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ABSTRACT
As to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.
Nomenclature
| A | = | coefficient matrix in Eq. (28) |
| aj | = | coefficients in polynomials |
| b | = | right-hand side in Eq. (28) |
| b1 | = | := BTb |
| bj | = | coefficients in polynomials |
| B | = | the matrix obtained from A by equilibrated method |
| cij | = | coefficients in the Pascal polynomials |
| c | = | vector of coefficients |
| D | = | := BTB |
| e(t) | = | := Tℓ(t) − T0(t) |
| f(x) | = | initial temperature |
| H(t) | = | time-dependent heat source |
| H(x, t) | = | general heat source |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of the rod |
| m | = | m − 1 highest order in the Pascal polynomials |
| m0 | = | number of test functions |
| m1 | = | m1 − 1 the highest order of polynomials |
| m2 | = | m2 − 1 the highest order of polynomials |
| n | = | = m1 + m2 for time-dependent heat source |
| n | = | m(m + 1)/2 for general heat source |
| nq | = | = 2m0 for time-dependent heat source |
| nq | = | = 3m0 for general heat source |
| p(t) | = | := Tm(t) − T0(t) |
| P | = | post conditioner in Eq. (31) |
| q(t) | = | := Tℓ(t) − Tm(t) |
| R(i) | = | random numbers |
| s | = | level of noise |
| = | multiple scales | |
| = | multiple scales | |
| sij | = | multiple scales in the Pascal polynomials |
| t | = | time |
| tf | = | final time |
| T(x, t) | = | temperature |
| T0(t) | = | left-boundary temperature |
| Tℓ(t) | = | right-boundary temperature |
| Tm(t) | = | measuring temperature at xm |
| u(x, t) | = | transformed temperature |
| uk(x, t) | = | adjoint Trefftz test functions in Ω |
| vk(x, t) | = | adjoint Trefftz test functions in Ω1 |
| wk(x, t) | = | adjoint Trefftz test functions in Ω2 |
| x | = | space variable |
| xm | = | measuring point |
| Greek symbols | = | |
| ε | = | converegence criterion |
| Ω | = | problem domain |
| Ω1 | = | sub domain |
| Ω2 | = | sub domain |
| Γ | = | boundary of Ω |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |
Nomenclature
| A | = | coefficient matrix in Eq. (28) |
| aj | = | coefficients in polynomials |
| b | = | right-hand side in Eq. (28) |
| b1 | = | := BTb |
| bj | = | coefficients in polynomials |
| B | = | the matrix obtained from A by equilibrated method |
| cij | = | coefficients in the Pascal polynomials |
| c | = | vector of coefficients |
| D | = | := BTB |
| e(t) | = | := Tℓ(t) − T0(t) |
| f(x) | = | initial temperature |
| H(t) | = | time-dependent heat source |
| H(x, t) | = | general heat source |
| ℋ | = | heat operator |
| ℋ* | = | adjoint heat operator |
| ℓ | = | length of the rod |
| m | = | m − 1 highest order in the Pascal polynomials |
| m0 | = | number of test functions |
| m1 | = | m1 − 1 the highest order of polynomials |
| m2 | = | m2 − 1 the highest order of polynomials |
| n | = | = m1 + m2 for time-dependent heat source |
| n | = | m(m + 1)/2 for general heat source |
| nq | = | = 2m0 for time-dependent heat source |
| nq | = | = 3m0 for general heat source |
| p(t) | = | := Tm(t) − T0(t) |
| P | = | post conditioner in Eq. (31) |
| q(t) | = | := Tℓ(t) − Tm(t) |
| R(i) | = | random numbers |
| s | = | level of noise |
| = | multiple scales | |
| = | multiple scales | |
| sij | = | multiple scales in the Pascal polynomials |
| t | = | time |
| tf | = | final time |
| T(x, t) | = | temperature |
| T0(t) | = | left-boundary temperature |
| Tℓ(t) | = | right-boundary temperature |
| Tm(t) | = | measuring temperature at xm |
| u(x, t) | = | transformed temperature |
| uk(x, t) | = | adjoint Trefftz test functions in Ω |
| vk(x, t) | = | adjoint Trefftz test functions in Ω1 |
| wk(x, t) | = | adjoint Trefftz test functions in Ω2 |
| x | = | space variable |
| xm | = | measuring point |
| Greek symbols | = | |
| ε | = | converegence criterion |
| Ω | = | problem domain |
| Ω1 | = | sub domain |
| Ω2 | = | sub domain |
| Γ | = | boundary of Ω |
| Subscripts and superscripts | = | |
| i | = | index |
| j | = | index |
| k | = | index |
| m | = | index |
| T | = | transpose |