\left(t\right),v),$$

(19)

where $(·,·)$ denotes the $L^2\left(\mathsf{\Omega }\right)$ inner product and

$$a(u,v)={(A^{\beta /4}u,A^{\beta /4}v)}_{L^2\left(\mathsf{\Omega }\right)}.$$

(20)

This bilinear form is symmetric, coercive, and continuous on $H^{\beta /2}\left(\mathsf{\Omega }\right)$ under the assumption $0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max}<\infty$.

Theorem 1.

Under the above assumptions on f, $u_0$, and $\kappa \left(x\right)$, the time–space fractional diffusion problem admits a unique weak solution

$$u\in L^2(0, T; H^{\beta /2}\left(\mathsf{\Omega }\right)),$$

(21)

which depends continuously on the data.

Proof. 

The proof follows from standard arguments combining fractional energy estimates with the theory of maximal monotone operators. The coercivity and boundedness of the bilinear form $a(·, ·)$ in $H^{\beta /2}\left(\mathsf{\Omega }\right)$ ensure spatial well-posedness. The properties of the Caputo derivative yield an appropriate fractional Grönwall inequality, which guarantees uniqueness and continuous dependence on the data [26,29]. Existence is obtained via Galerkin approximations and compactness arguments adapted to the fractional setting [14].    □

This well-posedness result establishes a rigorous analytical foundation for the numerical methods developed in the subsequent sections.

Throughout the numerical analysis, we assume that the exact solution u of problem (3) satisfies

$$u\in C([0,T];H^{\beta /2}\left(\mathsf{\Omega }\right)),{\partial }_{t}u\in L^1(0,T;H^{\beta /2}\left(\mathsf{\Omega }\right)),$$

(22)

and there exists a constant $C>0$ such that

$$\parallel {\partial }_t{u\left(t\right)\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}\le C{t}^{\alpha -1},t\in (0,T].$$

(23)

This weak singularity at the initial time is a characteristic feature of time-fractional diffusion equations and is essential for deriving sharp temporal error estimates.

4. Numerical Method and Discretization

This section presents a fully discrete numerical method for the time–space fractional diffusion problem (3). Particular attention is paid to the nonlocal nature of both the temporal and spatial operators, with the goal of constructing a scheme that is stable, convergent, and memory efficient, while remaining suitable for heterogeneous biological tissues.

4.1. Temporal Discretization

Let $0=t_0<t_1<\cdots <t_N=T$ be a uniform partition of the time interval $[0,T]$ with time step $\mathrm{\Delta }t=T/N$, and denote $u^n\left(x\right)\approx u(x,t_n)$. The Caputo fractional derivative at time $t=t_n$ is defined by

$${}_0{}^{C}D_t^{\alpha }u(x,t_n)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_0^{t_n}{(t_n-s)}^{-\alpha }{\displaystyle \frac{\partial u(x,s)}{\partial s}}ds.$$

(24)

A classical and robust approximation of this operator is provided by the $L^1$ scheme, which is derived by approximating ${\partial }_{t}u$ by a backward finite difference on each subinterval $[t_{k-1},t_k]$. This yields

$${}_0{}^{C}D_t^{\alpha }u(x,t_n)\approx {\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\sum _{k=0}^{n-1}{b}_k\left(u^{n-k}\left(x\right)-u^{n-k-1}\left(x\right)\right),$$

(25)

where the convolution weights are given by

$$b_k={(k+1)}^{1-\alpha }-k^{1-\alpha },k\ge 0.$$

(26)

The $L^1$ scheme is first-order accurate in time for sufficiently smooth solutions and has the advantage of unconditional stability when combined with implicit spatial discretizations. However, a well-known drawback is its $\mathcal{O}\left(N\right)$ memory requirement per spatial degree of freedom, which becomes prohibitive for long-time simulations typical in biological diffusion problems.

To reduce memory and computational costs, we adopt a memory-efficient convolution strategy based on truncation and compression of the history term. Specifically, the convolution sum in (25) is decomposed into a recent-history part, computed exactly, and a distant-history part, approximated using an SOE approximation representation of the kernel $t^{-\alpha }$. This approach preserves the accuracy and stability of the $L^1$ scheme while reducing the memory complexity from $\mathcal{O}\left(N\right)$ to $\mathcal{O}(logN)$, which is essential for realistic biomedical simulations over long time horizons.

4.2. Sum-of-Exponentials Approximation of the Memory Kernel

The main computational difficulty of the $L^1$ approximation of the Caputo derivative lies in the history term

$$C_0{D}_t^{\alpha }u\left(t_n\right)={\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\sum _{k=0}^{n-1}{b}_k\left(u^{n-k}-u^{n-k-1}\right),$$

(27)

which requires storing and processing all previous solution values. A direct implementation, therefore, entails $O\left(N\right)$ memory per spatial degree of freedom and $O\left(N^2\right)$ total operations over N time steps.

To overcome this limitation, we approximate the convolution kernel

$$\omega \left(t\right)=t^{-\alpha },0<\alpha <1,$$

(28)

by SOE,

$$t^{-\alpha }\approx \sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t},$$

(29)

where $w_j>0$ and ${\lambda }_j>0$ are suitably chosen weights and exponents, and J depends logarithmically on the desired tolerance.

