Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Fritz, Marvin; Rajendran, Mabel Lizzy; Wohlmuth, Barbara

doi:10.1016/j.camwa.2022.01.002

Cited by 19 publications

(10 citation statements)

References 94 publications

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“…Spatial discretizations of dilute polymer models were previously considered in [9,13,14]. In addition, weak solutions to time-fractional PDEs have been investigated using the Galerkin approach in the publications [25][26][27]. There have also been initial steps in the analysis of a decoupled time-fractional Fokker-Planck equation with time-dependent forces; see the papers [24,40,41,52,53].…”

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confidence: 99%

On a subdiffusive tumour growth model with fractional time derivative

Fritz

Kuttler

Rajendran

et al. 2021

IMA Journal of Applied Mathematics

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In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo–Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretized system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.

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confidence: 99%

On a subdiffusive tumour growth model with fractional time derivative

Fritz

Kuttler

Rajendran

et al. 2021

IMA Journal of Applied Mathematics

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“…As evidenced by the in vitro and in vivo experimental findings of Jiang et al (2014), tumors migrate via both traditional Fickian diffusion and subdiffusion. Fritz et al (2022Fritz et al ( , 2023 Tumor evolution models of phase-field type with nonlocal effects…”

Section: Nonlocal Phenomenamentioning

confidence: 99%

“…The chemical potential reads µ T = Ψ (φ T ) − ε 2 T ∆φ T in the case of the Ginzburg-Landau energy (2.1). This model is called the time-fractional Cahn-Hilliard equation (Fritz et al, 2022(Fritz et al, , 2023. If one selects the Dirichlet energy…”

Section: Nonlocal-in-space: Cell-to-cell and Cell-to-matrix Adhesionmentioning

confidence: 99%

“…As evidenced by the in vitro and in vivo experimental findings of Jiang et al (2014), tumors migrate via both traditional Fickian di↵usion and subdi↵usion. Fritz et al (2022Fritz et al ( , 2023 investigated the memory e↵ect in connection to the time-fractional Cahn-Hilliard equation with degenerating mobility. Additionally, Fritz et al (2021c) examined a fractional tumor model including subdi↵usion, nutritional couplings, and mechanical deformation.…”

Section: Three-dimensional Modelmentioning

confidence: 99%

“…In the works by (Fritz et al, 2021c(Fritz et al, , 2022 on time-fractional tumor growth models, a fractional linear multistep method is used as in Lubich (1986). Such a method is based on a convolution quadrature scheme, and it generalizes the standard linear multistep method for ODEs.…”

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confidence: 99%

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Tumor evolution models of phase-field type with nonlocal effects and angiogenesis

Fritz¹

2023

Preprint

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In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixeddimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor's evolution in response to various biological effects.

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Introduction

Rougirel

2024

Frontiers in Mathematics

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Time-fractional Cahn–Hilliard equation: Well-posedness, degeneracy, and numerical solutions

Cited by 19 publications

References 94 publications

On a subdiffusive tumour growth model with fractional time derivative

On a subdiffusive tumour growth model with fractional time derivative

Tumor evolution models of phase-field type with nonlocal effects and angiogenesis

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