Method of Lines for Third Order Partial Differential Equations ()
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)
![](https://www.scirp.org/html/htmlimages\5-1720085x\1bd84919-76fb-4c14-b869-42615235419c.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\67d7fabb-6525-406a-8c1f-3fd4feafea4d.png)
where
![](https://www.scirp.org/html/htmlimages\5-1720085x\f2c52ec6-671c-4516-a53b-d4c68f3ee976.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\5a63944c-75f9-4b35-bbb1-60dbb00e76db.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\178a65da-4717-419f-893c-3a8e511201a1.png)
I-unit matrix,
![](https://www.scirp.org/html/htmlimages\5-1720085x\196a0c8c-73fc-44c6-b60d-8672c07ce7b2.png)
The matrix
can be diagonalized as [5,6]
![](https://www.scirp.org/html/htmlimages\5-1720085x\cd08f60a-22ff-47c3-9c35-547b5a9d5182.png)
with
![](https://www.scirp.org/html/htmlimages\5-1720085x\7d38a12c-d543-4967-9bde-3aaea2162fdb.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\0f55f825-e928-4e3b-9c43-a92915b7befd.png)
Multiplying equation (9) on the left by
we have
(10)
(11)
(12)
where
![](https://www.scirp.org/html/htmlimages\5-1720085x\47f02fb8-15e8-4574-967f-5441f0177aab.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\bdb7d06e-a9c4-4be7-aa4c-e4ca7778d182.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\f32ccf45-c54d-4c2d-a281-2742c98fe297.png)
The solution of (10)-(12) containing the third order ordinary differential equation with constant coefficients can be explicitly found
![](https://www.scirp.org/html/htmlimages\5-1720085x\da513ca8-ab7f-44d5-adfa-817fade74e15.png)
where
![](https://www.scirp.org/html/htmlimages\5-1720085x\ffcb4012-1a77-442b-905c-fc020eb10503.png)
Therefore the solution of (5)-(8) can be expressed as
![](https://www.scirp.org/html/htmlimages\5-1720085x\bcf5d45f-8b97-4530-bf5b-cc027711e867.png)
where
.
Now we investigate the error of the approximate solution. For the error
we have the following boundary value problem:
![](https://www.scirp.org/html/htmlimages\5-1720085x\a135e0d9-5269-4bb1-99f0-62373fddda2d.png)
where
![](https://www.scirp.org/html/htmlimages\5-1720085x\585d1105-5fe3-4201-9f8b-cb058aa94e1e.png)
![](https://www.scirp.org/html/htmlimages\5-1720085x\717b10a0-a9a0-47ff-98c9-00dbd3e95a61.png)
or
![](https://www.scirp.org/html/htmlimages\5-1720085x\49ca9b38-89af-444a-84a5-b850740d1e56.png)
Next for
![](https://www.scirp.org/html/htmlimages\5-1720085x\e0c39967-af91-4b40-81d9-0f746f2c630c.png)
By the mean value theorem we have
![](https://www.scirp.org/html/htmlimages\5-1720085x\53b56602-59a6-45c4-8534-502ed8d58519.png)
Then
![](https://www.scirp.org/html/htmlimages\5-1720085x\a0f4a642-1205-4efc-9b81-2541a4623b19.png)
Since
then it follows that
![](https://www.scirp.org/html/htmlimages\5-1720085x\43978711-eca7-41bb-88a9-f1b2cd1b0e98.png)
Further, we note that
and
![](https://www.scirp.org/html/htmlimages\5-1720085x\85117baa-ee7f-4fa7-9036-42e9c47c83b6.png)
Hence
![](https://www.scirp.org/html/htmlimages\5-1720085x\d96e5ce5-8155-4916-b74c-dafd45cc43a5.png)
Using here the inequality
, and taking into account ![](https://www.scirp.org/html/htmlimages\5-1720085x\225195bf-329d-41a8-8bcf-b2e0c96beaa9.png)
it follows that
![](https://www.scirp.org/html/htmlimages\5-1720085x\0e6964e7-da4e-4093-ac94-c7d050c6dd33.png)
i.e., fourth order convergence for the approximate solution is established.