Keywords:Method of Lines; Partial Differential Equation; Convergence; Error Estimates
1. Introduction
We consider the boundary value problem for the third order differential equation in the domain
:
(1)
(2)
(3)
(4)
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:
(5)
(6)
(7)
(8)
Let we rewrite the problem (5)-(8) in the form
(9)
where
I-unit matrix,
The matrix 
can be diagonalized as [5,6]
with
Multiplying equation (9) on the left by 
we have
(10)
(11)
(12)
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.