A global random walk on spheres algorithm for transient heat equation and some extensions

2019 ◽  
Vol 25 (1) ◽  
pp. 85-96 ◽  
Author(s):  
Karl K. Sabelfeld
Keyword(s):  
Random Walk ◽  
Heat Equation ◽  
Symmetry Property ◽  
Forthcoming Paper ◽  
Diffusion Reaction ◽  
Advection Diffusion ◽  
The Green Function ◽  
Walk On Spheres Algorithm ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations

AbstractWe suggest in this paper a global Random Walk on Spheres (gRWS) method for solving transient boundary value problems, which, in contrast to the classical RWS method, calculates the solution in any desired family ofmprescribed points. The method uses onlyNtrajectories in contrast tomNtrajectories in the conventional RWS algorithm. The idea is based on the symmetry property of the Green function and a double randomization approach. We present the gRWS method for the heat equation with arbitrary initial and boundary conditions, and the Laplace equation. Detailed description is given for 3D problems; the 2D problems can be treated analogously. Further extensions to advection-diffusion-reaction equations will be presented in a forthcoming paper.

scholarly journals Estimating the extent of glioblastoma invasion

2021 ◽  
Vol 82 (1-2) ◽  
Author(s):  
Christian Engwer ◽  
Michael Wenske
Keyword(s):  
Analytic Solution ◽  
Glioma Cells ◽  
Diffusion Reaction ◽  
Kpp Equation ◽  
Advection Diffusion ◽  
Clinical Interest ◽  
Inhomogeneous Diffusion ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations ◽  
Tumor Extent

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.

scholarly journals Well-posedness of time-fractional advection-diffusion-reaction equations

2019 ◽  
Vol 22 (4) ◽  
pp. 918-944 ◽  
Author(s):  
William McLean ◽  
Kassem Mustapha ◽  
Raed Ali ◽  
Omar Knio
Keyword(s):  
Linear Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Reaction ◽  
Fractional Integrals ◽  
Well Posedness ◽  
Advection Diffusion ◽  
Initial Boundary Value ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations

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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