Models of Nanosystems and Methods of Their Investigation, Connected with Orthogonal Splines

Fourier method related with orthogonal splines in parabolic initial boundary value problem for domain with curvilinear boundary

“…where C𝑖𝑗 are constant coefficients, 𝑁 , 𝑗 1 , 𝑗 2 are natural numbers, the dependence of 𝑗 1 , 𝑗 2 on 𝑖 is determined by the sizes of the domain and the shape of the curvilinear boundary 𝜕𝑆 and α𝑖 (𝑥), β𝑗 (𝑦) are orthogonal splines [1]:…”

“…Compact supports of orthogonal (on each grid) splines [α 𝑖 (𝑥) β𝑗 (𝑦)] are rectangular subdomains. We note that orthogonal splines [1] admit also using compact supports consisting of triangles. The functions…”

“…, that is, to the problem on approximation of exact eigenfunctions 𝜙 (𝑘) (𝑥, 𝑦) by linear combinations 𝜙 (𝑘) 𝑁 (𝑥, 𝑦) of orthogonal splines. Such problem was solved in work [1], where there were provided appropriate theorems determining the approximation accuracy and the convergence rate as the number of the nodes in the grid increases depending on the types of particular systems of orthogonal splines.…”

“…As the number of the nodes in the grid increases, the number of the employed orthogonal splines does the same. The eigenvalues of homogeneous system of equations (4.5) in the case 𝑙 = 𝜋 are known, see, for instance, [1],…”