On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle
- PDF / 149,181 Bytes
- 6 Pages / 595.28 x 793.7 pts Page_size
- 21 Downloads / 233 Views
RESEARCH
Open Access
On an effective solution of the Riemann problem for the second-order improperly elliptic equation in the rectangle Armenak O Babayan1 and Seyed Mohammadali Raeisian2* *
Correspondence: [email protected] 2 Library of Institute of Mathematics (2nd floor), National Academy of Sciences of Armenia, Baghramyan 24b, Yerevan, Republic of Armenia Full list of author information is available at the end of the article
Abstract In this paper we present the numerical method for the solution of the Riemann problem for the second-order improperly elliptic equation. First, we reduce this problem to boundary value problems for properly elliptic equations, and after that we solve these problems by the grid method. MSC: 35G45; 35G15; 35J25; 35J57; 65N06; 65N20 Keywords: improperly elliptic equation; boundary value problem; Riemann problem; Bitzadze equation; grid method
1 Introduction Let D be a rectangle D = {(x, y) : a < x < b, c < y < d} in a complex plane with boundary = ∂D. We consider in D the equation n k=
Ak
∂ nu (x, y) = , ∂yn–k
∂xk
(x, y) ∈ D,
()
where Ak are complex constants (A = ) such that the characteristic equation n
Ak λn–k =
()
k=
has no real roots (i.e., elliptic equation). Let the roots of () λk with multiplicities mk satisfy the condition λk > and the roots μj with multiplicities lj satisfy the condition μj < . We suppose that k mk > j lk , so () is an improperly elliptic equation. It was shown in [] that for the equation uz¯ z¯ = (now known as a Bitzadze equation) the corresponding Dirichlet problem is not correct. It was shown later ([–]) that for arbitrary improperly elliptic equation () all of the classical boundary value problems are not correct (we say that the problem is correct if the corresponding homogeneous problem has a finite number of linearly independent solutions and the inhomogeneous problem is solvable if and only if the finite number of linearly independent conditions for the boundary functions are satisfied). Therefore another kind of boundary conditions must be introduced. In the works [–], different types of boundary conditions, whose number depends on the number of the roots of () with positive and negative imaginary parts, were introduced. In this © 2013 Babayan and Raeisian; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Babayan and Raeisian Advances in Difference Equations 2013, 2013:190 http://www.advancesindifferenceequations.com/content/2013/1/190
Page 2 of 6
paper we consider in D the Bitzadze equation ∂ u ∂ ∂ + i (x, y) ≡ u(x, y) = , ∂ z¯ ∂x ∂y
(x, y) ∈ D.
()
We seek the unknown function u ∈ C (D) ∩ C (,α) (D ∪ ) in the class of two times continuously differentiable in D functions which satisfy Hölder condition in D ∪ with their first degree deriva
Data Loading...
