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This paper proposes an explicit Fourier-Klibanov method as a new approximation technique for an age-dependent population PDE of Gompertz type in modeling the evolution of tumor density in a brain tissue. Through suitable nonlinear and linear transformations, the Gompertz model of interest is transformed into an auxiliary third-order nonlinear PDE. Then, a coupled transport-like PDE system is obtained via an application of the Fourier-Klibanov method, and, thereby, is approximated by the explicit finite difference operators of characteristics. The stability of the resulting difference scheme is analyzed under the standard 2-norm topology. Finally, we present some computational results to demonstrate the effectiveness of the proposed method.
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Simultaneous reconstruction of birth condition and mortality rate in an age-structured tumor model
Abstract This paper proposes a computational method to reconstruct both the birth and mortality coefficients in an age-structured population diffusive model. In mathematical oncology, solving this inverse problem is crucial for assessing the effectiveness of anti-cancer treatments and thus, gaining insights into the post-treatment dynamics of tumors. Through some linear and nonlinear transformations, the targeted inverse model is transformed into an auxiliary third-order nonlinear PDE. Subsequently, a coupled age-dependent quasi-linear parabolic PDE system is derived using the Fourier–Klibanov basis. The resulting PDE system is then approximated through the minimization of a cost functional, weighted by a suitable Carleman function. Ultimately, an analysis of the minimization problem is studied through a new Carleman estimate, and some computational results are presented to show how the proposed method works.
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- Award ID(s):
- 2451193
- PAR ID:
- 10606513
- Publisher / Repository:
- IOP Publishing
- Date Published:
- Journal Name:
- Inverse Problems
- Volume:
- 41
- Issue:
- 6
- ISSN:
- 0266-5611
- Format(s):
- Medium: X Size: Article No. 065019
- Size(s):
- Article No. 065019
- Sponsoring Org:
- National Science Foundation
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