4.2.1. Construction of the SOE Representation

The approximation (29) is constructed by exploiting the Laplace transform identity

$$t^{-\alpha }={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\infty }{e}^{-ts}{s}^{\alpha -1}ds,$$

(30)

and approximating the integral by a quadrature rule on a suitably truncated interval. Following standard fast convolution techniques, one introduces a geometric partition of the Laplace variable and applies a suitable quadrature formula, yielding

$$t^{-\alpha }\approx \sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t}\mathrm{such}\mathrm{that}|t^{-\alpha }-\sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t}|\le {\epsilon }_{\mathrm{SOE}}$$

(31)

for all $t\in [\mathrm{\Delta }t,T]$.

For a prescribed tolerance ${\epsilon }_{\mathrm{SOE}}>0$, the number of exponentials satisfies

$$J=O\left(log(T/\mathrm{\Delta }t)+|log{\epsilon }_{\mathrm{SOE}}|\right),$$

(32)

which ensures logarithmic memory complexity.

4.2.2. Fast Recursive Evaluation

Using representation (29), the distant-history contribution to the Caputo derivative can be written as

$$\sum _{k=0}^{n-1}\omega (t_n-t_k)\delta u^k\approx \sum _{j=1}^J{w}_j\sum _{k=0}^{n-1}{e}^{-{\lambda }_j(t_n-t_k)}\delta u^k,$$

(33)

where $\delta u^k=u^{k+1}-u^k$.

Define auxiliary variables

$$H_j^n=\sum _{k=0}^{n-1}{e}^{-{\lambda }_j(t_n-t_k)}\delta u^k.$$

(34)

These quantities satisfy the recurrence relation

$$H_j^n=e^{-{\lambda }_j\mathrm{\Delta }t}{H}_j^{n-1}+\delta u^{n-1},n\ge 1,$$

(35)

with initial value $H_j^0=0$.

Therefore, the history contribution can be updated recursively at each time step with $O\left(J\right)$ additional operations, without storing the entire solution history.

4.2.3. Decomposition of Recent and Distant History

To further enhance accuracy, the convolution sum is decomposed into

$$(\mathrm{recent}\mathrm{part})+(\mathrm{distant}\mathrm{part}),$$

(36)

where the recent part over a fixed number of time steps is computed exactly using the classical $L^1$ weights, while the distant part is approximated using the SOE representation. This hybrid strategy preserves first-order temporal accuracy while significantly reducing memory requirements.

4.2.4. Complexity and Accuracy

The resulting algorithm requires

$$\mathrm{Memory}=O\left(JM\right),\mathrm{Operations}\mathrm{per}\mathrm{time}\mathrm{step}=O\left(JM\right),$$

(37)

where M denotes the number of spatial degrees of freedom. As $J=O(logN)$, the total memory complexity reduces from $O\left(NM\right)$ to $O(MlogN)$, and the total computational cost becomes $O(NMlogN)$.

The fully discrete error analysis in Section 4.7 explicitly accounts for the additional approximation error ${\epsilon }_{\mathrm{SOE}}$, ensuring that memory compression does not compromise stability or convergence.

In practice, the parameters ${\{w_j,{\lambda }_j\}}_{j=1}^J$ are obtained using the algorithm proposed in [20], which constructs an accurate sum-of-exponentials approximation of the kernel $t^{-\alpha }$ on the interval $[\mathrm{\Delta }t,T]$.In our implementation we choose the tolerance

$${\epsilon }_{\mathrm{SOE}}={10}^{-8},$$

(38)

which results in a number of exponentials typically in the range $J\approx 20$–40 for the time horizons considered in the numerical experiments.

For completeness, we briefly outline the main idea of the construction. The kernel admits the Laplace transform representation

$$t^{-\alpha }={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_0^{\infty }{e}^{-ts}{s}^{\alpha -1}ds.$$

(39)

The above integral is truncated to a finite interval $[s_{min},s_{max}]$ depending on the prescribed tolerance ${\epsilon }_{\mathrm{SOE}}$, and then approximated by a quadrature rule with geometrically spaced nodes $s_j$. This leads to the sum-of-exponentials representation

$$t^{-\alpha }\approx \sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t},$$

(40)

where ${\lambda }_j=s_j$ and the weights $w_j$ incorporate the quadrature weights together with the factor $s_j^{\alpha -1}/\mathrm{\Gamma }\left(\alpha \right)$. This construction guarantees the uniform error bound

$$\underset{t\in [\mathrm{\Delta }t,T]}{sup}\left|t^{-\alpha }-\sum _{j=1}^J{w}_j{e}^{-{\lambda }_{j}t}\right|\le {\epsilon }_{\mathrm{SOE}}.$$

(41)

4.3. Spatial Discretization

We now describe the spatial discretization of the fractional diffusion operator. For clarity of exposition, we restrict the presentation to the one–dimensional domain $\mathsf{\Omega }=(a,b)$; the extension to higher spatial dimensions follows by analogous arguments.

In Section 2, the spatial operator was introduced through the spectral fractional power of the elliptic operator A, defined in Equation (5), posed on $\mathsf{\Omega }$ with homogeneous Dirichlet boundary conditions. Under the standard assumption

$$0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max},$$

(42)

the operator A is self–adjoint and positive definite in $L^2\left(\mathsf{\Omega }\right)$, and, therefore, admits a complete orthonormal system of eigenfunctions ${\left\{{\phi }_m\right\}}_{m=1}^{\infty }$ with associated eigenvalues ${\left\{{\lambda }_m\right\}}_{m=1}^{\infty }$.

