preprint 2011:23 analytical solutions for the pencil- beam
TRANSCRIPT
PREPRINT 2011:23
Analytical Solutions for the Pencil-beam Equation with Energy Loss and Straggling
MOHAMMAD ASADZADEH TOBIAS GEBÄCK Department of Mathematical Sciences Division of Mathematics
CHALMERS UNIVERSITY OF TECHNOLOGY UNIVERSITY OF GOTHENBURG Gothenburg Sweden 2011
Preprint 2011:23
Analytical Solutions for the Pencil-beam Equation with Energy Loss and Straggling
Mohammad Asadzadeh and Tobias Gebäck
Department of Mathematical Sciences Division of Mathematics
Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Gothenburg, October 2011
Preprint 2011:23
ISSN 1652-9715
Matematiska vetenskaper
Göteborg 2011
ANALYTICAL SOLUTIONS FOR THE PENCIL-BEAM EQUATION 9
z (cm)
E (
MeV
)
FEM
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
0.1
0.2
0.3
0.4
0.5
0.6
z (cm)
E (
MeV
)
Explicit (NESA)
0 2 4 6 8 10 12 14 160
10
20
30
40
50
60
70
0.1
0.2
0.3
0.4
0.5
0.6
Figure 2. Level curves for the numerical solution to equation (2.30) (top),and the analytical solution (2.36) under the narrow energy spectrum approx-imation (NESA) (bottom). The initial energy was E0 = 50 MeV. The cross-sections for electrons in water were used. The dashed thick lines are the curvesE = Ea(z), and the solid thick lines are the average energies for the respectivesolutions.
References
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10 M. ASADZADEH AND T. GEBACK
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E-mail address: [email protected]
E-mail address: [email protected]
† Department of Mathematics, Chalmers University of Technology and Goteborg University,
SE–412 96, Goteborg, Sweden
‡ Department of Oncology-Pathology, Karolinska Institute, SE-171 77 Stockholm, Sweden