The method of lines is applied to the boundary-value problem for third order partial differential equation. Explicit expression and order of convergence for the approximate solution are obtained.
Method of Lines; Partial Differential Equation; Convergence; Error Estimates1. Introduction
We consider the boundary value problem for the third order differential equation in the domain
:
where are sufficiently smooth functions.
The problems of type (1)-(4) arise in many mathematical and scientific applications [1-3]. In this study, we construct first order accurate differential difference scheme for this problem and give error estimate for its solutions. The approach to the construction of the discrete problem and the error analysis for the approximate solution are similar to those in [4].
Let the solution of the problem (1)-(4) have a bounded derivative in the domain.
2. Differential-Difference Algorithm and Convergence
We divide the domain into stripe by lines On this lines the problem (1)-(4) we approximate by the following differential difference problem:
Let we rewrite the problem (5)-(8) in the form
where
I-unit matrix,
The matrix can be diagonalized as [5,6]
with
Multiplying equation (9) on the left by we have
where
The solution of (10)-(12) containing the third order ordinary differential equation with constant coefficients can be explicitly found
where
Therefore the solution of (5)-(8) can be expressed as
where
.
Now we investigate the error of the approximate solution. For the error we have the following boundary value problem:
where
or
Next for
By the mean value theorem we have
Then
Since then it follows that
Further, we note that and
Hence
Using here the inequality, and taking into account
it follows that
i.e., fourth order convergence for the approximate solution is established.
REFERENCESReferencesA. I. Kozhanov, “Mixed Boundary Value Problem for Some Classes of Third Order Differential Equations,” Matematicheskii Sbornik, Vol. 118 (160), No. 4 (8), 1982, pp. 504-522. (in Russian)A. I. Kozhanov, “Mixed Problem for One Class of Quasilinear Equation of Third Order,” In: Boundary Value Problems for Nonlinear Equations, Novosibirsk, 1982, pp. 118-128. (in Russian)S. A. Gabov and A. G. Sveshnikov, “Problems of the Dynamics of Stratified Fluids,” Nauka, Moscow, 1986, p. 288. (in Russian)G. M. Amiraliyev and P. Okcu, “Error Estimates for Differential Difference Schemes to Pseudo-Parabolic Initial-Boundary Value Problem with Delay,” Computers & Mathematics with Applications, Vol. 18, No. 3, 2013, pp. 283-292.S. B. Nemchinov, “On the Finite Difference Method to the Elliptic Boundary Value Problems,” Journal of Computational and Applied Mathematics, Vol. 2, 1962, pp. 418-436. (in Russian)A. A. Samarskii, “The Theory of Difference Schemes,” Marcel Dekker, New York, 2001.
