On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

Abstract

In this paper, on the basis of the Carleman matrix, we explicitly construct a regularized solution of the Cauchy problem for the matrix factorization of Helmholtz’s equation in an unbounded two-dimensional domain. The focus of this paper is on regularization formulas for solutions to the Cauchy problem. The question of the existence of a solution to the problem is not considered—it is assumed a priori. At the same time, it should be noted that any regularization formula leads to an approximate solution of the Cauchy problem for all data, even if there is no solution in the usual classical sense. Moreover, for explicit regularization formulas, one can indicate in what sense the approximate solution turns out to be optimal.

Generalized Class of Finite Population Variance in the Presence of Random Nonresponse Using Simulation Approach

“…1 241.0492 T (1) 181.4473 T (4) 1 267.6401 T (4) 2 206.5321 T (4) 3 181.4576 T (5) 1 267.6965 T (5) 2 206.5532 T (5) 3 181.4663 T (6) 1 267.1691 T (6) 2 206.3557 T (6) 3 181.3844 T (7) 1 267.6403 T (7) 2 206.5322 T (7) 3 181.4567 T (8) 1 267.4689 T (8) 2 206.468 T (8) 3 181.431 T (9) 1 267.7158 T (9) 2 206.5604 T (9) 3 181.4693 T (10) 1 266.8310 T (10) 2 206.2289 T (10) 3 181.3318 171.5264 T (5) 1 259.3661 T (5) 2 210.6753 T (5) 3 171.5099 T (6) 1…”

“…T (7) 2 211.2946 T (7) 3 171.5173 T (8) 1 259.5887 T (8) 2 210.7888 T (8) 3 171.5888 T (9) 1 259.5026 T (9) 2 210.7449 T (9) 3 171.4911 T (10) 1 260.9819 T (10) 2 211.4958 T (10) 3 171.6693…”

“…Table 5: Data summary for Population-III and -IV. 754.335 T (5) 1 1828.449 T (5) 2 878.4392 T (5) 3 754.3343 T (6) 1 1828.516 T (6) 2 878.4476 T (6) 3 754.343 T (7) 1 1828.449 T (7) 2 878.4392 T (7) 3 754.3343 T (8) 1 1828.517 T (8) 2 878.4478 T (8) 3 754.3431 T (9) 1 1828.44 T (9) 2 878.4381 T (9) 3 754.3331 T (10) 1 18 8.6 6…”

“…T (10) 2 878.4616 T (10) 3 172.7624 T (5) 1 270.0321 T (5) 2 215.8111 T (5) 3 172.7651 T (6) 1040.117 T (5) 1 1858.08 T (5) 2 918.3413 T (5) 3 1040.073 T (6) 1 1859.334 T (6) 2 918.4983 T (6) 3 1040.289 T (7) 1 1858.185 T (7) 2 918.3545 T (7) 3 1040.092 T (8) 1 1858.957 T (8) 2 918.4511 T (8) 3 1040.224 T (9) 1 1857.946 T (9) 2 918.3246 T (9) 3 1040.05 T (10) 1 1860.361 T (10) 2 918.6 63 T (10) 3 1040. 466 10 Complexity T (6) 2 866.0122 T (6) 3 937.3265 T (7) 1 1742.708 T (7) 2 865.6568 T (7) 3 936.8599 T (8) 1 1743.53 T (8) 2 865.7443 T (8) 3 936.9748 T (9) 1 1742.197 T (9) 2 865.5748 T (9) 3 936.7523 T (10) 1…”

“…Sunday et al [6] adopted a variable step hybrid block method for the approximation of Kepler Problem by integrating the Lagrange polynomial with limits of integration selected at special points. Juraev et al [7,8] used regularization formula and matrix factorization for explicit form of the approximate solutions of the Cauchy Problem.…”