## Fourier regularization method for solving the surface heat flux from interior observations.(English)Zbl 1122.80016

From the text: In the present paper, a Fourier regularization method for solving the surface heat flux distribution from interior observations with some estimates is given. A numerical example is also provided.
The forward problem is the heat equation in the quarter plane: $\begin{cases} u_ t = u_ {xx}, \quad &t>0,\;x>0,\\ u(0, t) = f(t), \quad &t > 0,\\ u(x, 0) = 0, \quad &x > 0, \end{cases}$ where the boundary condition $$f$$ is assumed to be in the Sobolev space $$H^ p(\mathbb{R})$$ for some $$p \geq 0$$. The inverse problem is to find $$u(x,t)$$ and its gradient with respect to $$x$$, $$u_x(x,t)$$, for $$0\leq x<1$$.

### MSC:

 80M25 Other numerical methods (thermodynamics) (MSC2010) 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35R25 Ill-posed problems for PDEs 80A23 Inverse problems in thermodynamics and heat transfer 35R30 Inverse problems for PDEs
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### References:

 [1] Beck, J.V.; Blackwell, B.; Clair, S.R., Inverse heat conduction-ill-posed problem, (1985), Wiley · Zbl 0633.73120 [2] Carasso, A., Determining surface temperatures from interior observations, SIAM J. appl. math., 42, 558-574, (1982) · Zbl 0498.35084 [3] Seidman, T.; Eldén, L., An optimal filtering method for the sideways heat equation, Inverse problems, 6, 681-696, (1990) · Zbl 0726.35053 [4] Eldén, L., Numerical solution of the sideways heat equation by difference approximation in time, Inverse problems, 11, 913-923, (1995) · Zbl 0839.35143 [5] Eldén, L., Solving the sideways heat equation by a method of lines, J. heat. transfer, trns. ASME, 119, 406-416, (1997) [6] Tautenhaha, U., Optimal stable approximations for the sideways heat equation, J. inv. ill-posed problems, 5, 287-307, (1997) · Zbl 0879.35158 [7] Eldén, L.; Berntsson, F.; Regińska, T., Wavelets and Fourier methods for solving the sideways heat equation, SIAM J. sci. comput., 21, 2187-2205, (2000) · Zbl 0959.65107 [8] Qiu, C.Y.; Fu, C.L.; Zhu, Y.B., Wavelet and regularization of the sideways heat equation, Computers math. applic., 46, 5/6, 821-827, (2003) [9] Fu, C.L.; Qiu, C.Y.; Zhu, Y.B., A note on “sideways heat equation and wavelets” and constant e*, Computers math. applic., 43, 8/9, 1125-1134, (2002) · Zbl 1051.65090 [10] Fu, C.L.; Qiu, C.Y., Wavelet and error estimation of surface heat flux, J. comp. appl. math., 150, 143-155, (2003) · Zbl 1019.65074 [11] Fu, P.; Fu, C.L.; Xiong, X.T.; Li, H.F., Two regularization methods and the order optimal error estimates for a sideways parabolic equation, Computers math. applic., 49, 5/6, 777-788, (2005) · Zbl 1077.80005 [12] Háo, D.N.; Reinhardt, H.J.; Schneider, A., Numerical solution to a sideways parabolic equation, Engin. int. J. numer. meth. engng., 50, 1253-1267, (2001) · Zbl 1082.80003
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