The fractional power $A^{\beta /2}$, with $\beta \in (1,2]$, is defined spectrally as

$$A^{\beta /2}u=\sum _{m=1}^{\infty }{\lambda }_m^{\beta /2}(u,{\phi }_m){\phi }_m,$$

(43)

which provides a natural variational framework in the fractional Sobolev space $H^{\beta /2}\left(\mathsf{\Omega }\right)$. This formulation is particularly convenient for bounded domains and is consistent with finite element approximations of fractional elliptic operators (see, e.g., [19,22,23]).

The weak formulation associated with the spectral fractional operator can be written as

$$a(u,v)={\left(A^{\beta /4}u,A^{\beta /4}v\right)}_{L^2\left(\mathsf{\Omega }\right)},$$

(44)

which defines a symmetric and coercive bilinear form on $H^{\beta /2}\left(\mathsf{\Omega }\right)$.

In the finite element setting, the fractional operator is approximated through the spectral fractional power of the discrete elliptic operator obtained from the stiffness matrix, which yields a consistent variational approximation of the continuous operator $A^{\beta /2}$.

To approximate the solution numerically, we employ a standard finite element discretization. Let ${\mathcal{T}}_h$ be a conforming partition of $\mathsf{\Omega }$ with mesh size h, and let

$$V_h\subset H_0^1\left(\mathsf{\Omega }\right)$$

(45)

denote the space of continuous, piecewise linear finite elements associated with ${\mathcal{T}}_h$.

The semi-discrete formulation then reads: find $u_h\left(t\right)\in V_h$ such that for all $v_h\in V_h$

$$\left({}_0{}^{C}D_t^{\alpha }{u}_h\left(t\right),v_h\right)+a(u_h\left(t\right),v_h)=(f\left(t\right),v_h).$$

(46)

Finite Element Implementation

Let ${\mathcal{T}}_h$ be a quasi-uniform triangulation of the bounded domain $\mathsf{\Omega }\subset {\mathbb{R}}^d$ with mesh size h. We consider the standard conforming finite element space of continuous piecewise linear functions adapted to the prescribed boundary conditions,

$$V_h=\left\{v_h\in C^0\left(\overline{\mathsf{\Omega }}\right):v_h{|}_K\in {\mathbb{P}}_1\left(K\right),\forall K\in {\mathcal{T}}_h\right\}.$$

(47)

We denote by ${\left\{{\varphi }_i\right\}}_{i=1}^M$ the nodal basis of $V_h$, where M is the number of spatial degrees of freedom. The semi-discrete finite element approximation can be written as

$$u_h(x,t)=\sum _{i=1}^M{U}_i\left(t\right){\varphi }_i\left(x\right),$$

(48)

where $U_i\left(t\right)$ are time-dependent coefficients collected in the vector $U\left(t\right)={(U_1\left(t\right),\dots ,U_M\left(t\right))}^T$.

Substituting this representation into the weak formulation yields the semi-discrete system

$$M_{mass}{}_0{}^{C}D_t^{\alpha }U\left(t\right)+K_{\beta }U\left(t\right)=F\left(t\right),$$

(49)

$M_{mass}$ and $K_{\beta }$ denote the mass matrix and the stiffness matrix associated with the bilinear form defined in Equation (46). The vector $F\left(t\right)$ represents the load term with components

$$F_i\left(t\right)=(f\left(t\right),{\varphi }_i).$$

(50)

Due to the nonlocal character of the fractional operator, the matrix $K_{\beta }$ is generally dense. Efficient compression techniques are, therefore, required in large-scale simulations. Fast approaches such as hierarchical matrices [30] and the fast multipole method [31] are commonly employed, and recent developments continue to improve their efficiency for fractional and nonlocal operators [32].

In practice, the matrix $K_{\beta }$ is obtained from the spectral fractional power of the discrete elliptic operator. More precisely, $K_{\beta }$ corresponds to the spectral fractional power of the discrete operator $A_h=M_{mass}^{-1}K$ arising from the standard finite element discretization of A. Let K denote the standard finite element stiffness matrix associated with the operator A as in Equation (5). Assuming the generalized eigenvalue decomposition

$$K\mathsf{\Phi }=M_{mass}\mathsf{\Phi }\Lambda ,$$

(51)

where $\Lambda =\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_M)$ contains the eigenvalues and $\mathsf{\Phi }$ the corresponding $M_{mass}$-orthonormal eigenvectors, the discrete fractional operator can be written as

$$K_{\beta }=M_{mass}\mathsf{\Phi }{\Lambda }^{\beta /2}{\mathsf{\Phi }}^T.$$

(52)

In practical computations, the explicit eigenvalue decomposition is avoided, and the action of $K_{\beta }$ is evaluated using matrix-function techniques available in the FEniCS linear algebra backend.

4.4. Fully Discrete Scheme

Combining the $L^1$ approximation of the Caputo derivative (25) with the finite element spatial discretization introduced in Section 4.3 yields the following fully discrete scheme. Let $t_n=n\mathrm{\Delta }t$ for $n=0,1,\dots ,N$. For each time level $n=1,2,\dots ,N$, find $u_h^n\in V_h$ such that for all $v_h\in V_h$

$${\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\sum _{k=0}^{n-1}{b}_k\left(u_h^{n-k}-u_h^{n-k-1},v_h\right)+a(u_h^n,v_h)=(f^n,v_h),$$

(53)

where $f^n=f\left(t_n\right)$ and the coefficients

$$b_k={(k+1)}^{1-\alpha }-k^{1-\alpha }$$

(54)

are the classical $L^1$ convolution weights.

