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Numerical algorithm for parabolic problems with non-classical conditions. (English) Zbl 1190.65136

The authors consider the 1D heat equation with constant coefficients subject to initial conditions and nonlocal boundary conditions
\[ \lambda_iu(i,t)+\gamma_iu_x(i,t)=\int_0^1 p_i(x)u(x,t)\,dx+q_i(t) \text{ for } i \in \{0,1\}. \]
The numerical method is based on the truncation of a uniformly convergent series of the solution, which is assumed to lie in some reproducing kernel spaces. Numerical examples are provided.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
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References:

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