A non-separable solution of the diffusion equation based on the Galerkin’s method using cubic splines

• Published in: Applied mathematics and computation Vol. 217; no. 5; pp. 1831 - 1837
• Main Authors: Moon, B.S., Yoo, D.S., Lee, Y.H., Oh, I.S., Lee, J.W., Lee, D.Y., Kwon, K.C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01-11-2010; Elsevier
• Subjects: Approximation; Arrays; Bi-cubic splines; Calculus of variations and optimal control; Collocation method; Computation; Diffusion; Diffusion equation; Exact sciences and technology; Galerkin methods; Galerkin’s method; Mathematical analysis; Mathematical models; Mathematics; Non-separable solution; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Diffusion equation; Non-separable solution; Collocation method; Galerkin’s method; Bi-cubic splines; Initial condition; Square matrix; Galerkin's method; Cubic spline; Two dimensional equation; Numerical analysis; One-dimensional calculations; Applied mathematics; Approximation error; Galerkin method; Analytical solution
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2010.05.018

The two dimensional diffusion equation of the form
∂
2
u
∂
x
2
+
∂
2
u
∂
y
2
=
1
D
∂
u
∂
t
is considered in this paper. We try a bi-cubic spline function of the form
∑
i
,
j
=
0
N
,
N
C
i
,
j
(
t
)
B
i
(
x
)
B
j
(
y
)
as its solution. The initial coefficients
C
i,
j
(0) are computed simply by applying a collocation method;
C
i,
j
=
f(
x
i
,
y
j
) where
f(
x,
y)
=
u(
x,
y,
0) is the given initial condition. Then the coefficients
C
i,
j
(
t) are computed by
X(
t)
=
e
tQ
X(0) where
X(
t)
=
(
C
0,1,
C
0,1,
C
0,2,
…
,
C
0,
N
,
C
1,0,
…
,
C
N,
N
) is a one dimensional array and the square matrix
Q is derived from applying the Galerkin’s method to the diffusion equation. Note that this expression provides a solution that is not necessarily separable in space coordinates
x,
y. The results of sample calculations for a few example problems along with the calculation results of approximation errors for a problem with known analytical solution are included.