Galerkin Method, Single and Double Exponential Transformation of Sinc-Galerkin Method

Galerkin method, single and double exponential transformation of sinc-Galerkin method for solving single two-point boundary value problems

Galerkin method, single and double exponential transformation of sinc-Galerkin method for solving single two-point boundary value problems
Hossein Pourbashash1,2,*, Hossein Kheiri2 and AliasgharJodayreeAkbarfam2
1Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Email Address: , Funded by AdibanInstitue of Higher Education
2Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran , Email Address:,
ICM 2012, 11-14 March, Al Ain

Galerkin method, single and double exponential transformation of sinc-Galerkin method for solving single two-point boundary value problems

Galerkin method, single and double exponential transformation of sinc-Galerkin method for solving single two-point boundary value problems

[*]abstract

In many applicable problems, boundary value problems with singular solutions arise. Most of the numerical methods are not appropriate for solving these problems. They often cannot pass the singular point successfully. Sinc-Galerkin method is one of the best methods for overcoming on the singular points difficulties. In this paper we incorporate sinc-Galerkin method with the double exponential transformation (DE transformation), single exponential transformation (SE transformation) and Legendre Galerkin methods for these problems. Some examples are given for highlighting the accurate and the power of the proposed methods. In linear examples we show that Legendre Galerkin method is not suitable method then we don t use this method for nonlinear problems.Keywords: Sinc method, Singular, Double exponential transformation, single exponential transformation

1introductioN

The sinc method is a highly efficient numerical method that has been developed by Frank Stenger, the pioneer of this field, and his colleagues [1,2], it is widely used in various fields of numerical analysis, solution of integral, ordinary differential and partial differential equations [3-13]. Sinc-Galerkin is one of the sinc methods that used in this paper for solving boundary value problems with singular solutions. Despite most of the numerical methods, sinc-Galerkin method comprehends problems that have singular solutions. Conventional form of these methods is SE transformation. It is shown that in this case, error bound of the approximate solution is with , where N is number of terms in the Sinc approximation.

Takahasi and Mori [9] proposed the double exponential transformation for one dimensional numerical integration in 1974. The effectiveness of the DE transformation technique in numerical integration naturally suggests that the DE transformation technique could be useful in other numerical methods. In 1997, Sugihara [10] established the “meta-optimality” of the DE formula in a mathematically rigorous manner, and since then it has turned out that the DE transformation is also useful for other various kinds of numerical methods. Indeed, it has been demonstrated in [11-14] that, the use of the Sinc method incorporated with the DE transformation gives highly efficient numerical methods for functions approximation, indefinite numerical integration, and the solution of differential equations. It has been shown the error bound of the Sinc-Galerkin method based on the DE transformation to numerical solution of boundary value problems of second order is, c>0 [13,14] which converges to zero much faster than the method based on the SE transformation as N becomes large.

In this article we apply the DE and SE transformation sinc-Galerkin method to solve boundary-value problems:

(1)

Where , are analytic functions and has singular points on It is shown that, Both of these methods overcomes on the singular points difficultly but DE transformation Sinc-Galerkin method is more accurate.

2sinc bases

Galerkin method, single and double exponential transformation of sinc-Galerkin method for solving single two-point boundary value problems

Sinc function is demonstrated on by

(2)

This function is translated with evenly spaced nodes are given as

,.(3)

If is analytic on a strip domain

(4)

in the z-plane and as then, the series

(5)

converges, we call it whittaker cardinal expansion.

From [10] we can write

(6)

where d is half width of strip domain (4).

If be a real function, Sinc expansion (5) is defined on , while the equation that we want to solve is defined , and hence we need some transformation which the given interval transform onto. In many of applications of the sinc method transformation

.(7)

has been used. The map carries the eye-shaped region

,(8)

onto

.(9)

Define by

(10)

The is the mesh size in for the uniform grids The base functions on are given by

(11)

The sinc grid points in will be denoted bybecause they are real. The inverse images of the equispaced grids in SE transformation are

.(12)

or

(13)

In DE transformation, we can use

,(14)

,(15)

whichTakahasi and Mori proposed for numerical integration[9]. One of the best reasons for using (15) is optimalityof this transformation.It usually gives significantly faster convergence than (13)[9,10and 15].

Definition 1 Let be a simply-connected domain which satisfies, and let be positive constants. Then denotes the family of all functions that satisfy the following conditions: (i) is analytic in ; (ii) there exists a constant C such that

(16)

holds for all in . For later convenience, let us denote and introduce a function . If

, (17)

when for some positive constants and , the next theorem guarantees the exponential convergence of the SE-sinc approximation.

Theorem 1 Let for with Let also be a positive integer, and be given by (10), Then there exists a constant independent of , such that

. (18)

Proof: Ref [1].

□

In DE transformation, if

, (19)

and we have:

Theorem 2 Let for with, let be a positive integer, and let be selected by the formula

(20)

Then there exists a constant which is independent of , such that

. (21)

Proof: Ref [17, 18].

