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A Comparison between Solving Two Dimensional Integral Equations by the Traditional Collocation Method and Radial Basis Functions on project
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Introduction
Two dimensional integral equations provide an important tool for modeling
a numerous problems in engineering and mechanics [2, 11]. There are many
different numerical methods for solving integral equations. Some of them can
be used for solving double integral equations. Computational complexity of
mathematical operations is the most important obstacle for solving integralTwo dimensional integral equations provide an important tool for modeling
a numerous problems in engineering and mechanics [2, 11]. There are many
different numerical methods for solving integral equations. Some of them can
be used for solving double integral equations. Computational complexity of
mathematical operations is the most important obstacle for solving integral
Radial basis functions
Definition(Basic RBF Method): Given a set of n distinct data points
{pj}n
j=0 and corresponding data value {fj}n
j=0 , f(x) can be written the follow-ing form
Φ(p) = n
j=0
λjφ(p − pj), (2)
where . is Euclidean norm, p, pj ∈ Rd(d is positive finite integer) and fj is
scalar. Also φ®,r >= 0, is some radial basis functions. The coefficient λj
must be determined as unknown. Collocating of {xj}n
j=0 in (2) leads us to
following symmetric linear system
⎡
⎣ A
⎤
⎦
⎡
⎣λ
⎤
⎦ =
⎡
⎣f
⎤
⎦, (3)
where the entries of A are given by
ajk = φ(pj − pk). (4)
Micchelli [16] gave sufficient conditions for φ® in (3) to guarantee that A
matrix in (4) is unconditionally nonsingular, and thus that the basic RBF
method is uniquely solvable. Some common infinitely smooth example of the
φ® that lead to a uniquely solvable method are the following forms
Linea
Solution of linear two dimensional integral equation
Consider the two dimensional linear Fredholm integral equation as follows
u(x, t) −
1
−1
1
−1
k(x, t, y, z)u(y, z)dydz = f(x, t), (x, t) ∈ [−1, 1] × [−1, 1], (5)where k(x, t, y, z) and f(x, t) are continuous functions on [−1, 1]4 and [−1, 1]2
respectively. For the case which integration domain is [a, b]×[c, d], we can use
suitable change of variable to obtain this intervals.
conclusion
Analytical solution of the two dimensional integral equations are usually
difficult. In many cases, it is required to approximate solutions. In this work,
the two dimensional linear integral equations of the second kind is solved and
compared by using Chebyshev polynomials and RBF method. The illustrative
examples confirm the spectral convergence in both of conventional spectral
method and RBF method. however, the best choice must be more effective in
higher dimensions. There are a lot of sources that recommend to use radial
basis functions in higher dimensions.