Let ${\left\{{\varphi }_i\right\}}_{i=1}^M$ denote the nodal basis of $V_h$, where M is the number of spatial degrees of freedom. The finite element approximation at time level $t_n$ is written as

$$u_h^n\left(x\right)=\sum _{i=1}^M{U}_i^n{\varphi }_i\left(x\right),$$

(55)

where $U_i\left(t\right)$ are time-dependent coefficients collected in the vector $U\left(t\right)={(U_1\left(t\right),\dots ,U_M\left(t\right))}^T$. Substituting this representation into the weak formulation (53) yields the linear system

$$\left({\displaystyle \frac{b_0}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}{M}_{mass}+K_{\beta }\right)U^n=F^n+H^n,$$

(56)

for $M_{mass}$ the mass matrix, $K_{\beta }$ the stiffness matrix associated with the fractional bilinear form $a(·,·)$, and

$$F_i^n=(f^n,{\varphi }_i)$$

(57)

represents the source term.

The vector $H^n$ contains the history contribution arising from the $L^1$ discretization,

$$H^n={\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\sum _{k=1}^{n-1}{b}_k{M}_{mass}\left(U^{n-k}-U^{n-k-1}\right).$$

(58)

Introducing the constant

$$c_{\alpha }={\displaystyle \frac{b_0}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}},$$

(59)

the linear system can be written compactly as

$$(c_{\alpha }{M}_{mass}+K_{\beta })U^n=G^n,$$

(60)

where $G^n=F^n+H^n$.

The resulting scheme is implicit with respect to the spatial fractional operator and is, therefore, expected to be unconditionally stable.

The numerical implementation was carried out using the FEniCS finite element framework, which provides high-level tools for the automated assembly of variational forms and sparse matrices. The finite element space $V_h$ was constructed using continuous piecewise linear elements on a uniform mesh of $\mathsf{\Omega }$. Define the system matrix

$$A_h=c_{\alpha }{M}_{mass}+K_{\beta }.$$

(61)

As the system matrix does not depend on the time level, its sparse LU factorization was computed once and reused at each time step. This strategy significantly reduces the computational cost of the time-stepping procedure.

For the problem sizes considered here, the resulting linear systems were solved using the sparse direct solvers available in the FEniCS linear algebra backend.

4.5. Computational Complexity

We conclude this section by analyzing the computational cost of the proposed method.

Let M denote the number of spatial degrees of freedom and N the number of time steps. A direct implementation of the $L^1$ scheme would require $\mathcal{O}\left(MN\right)$ storage and $\mathcal{O}\left(MN\right)$ operations. By employing a compressed SOE-based history approximation, the memory requirement is reduced to $\mathcal{O}(MlogN)$ and the total computational cost to $\mathcal{O}(MNlogN)$.

Importantly, the spatial discretization cost is dominated by the solution of a linear system with a dense but structured matrix. In practice, this cost can be significantly reduced using hierarchical matrices or fast multipole-type techniques, which are particularly effective for fractional operators.

Overall, the proposed numerical method achieves a favorable balance between accuracy, stability, and computational efficiency, making it suitable for large-scale simulations of anomalous diffusion in heterogeneous biological tissues.

Theorem 2 

(Computational complexity of the proposed algorithm). Let M denote the number of spatial degrees of freedom and N the number of time steps. Assume that the Caputo convolution kernel is approximated by SOE representation with $J=\mathcal{O}(logN)$ exponential terms. Then the fully discrete algorithm defined by scheme (53) requires

$$Memory=\mathcal{O}(MlogN),Totalcomputationalcost=\mathcal{O}(MNlogN).$$

(62)

In particular, the use of the SOE approximation reduces the memory requirement of the classical $L^1$ scheme from $\mathcal{O}\left(MN\right)$ to $\mathcal{O}(MlogN)$ while preserving the first-order temporal accuracy of the discretization.

4.6. Stability of the Fully Discrete Scheme

Theorem 3 

(Discrete energy structure preservation). Let $u_h^n$ denote the fully discrete solution obtained from fully discrete scheme (53). Under the assumptions

$$0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max},$$

(63)

the numerical scheme satisfies the discrete energy identity

$$E^n+\mathrm{\Delta }{t}^{\alpha }\sum _{k=1}^{n}a(u_h^k,u_h^k)\le E^0+C\sum _{k=1}^n{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2,$$

(64)

where the discrete energy is defined by

$$E^n={\parallel u_h^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.$$

(65)

Consequently, the proposed numerical method preserves the coercive dissipative structure of the continuous fractional diffusion problem.

Theorem 4 

(Unconditional stability in $L^2\left(\mathsf{\Omega }\right)$). Let $u_h^n\in V_h$ be the solution of the fully discrete scheme (53). Assume that the diffusion coefficient $\kappa \left(x\right)$ satisfies

$$0<{\kappa }_{min}\le \kappa \left(x\right)\le {\kappa }_{max}.$$

(66)

Then the fully discrete scheme is unconditionally stable in the sense that

$$\parallel u_h^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\mathrm{\Delta }{t}^{\alpha }\sum _{k=1}^n{\parallel u_h^k\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}^2\le C\left(\parallel u_0{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\underset{1\le k\le n}{max}{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right),$$

(67)

where the constant $C>0$ is independent of the spatial mesh size h, the time step $\mathrm{\Delta }t$, and the time level n.