□

By comparing (18) and (21) it is observed that convergence by the DE transformation as become large is much faster than that by SE transformation. In fact it is proved that DE sinc approximation is optimal in some sense in the approximation [19, 20].

3Sinc-Galerkin method

For solving problem (1) with DE sinc-Galerkin method, we need some information.

Lemma 1. Let be the conformal one-to-one mapping of the simply connected domain to Given by (21). Then

(22)

(23)

(24)

Proof: Ref [1].

□

In linear problem we have

(25)

We consider equation (25) and its approximation solution by

,(26)

where is the function for some fixed step size h. The unknown coefficients is determined that

(27)

or

.(28)

The used inner product is defined by

,(29)

where[1].

For solving achieved system we can use following theorem

Theorem 3. The following relations hold

(30)

(31)

and

,(32)

where

, , ,

,

and

or.

Proof: Ref [16].

□

If use theorem 3 for replacing in inner product (27) we obtain following theorem:

Theorem 4. If the assumed approximate solution of the boundary-value problem (25) is (26), then the discrete Sinc-Galerkin system for the determination of the unknown coefficients is given by

(33)

We can rewrite(33) in following system:

(34)

where

, the matrices whose jk-th entry is given by (16)-(18), be the m-vector with i-th component given by , 1 be the m-vector each of whose components is 1 and the functions are given by:

,

,

.

Proof: Ref [3,16].

□

By solving the obtained algebraic linear system, the vector and so the approximation solution is determined.

Now we consider nonlinear boundary value problem

(35)

If (26) be a approximation solution of (35), the unknown coefficients is determined that

. (36)

Lemma 2.we have

Proof: Reff [3,16].

□

If use theorem 3 and lemma 2 for replacing in inner product (36) we obtain following theorem:

Theorem 4If the assumed approximate solution of the boundary-value problem (35) is (26), then the discrete Sinc-Galerkin system for the determination of the unknown coefficients is given by

(37)

We can rewrite (37) in following system:

(38)

where

and is given by (34). For solving nonlinear system (38), we can use Newton s method.

4Numerical examples

Here we present some examples that are solved by DE and SE Sinc-Galerkin methods and Legendre Galerkin method. These examples have singular point in solutions. Comparisons show that Legendre Galerkin method is not appropriate method for solving these problems. Sinc-Galerkin methods are suitable methods for solving problems with singular solutions. The DESinc-Galerkinmethod gives better results than SE Sinc-Galerkin method. The problems are solved with Matlab on a personal computer.

In these examples, the maximum absolute error at sinc points is taken as

where

,

with and

where

.

In tables 1, 3, 5 and 7 we give the absolute errors in some points using proposed methods.In tables 2,4, 6 and 8 we present maximum error in SE sinc-Galerkinmethod and DE sinc-Galerkin method in all sinc points. We use in SE sinc-Galerkin method and in DE sinc-Galerkin method. It is observed that although in DE transformation is smaller but it’s better than SE transformation. In Legendre Galerkin method our bases functions are Legendre polynomials and we cannot choose a high value . Thus,we choice in our examples is

Example 1. Consider the equation

with exact solution

In this problem the singular pointis.

Table 1.The error of solving example 1

Error in Galerkin method / Error in SE / in SE / Error in / in DE
4.004 / 6.73 e-008 / -0.961 / 5.42
e-014 / -0.963
15.297 / 2.57 e-007 / -0.852 / 3.14
e-013 / -0.857
38.770 / 6.58 e-007 / -0.605 / 7.8
e-013 / -0.608
35.087 / 9.04 e-007 / -0.398 / 1.06
e-012 / -0.375
36.434 / 9.58 e-007 / -0.273 / 1.17
e-012 / -0.286
37.616 / 9.84 e-007 / -0.139 / 1.11
e-012 / -0.097
37.090 / 9.72 e-007 / 0 / 1.07
e-012 / 0
32.150 / 8.42 e-007 / 0.273 / 9.47
e-013 / 0.286
28.269 / 7.40 e-007 / 0.398 / 9.44
e-013 / 0.375
19.889 / 5.20 e-007 / 0.605 / 6.34
e-013 / 0.608
7.7863 / 2.04 e-007 / 0.852 / 2.22
e-013 / 0.857
2.0366 / 5.34 e-008 / 0.961 / 6.01
e-014 / 0.963

Table 2.Maximum error in SE sinc-Galerkinmethod and DE sinc-Galerkin method

/ / N
8.61 e-008 / 0.0855 / 25
1.15 e-012 / 6.85 e-004 / 50
1.93 e-011 / 2.76 e-004 / 75
3.41 e-011 / 1.16 e-005 / 100
2.90e-011 / 9.84 e-007 / 125

Example 2. Consider the equation

with exact solution

In this problem the singular pointis.

