Applications of the boundary integral equation method to real world problems often require that field values should be obtained near boundary surfaces.
The boundary value condition should be supplemented definitely on the nondegenerate boundary and the weakly degenerate boundary although the equation is degenerate on this portion of the degenerate boundary.
We conjectured that, to ensure the well-posedness of the solutions, a partial boundary value condition should be imposed on equation (1.4).
By the way, for the following reaction diffusion equation begin{aligned} frac{partial v}{partial t}=operatorname{div} bigl(a v,x,t nabla v bigr) +operatorname{div} bigl(b v) bigr),quad (x,t in Omegatimes 0,T), end{aligned} (1.24) with begin{aligned} a v,x,t)|_{xinpartialOmega}=0, end{aligned} (1.25) we had conjectured that a partial boundary value condition should be imposed.
By the aid of Fichera Oleinik theory, we conjectured that the partial boundary value condition (1.5) should be u x,t)=0,qquad (x,t inSigma_{1}times 0,T),qquad Sigma_{1}=bigl{ xinpartial Omega: b_{i}(0)n_{i}< 0 bigr}, (1.10) where (vec{n}={n_{i}}) is the inner normal vector of Ω, and proved the following theorem.. Suppose that (A s)) is (C^{2}) and (b_{i}(s,x,t equiv b_{i}(s)) is (C^{1}).
Before running parameter estimation, initial parameter values and boundaries should be set within physically plausible ranges.
The advantage of the Chinese way lies in the fact that one can figure out on which portion of the boundary should be imposed the boundary value, whereas the rest of the boundary is free from any limitation.
But if equation (1.1) is strongly degenerate, we shall show that only a portion of the boundary should be given the boundary value.