Proof. 

We test the fully discrete scheme (53) with $v_h=u_h^n$. This yields

$${\displaystyle \frac{1}{\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\sum _{k=0}^{n-1}{b}_k\left(u_h^{n-k}-u_h^{n-k-1},u_h^n\right)+a(u_h^n,u_h^n)=(f^n,u_h^n).$$

We emphasize that the positivity property of the $L^1$ discrete convolution operator remains valid in the present setting, as the temporal discretization acts on the solution coefficients independently of the spatial operator, which enters only through the coercive bilinear form (see [33,34]). Consequently, we obtain

$${\displaystyle \frac{1}{2\mathrm{\Gamma }(2-\alpha ){\left(\mathrm{\Delta }t\right)}^{\alpha }}}\left(\parallel u_h^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2-{\parallel u_h^{n-1}\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right)+a(u_h^n,u_h^n)\le (f^n,u_h^n).$$

The coercivity of the bilinear form $a(·,·)$ implies

$$a(u_h^n,u_h^n)\ge c_0{\parallel u_h^n\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}^2$$

for some constant $c_0>0$ depending only on ${\kappa }_{min}$.

Using the Cauchy–Schwarz and Young inequalities, we obtain

$$(f^n,u_h^n)\le {\displaystyle \frac{1}{2}}\parallel f^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\displaystyle \frac{1}{2}}{\parallel u_h^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.$$

Combining the above estimates yields a discrete energy inequality of the form

$$\parallel u_h^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2-\parallel u_h^{n-1}{\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\mathrm{\Delta }{t}^{\alpha }\parallel u_h^n{\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}^2\le C{\parallel f^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.$$

Summing the above inequality for $k=1,\dots ,n$ gives

$${\parallel u_h^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\mathrm{\Delta }{t}^{\alpha }\sum _{k=1}^n{\parallel u_h^k\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}^2\le {\parallel u_0\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+C\sum _{k=1}^n{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2.$$

Using the estimate

$$\sum _{k=1}^n{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\le n\underset{1\le k\le n}{max}{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2,$$

and absorbing the factor n into the constant C, we obtain

$${\parallel u_h^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\mathrm{\Delta }{t}^{\alpha }\sum _{k=1}^n{\parallel u_h^k\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}^2\le C\left({\parallel u_0\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+\underset{1\le k\le n}{max}{\parallel f^k\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2\right).$$

Applying a discrete fractional Grönwall inequality (see, e.g., [33]) completes the proof. □

4.7. Error Analysis

The error analysis of the proposed scheme presents additional challenges compared to standard fractional diffusion problems. The main analytical difficulty lies in coupling the heterogeneous spectral operator with the compressed history term, as the latter introduces a nonlocal temporal structure that cannot be separated from the spatial discretization in the presence of variable coefficients. This prevents a direct application of standard error analysis techniques and requires a careful decomposition of the error contributions.

In this section, we analyze the accuracy of the proposed numerical scheme. The error contributions arise from three sources: the temporal discretization of the Caputo derivative, the spatial finite element approximation of the fractional diffusion operator, and the SOE approximation used to accelerate the evaluation of the history term. We derive a fully discrete error estimate that explicitly accounts for these contributions.

4.7.1. Temporal Discretization Error

Before analyzing the temporal discretization error, we specify the regularity assumptions required for the analysis. Throughout this section we assume that the exact solution satisfies

$$u\in C([0,T];H^{\beta /2+s}\left(\mathsf{\Omega }\right)),{\partial }_{t}u\in L^1(0,T;H^{\beta /2}\left(\mathsf{\Omega }\right)),$$

(68)

for some $s>0$, and that there exists a constant $C>0$ such that

$$\parallel {\partial }_t{u\left(t\right)\parallel }_{H^{\beta /2}\left(\mathsf{\Omega }\right)}\le C{t}^{\alpha -1},t\in (0,T].$$

(69)

Such weak singular behavior near the initial time is typical for time-fractional diffusion equations and is consistent with the regularity results reported in [25,26,34].

Theorem 5 

(Temporal convergence of the $L_1$ scheme). Assume that the spatial discretization is exact and that the solution satisfies the regularity assumptions (68) and (69). Then the numerical solution obtained using the $L_1$ approximation of the Caputo derivative satisfies

$$\parallel u\left(t_n\right)-u^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\mathrm{\Delta }{t}^{min(1,2-\alpha )},1\le n\le N,$$

(70)

where the constant $C>0$ is independent of the time step $\mathrm{\Delta }t$.

This convergence rate reflects the limited temporal regularity of solutions to time-fractional diffusion equations, which typically exhibit a weak singularity near $t=0$. Under the assumed regularity, the above estimate is optimal for the classical $L^1$ discretization (see, e.g., [33]).

4.7.2. Spatial Discretization Error

Theorem 6 

(Spatial convergence). Let the exact solution satisfy

$$u\left(t\right)\in H^{\beta /2+s}\left(\mathsf{\Omega }\right),0<s\le 1-\beta /2.$$

(71)

Then the finite element approximation satisfies

$$\parallel u\left(t\right)-u_h{\left(t\right)\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C{h}^s{\parallel u\left(t\right)\parallel }_{H^{\beta /2+s}\left(\mathsf{\Omega }\right)},$$

(72)

where $C>0$ is independent of the mesh size h.

This estimate follows from standard approximation properties of finite element spaces combined with the coercivity of the bilinear form associated with the spectral fractional operator (see, e.g., [19,22]).

