http://orcid.org/0000-0001-6366-3539
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ABSTRACT
In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.
Nomenclature
| aj | = | expansion coefficients in Eq. (11) |
| A | = | matrix defined in Eq. (19) |
| bj | = | expansion coefficients in Eq. (33) |
| B | = | matrix defined in Eq. (19) |
| C | = | matrix defined in Eq. (19) |
| Dj | = | coefficients in Eq. (38) |
| Ej | = | coefficients in Eq. (16) |
| e(H) | = | relative root mean square error |
| f(x) | = | initial value of u |
| Fj | = | coefficients in Eq. (15) |
| F(t) | = | time-dependent heat source |
| g(x) | = | final value of u |
| Gj | = | coefficients in Eq. (36) |
| h(x) | = | extra temperature at t = t0 |
| H(x) | = | space-dependent heat source |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of space |
| m | = | the highest order of expansion series |
| n | = | the number of unknown coefficients |
| R(i) | = | random noise |
| S(x, t) | = | heat source |
| s | = | relative noise level |
| t | = | time |
| tf | = | final time |
| t0 | = | a measured time |
| u(x, t) | = | temperature |
| u0(t) | = | left-boundary value of u |
| uℓ(t) | = | right-boundary value of u |
| vj(x, t) | = | adjoint Trefftz test function |
| wj(x, t) | = | another adjoint Trefftz test function |
| x | = | space variable |
| α | = | heat conduction coefficient |
| β | = | regularization factor |
| δjk | = | Kronecker delta symbol |
| Ω | = | a bounded region |
| Γ | = | the boundary of Ω |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
Nomenclature
| aj | = | expansion coefficients in Eq. (11) |
| A | = | matrix defined in Eq. (19) |
| bj | = | expansion coefficients in Eq. (33) |
| B | = | matrix defined in Eq. (19) |
| C | = | matrix defined in Eq. (19) |
| Dj | = | coefficients in Eq. (38) |
| Ej | = | coefficients in Eq. (16) |
| e(H) | = | relative root mean square error |
| f(x) | = | initial value of u |
| Fj | = | coefficients in Eq. (15) |
| F(t) | = | time-dependent heat source |
| g(x) | = | final value of u |
| Gj | = | coefficients in Eq. (36) |
| h(x) | = | extra temperature at t = t0 |
| H(x) | = | space-dependent heat source |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of space |
| m | = | the highest order of expansion series |
| n | = | the number of unknown coefficients |
| R(i) | = | random noise |
| S(x, t) | = | heat source |
| s | = | relative noise level |
| t | = | time |
| tf | = | final time |
| t0 | = | a measured time |
| u(x, t) | = | temperature |
| u0(t) | = | left-boundary value of u |
| uℓ(t) | = | right-boundary value of u |
| vj(x, t) | = | adjoint Trefftz test function |
| wj(x, t) | = | another adjoint Trefftz test function |
| x | = | space variable |
| α | = | heat conduction coefficient |
| β | = | regularization factor |
| δjk | = | Kronecker delta symbol |
| Ω | = | a bounded region |
| Γ | = | the boundary of Ω |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
Acknowledgments
The Thousand Talents Plan of China under the Grant Number A1211010 for the financial support to the first author is highly appreciated. Chein-Shan Liu to be a Chair Professor of National Taiwan Ocean University is acknowledged.