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Research Article
 

Solution of Pseudoparabolic Equation by Finite-Element Method



R. Lotfikar and A. Farmany
 
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ABSTRACT

In this study, an approximate solution for the initial-boundary value problem for the pseudoparabolic equation using finite-element method is obtained. It is proved that the constructed sequence converges to the exact solution is possible.

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  How to cite this article:

R. Lotfikar and A. Farmany, 2010. Solution of Pseudoparabolic Equation by Finite-Element Method. Journal of Applied Sciences, 10: 781-783.

DOI: 10.3923/jas.2010.781.783

URL: https://scialert.net/abstract/?doi=jas.2010.781.783
 

INTRODUCTION

Let, Ω⊂Rn be a bounded domain with the smooth boundary and t>0.

The following initial-boundary value problem (Showalter, 1996):

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(1)

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(2)

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(3)

Where:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

and bij(x) = bji(x), aij(x) = aji(x) (i,j = 1,2,...,n) are continuous functions in Image for - Solution of Pseudoparabolic Equation by Finite-Element Methodand the following inequality is valid:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

For ∀x ∈ Image for - Solution of Pseudoparabolic Equation by Finite-Element Method, ∀ξ ∈ ún was investigated by Petrosyan and Hakobyan (2008).

The problem in Eq. 1-3 was investigated in the case in which L is linear, M is nonlinear where, L and M are degenerated operators (Petrosyan and Hakobyan, 2008). The solution of a general case in which L and M are nonlinear is considered by Gaevskii et al. (1978).

Quarteroni et al. (2000) proved (using Galerkin’s method) that the solution of the problem 1-3 exists (Mamikonyan, 2006).

In this study, we construct an approximate solution of the problem 1-3 using finite-element method for the case in which Ω∈(0,1)x(0,1)⊂ú2, Lu = -Δu,

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

Definition: The function:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

may be a weak solution of the problem in Eq. 1-3 if:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

and for:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

the Eq. 4 is valid:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(4)

Petrosyan and Hakobyan (2008) proved that the Eq. 4 has a unique solution.

Let, we construct an approximate solution for the problem in Eq. 4 using the finite-element method.

Suppose the partition domain Ω = (0,1)2 with a uniform triangulation of mesh size h with respect to x and y as Fig. 1.


Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
Fig. 1:

Partition domain Ω = (0,1)2 with a uniform triangulation of mesh size h with respect to x and y


xi+1 -xi = h,i,j = 1,2,...,n,
yj+1-yj = hh = 1/n

We construct the piecewise linear functions φij(x,y) by the following rule:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

and linear in the domain of every triangle. In the remained triangles of the square [0,1]x[0,1] we assume φij (x,y) = 0.

In continue, we set N = (n-1)2 as the basis functions. Let, ωn = {(ih, jh); i, j = 1,2,...,n-1}. If we number the points of the set ωn for example:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

then, the basis functions φij will be renumbered, i.e., by constructing ψk(Ar) δkr (k,r = 1,2,...,n-1) we get the system ψ12,...,ψn.

Note that Sn is the linear space generated by the functions ψi = (i = 1,2,...,N) and dimSn = N and Image for - Solution of Pseudoparabolic Equation by Finite-Element Methodwhere, v is linear in every triangle and v = 0 on ∂Ω}.

It is easy to see that:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

is a subspace. To calculate,

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

we use the Table 1.


Table 1:

The derivatives of the basis functions ψi

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

Denote:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

To find the weak solution of the problem 1-3 we use the Galerkins method:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

which is equivalent to:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(5)

We can rewrite the Eq. 5 in the matrix form:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(6)

where, βN = (α12,...,αN),

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

It is easy to check that the matrix MN has the following form:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

where, E is the unit matrix and A is the following matrix of order of (n-1)x(n-1):

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

The solution of the system of differential Eq. 6 with the conditions:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(7)

where, ci are the coefficients of the expansion of the function u0(x,y) by the basis ψi(x,y) (i =1,...,N) may be denoted by using αi(t) (i = 1,...,). Thus, we obtain the following sequence of the functions:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

This sequence:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

converges in:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

norm to the weak solution of the problem in Eq. 1-3.

To find the numerical solution of the system 6 we use the θ method (Braess, 2001). Suppose the partition [0,T] into equal parts with the step Δt denote by:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

Now we replace the system 6 by the following different system:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(8)

where, Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

For every k, we get the linear system of equations.

We choose the parameter θ such that the matrix:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

will be positive,

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

Then we can represent the system of Eq. 8 in the following form:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method
(9)

where, K = HTH.

Denote by:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

It is easy to verify that:

Image for - Solution of Pseudoparabolic Equation by Finite-Element Method

CONCLUSION

The initial boundary value problem is investigated for the pseudoparabolic equation with nonlinear operators. An approximate solution for this problem is obtained using the finite element method. Finally it is proved that the constructed sequence converges to the exact solution is possible.

REFERENCES
1:  Braess, D., 2001. Finite Elements. Cambridge University Press, Cambridge.

2:  Gaevskii, K., K. Greger and K. Zakharis, 1978. Nonlinear operator equations and operator differential equations. Mir, Moscow, (In Russian).

3:  Mamikonyan, H.A., 2006. The Galerkin Method for Some Class of Nonlinear Degenerate Pseudoparabolic Equations. EGU, Austria, pp: 33-40.

4:  Petrosyan, A.A. and G.S. Hakobyan, 2008. On a generalization of nonlinear pseudoparabolic variational inequalities. J. Contemporary Math. Anal., 43: 118-125.
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5:  Quarteroni, A., R. Sacco and F. Saleri, 2000. Numerical Mathematics. 1st Edn., Springer, USA.

6:  Showalter, R.E., 1996. Monotone Operators in Banach space and Nonlinear Partial Differential Equations (Mathematical Surveys and Monographs). American Mathematical Society, Amrica, pp: 278.

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