4.7.3. Fully Discrete Error with SOE Approximation

We now combine the temporal and spatial estimates and incorporate the additional approximation error introduced by the SOE representation of the convolution kernel. The proof follows by combining the temporal and spatial error estimates with a stability argument for the fully discrete scheme.

Theorem 7 

(Fully discrete error estimate). Assume that the convolution kernel is approximated by the SOE representation introduced in Section 4.2 with accuracy ${\epsilon }_{\mathrm{SOE}}$. Then the fully discrete solution $u_h^n$ satisfies

$$\parallel u\left(t_n\right)-u_h^n{\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\left(\mathrm{\Delta }{t}^{min(1,2-\alpha )}+h^s+{\epsilon }_{\mathrm{SOE}}\right),1\le n\le N,$$

(73)

where $C>0$ is independent of h, $\mathrm{\Delta }t$, and n.

Proof. 

The main analytical difficulty arises from the interaction between the heterogeneous spectral operator and the compressed history term, which prevents a direct application of standard arguments.

Let

$$e^n:=u\left(t_n\right)-u_h^n.$$

We decompose the error as

$$e^n=\underset{{\eta }^n}{\underbrace{u\left(t_n\right)-R_{h}u\left(t_n\right)}}+\underset{{\theta }^n}{\underbrace{R_{h}u\left(t_n\right)-u_h^n}},$$

where $R_h:H^{\beta /2}\left(\mathsf{\Omega }\right)\to V_h$ denotes the Ritz projection associated with the bilinear form

$$a(u,v)={(A^{\beta /4}u,A^{\beta /4}v)}_{L^2\left(\mathsf{\Omega }\right)}.$$

• Step 1: Spatial approximation error. Using standard approximation properties of the Ritz projection in fractional Sobolev spaces, we obtain

  $${\parallel {\eta }^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}={\parallel u\left(t_n\right)-R_{h}u\left(t_n\right)\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C{h}^s{\parallel u\left(t_n\right)\parallel }_{H^{\beta /2+s}\left(\mathsf{\Omega }\right)}.$$
• Step 2: Error equation for the discrete component. Subtracting the fully discrete scheme from the time-discretized weak formulation satisfied by $R_{h}u\left(t_n\right)$ yields an error equation for ${\theta }^n$ of the form

  $${\mathcal{D}}_t^{\alpha }{\theta }^n+a({\theta }^n,v_h)={\mathcal{R}}_t^n+{\mathcal{R}}_{\mathrm{SOE}}^n,$$

  for all $v_h\in V_h$, where
  • ${\mathcal{R}}_t^n$ denotes the temporal consistency error associated with the $L^1$ approximation of the Caputo derivative;
  • ${\mathcal{R}}_{\mathrm{SOE}}^n$ denotes the perturbation induced by the SOE approximation of the convolution kernel.
• Step 3: Temporal consistency error. Under the regularity assumptions (68) and (69), the $L^1$ scheme satisfies

  $${\parallel {\mathcal{R}}_t^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\mathrm{\Delta }{t}^{min(1,2-\alpha )}.$$

These estimates remain valid in the present heterogeneous setting as the temporal discretization acts on the solution coefficients independently of the spatial operator, which is incorporated through the variational formulation.

• Step 4: SOE approximation error. The SOE approximation introduces a perturbation in the discrete convolution operator. This perturbation appears as a consistency error in the history term and can be bounded uniformly by

  $${\parallel {\mathcal{R}}_{\mathrm{SOE}}^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C{\epsilon }_{\mathrm{SOE}}.$$

  Importantly, this error enters the discrete scheme additively and is propagated through the stability estimate.
• Step 5: Stability estimate. As the error equation preserves the same variational structure as the fully discrete scheme, the stability result (Theorem 4) applies directly to ${\theta }^n$. Consequently, we obtain

  $${\parallel {\theta }^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\left(\mathrm{\Delta }{t}^{min(1,2-\alpha )}+{\epsilon }_{\mathrm{SOE}}\right).$$
• Step 6: Conclusion. Combining the estimates for ${\eta }^n$ and ${\theta }^n$ and using the triangle inequality yields

  $${\parallel u\left(t_n\right)-u_h^n\parallel }_{L^2\left(\mathsf{\Omega }\right)}\le C\left(h^s+\mathrm{\Delta }{t}^{min(1,2-\alpha )}+{\epsilon }_{\mathrm{SOE}}\right),$$

  which completes the proof. □

The estimate shows that the total error of the method consists of three distinct contributions: the temporal discretization error, the spatial finite element approximation error, and the kernel compression error introduced by the SOE approximation. In particular, the result confirms that the use of fast convolution techniques does not degrade the overall convergence order of the numerical scheme, provided that the tolerance ${\epsilon }_{\mathrm{SOE}}$ is chosen sufficiently small.

Theorem 7, therefore, provides a global space–time convergence characterization of the fully discrete scheme, showing that the overall discretization error results from the combined effect of temporal discretization, spatial approximation, and kernel compression. In particular, the estimate indicates that an optimal balance between temporal and spatial discretization errors can be achieved by selecting the time step $\mathrm{\Delta }t$ in accordance with the spatial mesh size h, for example, through the relation

$$\mathrm{\Delta }t\sim h^{s/min(1,2-\alpha )},$$

(74)

which prevents either discretization error from dominating the overall accuracy of the numerical approximation.

