Uniqueness of the Two Functional Coefficients in the Population-Model Boundary Conditions
Tóm tắt
We consider a model of biological population dynamics allowing for age structure. One of the boundary conditions in the model’s mixed initial–boundary-value problem is integral and nonlocal. The model is used for the inverse problem that simultaneously reconstructs the two coefficients in the model boundary conditions — the initial and the integral — given additional information on the solution of the forward problem in the form of two functions specifying the solution on the boundary. The uniqueness conditions for the inverse problem are proved. The integral relationships used to prove the uniqueness for the inverse problem lead to an iterative procedure to derive an approximate numerical solution of the inverse problem.
Tài liệu tham khảo
O. Angulo, J. C. López-Marcos, and F. A. Milner, “The application of an age-structured model with unbounded mortality to demography,” Math. Biosci., 208, No. 2, 495–512 (2007).
A. Gandolf, M. Iannelli, and G. Marinoschi, “An age-structured model of epidermis growth,” J. Math. Biol., 62, No. 1, 111–141 (2011).
M. Martcheva, An Introduction to Mathematical Epidemiology, Texts in Applied Mathematics (Springer, New York), Vol. 61 (2015).
T. Kuniya, “Stability analysis of an age-structured sir epidemic model with a reduction method to ODEs,” Mathematics, 6, No. 9, 147–156 (2018).
A. M. Denisov and A. S. Makeev, “Iterative methods for solving an inverse problem for a population model,” J. Computational Mathematics and Mathematical Physics (Pleiades Publishing), 44, No. 8, 1404–1413 (2004).
A. M. Denisov and A. S. Makeev, “Numerical method for solving an inverse problem for a population model,” J. Computational Mathematics and Mathematical Physics (Pleiades Publishing), 46, No. 3, 470–480 (2006).
A. S. Makeev, “Application of Tikhonov’s regularization method to solve inverse problems for two population models,” Computational Mathematics and Modeling (Consultants Bureau), 18, No. 1, 1–9 (2007).
D. V. Churbanov, “Uniqueness of finding the coefficient of the derivative in a first order nonlinear equation,” J. Moscow University Computational Mathematics and Cybernetics (Allerton Press), 37, No. 1, 8–13 (2013).
A. M. Denisov and S. I. Solov’eva, “Numerical determination of the initial condition in Cauchy problems for a hyperbolic equation with a small parameter,” J. Computational Mathematics and Modeling (Consultants Bureau), 29, No. 1, 1–9 (2018).
A. V. Baev and S. V. Gavrilov, “An iterative way of solving the inverse scattering problem for an acoustic system of equations in an absorptive layered nonhomogeneous medium,” J. Moscow University Computational Mathematics and Cybernetics (Allerton Press), 42, No. 2, 55–62 (2018).
I. V. Tikhonov and Vu Nguyen Son Tung, “Solvability of a nonlocal problem for an evolution equation with a superstable semgroup,” J. Differential Equations (Pleiades Publishing), 56, No. 4, 478–498 (2020).
S. G. Golovina and E. V. Zakharov, “A numerical method for a system of integral equations determining the boundaries of local nonhomogeneities,” J. Computational Mathematics and Modeling (Consultants Bureau), 31, No. 1, 8–12 (2020).