## 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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