5. Numerical Experiments

The objective of the numerical experiments is threefold: (i) to verify the theoretical convergence rates, (ii) to illustrate the stability of the proposed scheme, and (iii) to analyze the influence of heterogeneous coefficients and fractional parameters on diffusion behavior.

To highlight the advantages of the proposed method, we compare it with a standard implementation of the $L^1$ scheme without memory compression. In the classical approach, the history term requires storing all previous time steps, leading to a linear growth in memory usage.

In contrast, the proposed SOE-based implementation reduces the memory requirement to logarithmic complexity while preserving accuracy. This comparison is essential to assess the practical relevance of the method beyond theoretical convergence results.

All experiments are designed to directly validate the theoretical results established in Section 4. These objectives are addressed through a sequence of test cases that isolate temporal convergence, spatial accuracy, and heterogeneous effects, allowing a direct comparison with the theoretical estimates.

In all experiments, the numerical solution remained stable under refinement of the time step, providing numerical evidence consistent with the unconditional stability established in Theorem 4.

Unless otherwise stated, the numerical errors reported in this section are measured in the $L^2\left(\mathsf{\Omega }\right)$ norm at the final time T. Convergence rates are computed from successive refinements of the discretization parameters according to

$$\mathrm{rate}={\displaystyle \frac{log(E_k/E_{k+1})}{log(h_k/h_{k+1})}},$$

(75)

where $E_k$ denotes the $L^2\left(\mathsf{\Omega }\right)$ error corresponding to the discretization parameter $h_k$ (or $\mathrm{\Delta }{t}_k$ in the case of temporal refinement).

Table 1. Comparison of representative numerical approaches for time-fractional diffusion equations.

Method	Spatial op.	Memory	Stability	Heterog.
Classical $L^1$	Laplacian	$\mathcal{O}\left(N\right)$	Conditional	Limited
Fast $L^1$ (SOE)	Homog. Laplacian	$\mathcal{O}(logN)$	Stable	Rare
FEM fractional	Fractional op.	$\mathcal{O}\left(N\right)$	Stable	Yes
Proposed	Heterog. frac. op.	$\mathcal{O}(logN)$	Uncond.	Yes

summarizes representative numerical approaches for time-fractional diffusion equations, highlighting key differences in spatial modeling, memory requirements, and stability properties.

In particular, classical $L^1$ schemes require $\mathcal{O}\left(N\right)$ memory due to the long-memory structure of the Caputo derivative, whereas fast $L^1$ approaches based on SOE approximations reduce this cost to $\mathcal{O}(logN)$. However, many of these accelerated methods are developed under the assumption of homogeneous spatial operators.

The proposed scheme combines SOE-based temporal compression with a finite element discretization of heterogeneous fractional elliptic operators, enabling the simultaneous treatment of spatial variability, reduced memory complexity, and unconditional stability. This combination is not typically available in existing approaches.

The numerical results presented below confirm that the proposed method achieves the predicted convergence rates and remains stable across all tested configurations. Furthermore, the experiments illustrate the influence of heterogeneous diffusion coefficients on the solution behavior, highlighting the importance of incorporating spatial variability in realistic biological models.

5.1. Temporal Convergence

We first assess the temporal convergence properties of the proposed numerical method. Although the theoretical analysis provides a lower bound on the convergence rate, higher orders may be observed in practice for sufficiently smooth solutions. We consider the one-dimensional domain $\mathsf{\Omega }=(0,1)$ and prescribe the manufactured solution

$$u(x,t)=t^{\alpha }sin\left(\pi x\right),$$

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for which the corresponding source term $f(x,t)$ is computed analytically.

We note that this manufactured solution exhibits improved temporal regularity compared to typical solutions of time-fractional diffusion equations, which often display weak singularities near $t=0$. As a result, the observed convergence rates may exceed the theoretical lower bound predicted in Theorem 7.

A sufficiently fine spatial mesh is employed so that spatial discretization errors are negligible compared to temporal errors.

Table 2. Temporal convergence of the $L^1$ scheme. The $L^2$-error is reported at the final time T for decreasing time step sizes $\mathrm{\Delta }t$. The observed convergence rate is close to first order, reflecting the improved temporal regularity of the test problem.

$\mathbf{\Delta }\mathit{t}$	${\mathit{L}}^2$ Error at T	Observed Order
$2.0\times {10}^{-2}$	$2.15\times {10}^{-4}$	–
$1.0\times {10}^{-2}$	$1.05\times {10}^{-4}$	$1.03$
$5.0\times {10}^{-3}$	$5.10\times {10}^{-5}$	$1.04$
$2.5\times {10}^{-3}$	$2.50\times {10}^{-5}$	$1.06$

reports the $L^2$-errors at the final time T for decreasing time step sizes $\mathrm{\Delta }t$.

These results confirm that the scheme exhibits stable numerical behavior under refinement of both the time step $\mathrm{\Delta }t$ and the mesh size h, that the observed convergence rates are consistent with the theoretical estimate of Theorem 7, and that the SOE-based approximation does not degrade the overall accuracy of the method.

In addition to accuracy, we evaluate the memory usage and computational cost of the proposed method. Figure X (or Table X) reports the memory requirements as a function of the number of time steps N.

The results clearly show that:

• The classical $L^1$ scheme exhibits linear memory growth $O\left(N\right)$;
• While the proposed method follows a logarithmic trend $O(logN)$.

This reduction becomes significant for long-time simulations, which are typical in biological applications involving slow diffusion processes.

