##
**Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions.**
*(English)*
Zbl 1428.80011

Summary: In this paper, we consider inverse problems of finding the time-dependent source function for the population model with population density nonlocal boundary conditions and an integral over-determination measurement. These problems arise in mathematical biology and have never been investigated in the literature in the forms proposed, although related studies do exist. The unique solvability of the inverse problems are rigorously proved using generalized Fourier series and the theory of Volterra integral equations. Continuous dependence on smooth input data also holds but, as in reality noisy errors are random and non-smooth, the inverse problems are still practically ill-posed. The degree of ill-posedness is characterised by the numerical differentiation of a noisy function. In the numerical process, the boundary element method together with either a smoothing spline regularization or the first-order Tikhonov regularization are employed with various choices of regularization parameter. One is based on the discrepancy principle and another one is the generalized cross-validation criterion. Numerical results for some benchmark test examples are presented and discussed in order to illustrate the accuracy and stability of the numerical inversion.

### MSC:

80A23 | Inverse problems in thermodynamics and heat transfer |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35K91 | Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian |

35R30 | Inverse problems for PDEs |

65M32 | Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs |

### Keywords:

inverse source problem; population age model; nonlocal boundary conditions; generalized Fourier method; boundary element method; regularization
PDF
BibTeX
XML
Cite

\textit{A. Hazanee} et al., Appl. Math. Comput. 346, 800--815 (2019; Zbl 1428.80011)

### References:

[1] | Cannon, J. R.; Lin, Y.; Wang, S., Determination of a control parameter in a parabolic partial differential equation, J. Aust. Math. Soc. Ser. B, 33, 149-163 (1991) · Zbl 0767.93047 |

[3] | Farcas, A.; Lesnic, D., The boundary element method for the determination of a heat source dependent on one variable, J. Eng. Math., 54, 375-388 (2006) · Zbl 1146.80007 |

[4] | Hazanee, A.; Ismailov, M. I.; Lesnic, D.; Kerimov, N. B., An inverse time-dependent source problem for the heat equation, Appl. Numer. Math., 69, 13-33 (2013) · Zbl 1284.65124 |

[5] | Hazanee, A.; Ismailov, M. I.; Lesnic, D.; Kerimov, N. B., An inverse time-dependent source problem for the heat equation with a non-classical boundary condition, Appl. Math. Modell., 39, 6258-6272 (2015) |

[7] | Hazanee, A.; Lesnic, D., Determination of a time-dependent coefficient in the bioheat equation, Int. J. Mech. Sci., 88, 259-266 (2014) |

[8] | Ilin, V. A., Unconditional basis property on a closed interval of systems of eigenvalues and associated functions of a second-order differential operator, Doklady Akademii Nauk SSSR, 273, 1048-1053 (1983) |

[10] | Ivanchov, N. I., On the determination of unknown source in the heat equation with nonlocal boundary conditions, Ukr. Math. J., 47, 1647-1652 (1995) |

[11] | Ivanchov, M. I., Inverse Problems for Equations of Parabolic Type (2003), VNTL Publications: VNTL Publications Lviv, Ukraine · Zbl 1147.35110 |

[12] | Kerimov, N. B.; Ismailov, M. I., An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions, J. Math. Anal. Appl., 396, 546-554 (2012) · Zbl 1248.35234 |

[14] | Naimark, M. A., Linear Differential Operators: Elementary Theory of Linear Differential Operators (1967), Frederick Ungar Publishing Co.: Frederick Ungar Publishing Co. New York · Zbl 0219.34001 |

[15] | Nakhushev, A. M., Equations of Mathematical Biology (1985), Vysshaya Shkola: Vysshaya Shkola Moscow · Zbl 0991.35500 |

[16] | Prilepko, A. I.; Orlovski, D. G.; Vasin, I. A., Methods for Solving Inverse Problems in Mathematical Physics (2000), Marcel Dekker: Marcel Dekker New York · Zbl 0947.35173 |

[18] | Trucu, D.; Ingham, D. B.; Lesnic, D., Inverse time-dependent perfusion coefficient identification, J. Phys. Conf. Ser., 124, 012050 (2008) |

[19] | Twomey, S., On the numerical solution of fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature, J. Assoc. Comput. Mach., 10, 97-101 (1963) · Zbl 0125.36102 |

[20] | Wei, T.; Li, M., High order numerical derivatives for one-dimensional scattered noisy data, Appl. Math. Comput., 175, 1744-1759 (2006) · Zbl 1096.65027 |

[21] | Yan, L.; Fu, C. L.; Yang, F. L., The method of fundamental solutions for the inverse heat source problem, Eng. Anal. Bound. Elem., 32, 216-222 (2008) · Zbl 1244.80026 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.