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....converges also in L norm. It remains to show that Eq. 12) is sufficient to ensure (11) i.e. more generally, we have to show the completeness of the eigenfunction function system. For details, see next section 5. Remark 1. The boundary value system (8) is of Sturm Liouville type (e.g. see [1, 2] ) But (to the knowledge of the authors) the dynamic boundary condition (8c) has not been treated before within that context. Remark 2. Let denote H : H 0; 0; 1] the Hilbert space of functions with (0) 0 and [ h the corresponding dotproduct. A function 2 H is a weak ....
....solution of the system (4) for a fixed . Computation of the Fourier series (Eq. 5) in Mathematica u[kappa] is the an expression containing the variables x and t ) u[kappa ] Module[ predictor, lambda, coeff, newtonstp , predictor[n ] l . Solve[l kappa(l n Pi) 1 (l n Pi)2 2) 0,l] [2]] f = l kappa Sin[l] Cos[l] newtonstp[l ] N[l f D[f,l] 20] lambda[n ] FixedPoint[newtonstp, predictor[n] coeff[n ] 2 Sin[lambda[n x] lambda[n] 1 kappa Sin[lambda[n] 2) Sum[coeff[n] Exp[ lambda[n]2 t] n, 0, 20 ] 4 Estimation of the diffusion constant Except for a short initial ....
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