Table 3. The error of solving example 2

in DE trasformation / Error in / in SE / Error in SE / Error in Galerkin method
-0.999949158093507 / 1.31
E-16 / -0.999938937287810 / 7.83 E-11 / 1.91
E-06
-0.672078625326399 / 2.30
E-13 / -0.687383065290916 / 5.70 E-7 / 7.80
E-2
-0.375635017328184 / 6.57
E-13 / -0.398183994844089 / 1.60 E-6 / 0.1804
0 / 1.91
E-12 / 0 / 4.80 E-5 / 0.4333
0.286634259852092 / 3.76
E-12 / 0.273823495469893 / 9.62 E-6 / 0.9274
0.537116659907527 / 6.93
E-12 / 0.509448868030365 / 1.72 E-5 / 1.9036
0.672078625326399 / 1.01
E-11 / 0.687383065290916 / 2.66 E-5 / 3.8825
0.778512341446609 / 1.32
E-11 / 0.754565867539812 / 3.13 E-5 / 5.3996
0.888259468294954 / 1.59
E-11 / 0.886419699493427 / 4.29 E-5 / 12.1081
0.950316761845466 / 1.48
E-11 / 0.949477784387200 / 2.61 E-5 / 19.9905
0.986601816069927 / 2.96
E-12 / 0.987362142465592 / 6.81 E-6 / 30.4636
0.999994677382579 / 3.32
E-15 / 0.999995130475963 / 3.33 E-9 / 1.35 E-2

Table 4. Maximum error in SE sinc-Galerkin method and DE sinc-Galerkin method

/ / N
2.27 e-006 / 5.2519 / 25
1.88 e-011 / 0.65403 / 50
6.08 e-011 / 7.02 e-003 / 75
4.08 e-010 / 7.28 e-004 / 100
4.39 e-010 / 4.29 e-005 / 125

Example 3. Consider the nonlinear equation

with exact solution

In this problem the singular point is .

Table 5. The error of solving example 3

in DE trasformation / Error in / in SE / Error in SE
-0.999999986180038 / 6.32 E-17 / -0.999999982347064 / 4.80
E -15
-0.821345508641236 / 8.62 E-15 / -0.808945514479661 / 6.82
E -09
-0.608126245913561 / 1.84 E-14 / -0.605940504455271 / 1.42
E -08
-0.193494262754776 / 6.66 E-14 / -0.139579107197183 / 3.25 E -08
0.286634259852092 / 7.69 E-14 / 0.273823495469893 / 5.95 E -08
0.608126245913561 / 1.84
E-13 / 0.605940504455271 / 1.46 E -07
0.672078625326399 / 4.19
E-13 / 0.687383065290916 / 7.6 E -08
0.888259468294954 / 2.37
E-14 / 0.886419699493427 / 3.01 E -08
0.980891312711878 / 4.84
E-15 / 0.983296113344497 / 4.46 E -09
0.999814417834297 / 2.25
E-16 / 0.999858140267242 / 3.24 E -11

Table 6.Maximum error in SE sinc-Galerkin method and DE sinc-Galerkin method

/ / N
1.34 e-008 / 0.2200 / 25
2.27 e-013 / 0.0030 / 50
2.23 e-012 / 8.5041e-005 / 75
1.19 e-012 / 2.3067e-006 / 100
3.55 e-012 / 1.4657e-007 / 125

Example 4. Consider the nonlinear equation

with exact solution

In this problem the singular point is .

Table 7. The error of solving example 4

in DE / Error in / in SE / Error in SE
-0.999999986180038 / 6.27
E-17 / -0.999999982347064 / 1.05 E -14
-0.821345508641236 / 2.38
E-14 / -0.808945514479661 / 2.39 E -08
-0.375635017328184 / 8.71
E-14 / -0.398183994844089 / 7.96 E -08
-0.193494262754776 / 2.13
E-13 / -0.139579107197183 / 1.22 E -08
0.286634259852092 / 2.52
E-13 / 0.273823495469893 / 2.36 E -07
0.608126245913561 / 1.02
E-12 / 0.605940504455271 / 8.87 E -07
0.672078625326399 / 9.59
E-13 / 0.687383065290916 / 6.17 E -07
0.888259468294954 / 1.50
E-13 / 0.886419699493427 / 2.41 E -07
0.999814417834297 / 2.73
E-16 / 0.999858140267242 / 3.75 E -10

Table 8. Maximum error in method and method

/ / N
6.60 e-007 / 1.1645 / 25
3.06 e-012 / 0.0209 / 50
2.05 e-008 / 2.7825e-004 / 75
7.99 e-008 / 8.9257e-006 / 100
8.60 e-008 / 8.8722e-007 / 125

5Conclusion

In this paper we compared the DE Sinc-Galerkin and SE Sinc-Galerkinand Legendre Galerkinmethods for solving boundary value problems with singular point solutions.the DE Sinc-Galerkin and SE Sinc-Galerkinare appropriate andThe Galerkin method is not suitable. It was observed that DE Sinc-Galerkin with small gives better results than SE Sinc-Galerkin with bigger. These results highlight the accuracy and potency of DE transformation

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[*]Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran, Email Address: , Tel: +989125088385