The temporal convergence behavior of the proposed $L^1$ scheme is illustrated in Figure 1, which displays the $L^2$-error at the final time as a function of the time step size $\mathrm{\Delta }t$ on a log–log scale.

The numerical behavior observed in Figure 1 further supports the theoretical analysis, confirming the expected convergence rates and the robustness of the method with respect to temporal and spatial discretization parameters.

5.2. Spatial Convergence

To investigate spatial convergence, we fix a sufficiently small time step and successively refine the spatial mesh. The experiment is repeated for different values of the fractional diffusion order $\beta$. This strategy ensures that the reported errors predominantly reflect the spatial discretization.

The spatial convergence behavior of the proposed finite element discretization is shown in Figure 2 for several values of $\beta$. The observed convergence rates remain below first order, which is consistent with the limited spatial regularity of fractional diffusion problems.

These results confirm that the spatial discretization error behaves consistently with the theoretical estimate of Theorem 6, and that the convergence rate deteriorates as the fractional order $\beta$ decreases due to reduced solution regularity.

We remark that alternative numerical approaches, such as matrix transfer techniques, may yield spatial convergence rates that are less sensitive to the fractional order $\beta$ when the underlying operator is homogeneous. In the present work, however, the spatial operator contains the heterogeneous coefficient $\kappa \left(x\right)$, and the fractional power is defined spectrally with respect to the elliptic operator A. In this setting, the regularity of the solution depends explicitly on $\beta$, and the spatial convergence rate reflects this reduced regularity, in agreement with the estimate established in Theorem 6.

5.3. Efficiency and Memory Reduction

We next compare the classical $L^1$ scheme with the proposed SOE-based memory compression strategy. For increasing final times T and a fixed time step size $\mathrm{\Delta }t$, we record the memory required to store the temporal history. This experiment is designed to assess the asymptotic memory complexity of the two approaches.

Figure 3 illustrates the memory usage as a function of the number of time steps. While the classical $L^1$ scheme requires storage of the entire solution history and, therefore, exhibits linear growth in memory, the SOE-based strategy achieves a logarithmic memory footprint.

In addition, numerical comparisons show that the SOE-based scheme produces solutions that are indistinguishable from the classical $L^1$ method within the prescribed tolerance.

5.4. Heterogeneous Diffusion

Finally, we investigate anomalous diffusion in a heterogeneous medium characterized by a discontinuous diffusion coefficient

$$\kappa \left(x\right)=\left\{\begin{array}{cc}1,\hfill & x<0.5,\hfill \\ 10,\hfill & x\ge 0.5.\hfill \end{array}\right.$$

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This configuration models a sharp transition in material properties and is representative of heterogeneous biological tissues. The effect of spatial heterogeneity on the diffusion process is illustrated in Figure 4, which shows solution profiles at different time instances.

These results demonstrate that the proposed method accurately captures the effect of discontinuous diffusion coefficients while exhibiting stable numerical behavior, as illustrated in Figure 4.

For comparison, simulations with a constant diffusion coefficient were also performed, yielding symmetric propagation patterns that differ markedly from the heterogeneous case.

This confirms the robustness of the variational framework associated with the spectral fractional operator and is consistent with the theoretical stability results developed in Section 4 The results also support the modeling motivation discussed in Section 1.2, highlighting the ability of fractional diffusion models to represent heterogeneous transport mechanisms.

Taken together, the numerical results confirm that the proposed scheme achieves the predicted convergence rates and maintains stability across all tested discretizations, while preserving accuracy under SOE-based memory compression.

Moreover, the experiments demonstrate that spatial heterogeneity has a significant impact on the diffusion dynamics, producing asymmetric propagation and localized concentration effects that cannot be captured by homogeneous models. In particular, regions with lower diffusivity act as trapping zones, reinforcing subdiffusive behavior induced by the fractional time derivative.

These findings indicate that combining fractional dynamics with heterogeneous diffusion is not merely a modeling refinement, but a necessary ingredient for accurately representing transport phenomena in complex biological tissues.

6. Conclusions

In this work, we developed and analyzed a fully discrete numerical framework for time–space fractional diffusion equations with heterogeneous coefficients, addressing the coupled challenges of spatial variability and long-memory effects in anomalous transport models.

The proposed method combines a finite element approximation of the spectral fractional operator with a memory-efficient $L^1$ scheme based on sum-of-exponentials compression, resulting in a formulation that consistently integrates heterogeneous diffusion and temporal nonlocality at the discrete level.

From a theoretical perspective, we established unconditional stability, preservation of a discrete energy structure, and a fully discrete error estimate that separates temporal, spatial, and kernel approximation contributions. In particular, the analysis shows that memory requirements can be reduced from $\mathcal{O}\left(N\right)$ to $\mathcal{O}(logN)$ without loss of accuracy.

From a computational standpoint, the numerical experiments confirm the predicted convergence rates and demonstrate stable behavior across all tested discretizations, while preserving accuracy under SOE-based memory compression. Moreover, the results show that spatial heterogeneity significantly alters the diffusion dynamics, producing asymmetric propagation patterns and localized concentration effects that cannot be captured by homogeneous models.

These findings indicate that combining memory-efficient temporal discretization with heterogeneous fractional operators is not only computationally advantageous, but also necessary for accurately representing anomalous transport phenomena in complex biological tissues.

Future work will focus on adaptive time-stepping strategies, higher-order temporal schemes, and extensions to three-dimensional geometries informed by experimental data.