An operator splitting algorithm for coupled one-dimensional advection-diffusion-reaction equations

1995 ◽  
Vol 127 (1-4) ◽  
pp. 181-201 ◽  
Author(s):  
Liaqat Ali Khan ◽  
Philip L.-F. Liu
2021 ◽  
Vol 82 (1-2) ◽  
Author(s):  
Christian Engwer ◽  
Michael Wenske

AbstractGlioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.


2019 ◽  
Vol 22 (4) ◽  
pp. 918-944 ◽  
Author(s):  
William McLean ◽  
Kassem Mustapha ◽  
Raed Ali ◽  
Omar Knio

Abstract We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.


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