Numerical-analytic technique for investigation of solutions of some nonlinear equations with Dirichlet conditions

Abstract

The article deals with approximate solutions of a nonlinear ordinary differential equation with homogeneous Dirichlet boundary conditions. We provide a scheme of numerical-analytic method based upon successive approximations constructed in analytic form. We give sufficient conditions for the solvability of the problem and prove the uniform convergence of the approximations to the parameterized limit function. We provide a justification of the polynomial version of the method with several illustrating examples.

A new approach to non-local boundary value problems for ordinary differential systems

“…Version 2 (Polynomial interpolation) Formula (52) is modified so that the polynomial approximations of the integrands are used, i. e., instead of (9), one uses the formula v mþ1 ðt; n; gÞ :¼ u 0 ðt; n; gÞ þ where l is fixed and p l y stands for the polynomial of degree l interpolating the function y at l suitably chosen nodes. The substantiation is similar to other similar cases (see, e.g., [14] where Dirichlet problems for systems of two equations are considered). In this case, one assumes that f satisfies the Dini condition in the time variable [15].…”

“…Let ðn; gÞ 2 D 0 Â D 1 be arbitrary. In view of (14), it follows immediately from (8) that u 0 ðt; n; gÞ 2 X for any t 2 ½a; b, i.e., (24) holds for m ¼ 0.…”

“…R 2 and vector d.We need to choose some domains where the values of a solution at 0 and 1=2 should belong. Let us put, e.g., D 0 :¼ ðu 1 ; u 2 Þ : À0:55 6 u 1 0:45; À0:2 6 u 2 0:15f g ; D 1 :¼ D 0 :ð58ÞIt is clear from (4) that BðD 0 ; D 0 Þ ¼ D 0 and, therefore, according to(14), we have X ¼ D 0 in this case. Putting.…”

On boundary value problems with prescribed number of zeroes of solutions

“…We use here the numerical-analytic approach from [19], which allows one to constuct approximate the solutions of problem (4.4), (4.5) and, moreover, rigorously prove the existence of an exact solutions by using the results of computation. More details on the techniquees discussed and used here can be found, e. g., in [15][16][17][18]. We describe how these techniques can be efficiently used to find either sign-constant or sign-changing solutions.…”

“…In [19], we mention two natural modifications of this kind, namely the version of "frozen" parameters and the version with polynomial interpolation used in [16]. Let us apply here the "frozen" parameters version, which means that the approximations X m and Y m obtained on the mth step are used directly instead of x m .…”