2009 Method of Particular Solutions 1 A Study on Particular Solutions of Coupled PDE System...

1

2009 Method of Particular Solutions

A Study on Particular Solutions of Coupled PDE System

Chia-Cheng Tasi 蔡加正

Department of Information Technology Toko University, Chia-Yi County, Taiwan

2


Motivation and Introduction

Method of Particular Solutions (MPS)

Particular solutions of Chebyshev polynomials

Numerical example I

Particular solutions of spline

Numerical example II

Conclusions

Overview

3


陳建宏柯永澤陳柏台辛敬業劉德源張建仁

周宗仁林炤圭蕭松山岳景雲翁文凱臧效義

郭世榮葉為忠曹登皓劉進賢陳正宗范佳銘

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河工機械

林俊華

尹彰

電機

河工

You are

Welcome !

NTOU is a Kingdom of BEM

4


Motivation and Introduction1. Boundary-type numerical method: BEM, Treffz meth

od, MFS

2. Advantage: Reduction of dimensionalities

3. Disadvantage: Domain integration => the method of particular solutions (MPS) or the dual reciprocity method (DRM)

4. Active research fields of BEM: Singularity and Domain integration

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7


Innovation

Application

DBEM

MPS MRM

Hypersingularity

121, , , , , ( ) ( )

2 2

2 2

( ) 0

0 ( )

0

x

y

x y


How to prove?

BIEM

8


RBF

MFSMPS with

Chebyshev Polynomial

s

exponential convergence

Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)


Fig. 4: Geometry configuration of the MFS-MPS.

ls lx

c

1 2( , )b bx x

1 2( , )a ax x

Golberg (1995)

Chebyshev

9



10


( ) ( );pu f L x x

1

( ) ( )n

i ii

f

x x

( ) ( )i i L x

1

( ).n

p i ii

u

x

Method of particular solutions

11



( ) ( );u f L x x

( ) ( );u g B x x

p hu u u

( ) ( );pu f L x x

( ) 0;hu L x

( ) ( ) ( );h pu g u B Bx x


Method of fundamental solutions, Trefftz method, boundary element metho

d, et al.

12


1

( ) ( )n

i ii

f

x x

2 2

2 1 2

1

: ( )

( ( )) : ( ) (3 ), ln (2 )

: ( ) (1 ) ( )

nn n

i ii n

MQ r r c

RBF r APS r r D r r D

CSRBF r r g r

,

, , ,,

,

: ( ) sin sin

( ( )) : ( )

: ( ) ( ) ( )

m n

m nm n m n m n

m n

m n m n

Tri m x n y

Global Polynomials x y

Chebyshev T x T y

x

x x

x

1 , 1(1 )

0, 1

r rr

r

( (?) )iL x

Method of particular solutions (basis functions)

13


Coupled PDE system

Product operator

Polyharmonic operator Poly-Helmholtz operator

Helmholtz operatorLaplacian operator

Hörmander operator decomposition technique

Partial fraction decomposition

Generating theorem?


14


1 21 2( ) ( ) ( )

L

1 2, , ,

1 2, ,...,

( ) ( )i i L x

Particular solutions for

the engineering problems

Method of particular solutions (Hörmander Operator Decomposition technique)

15


Example

2 22 2

2

2 22 2

2

2 2 2 2 2 2 2 2

( )

( )

( ) ( ) ( )

adj

y x y x

x y x y

x y

L

2 2 2( )adj LL I

2 2 111

2 2 121

11 21

( )

( ) 0

0

pu

xp

uy

u u

x y

2 2 212

2 2 222

12 22

( ) 0

( )

0

pu

xp

uy

u u

x y

2 2 313

2 2 323

13 23

( ) 0

( ) 0

pu

xp

uy

u u

x y

11 12 13

21 22 23

1 2 3

0 0

0 0

0 0

u u u

u u u

p p p

L

2 2

2 2

( ) 0

0 ( )

0

x

Ly

x y

16


Example

11 12 13

21 22 23

1 2 3

0 01

0 0

0 0

adj

u u u

u u u

p p p

L

0 0 0 01

( ) 0 0 0 0

0 0 0 0

adj

L L11 12 13

21 22 23

1 2 3

0 0

0 0

0 0

u u u

u u u

p p p

L

2 2 2( )

17


Other examples

2

11 12 132

21 22 23

1 2 3

00 0

0 0 0

0 0

0

x u u u

u u uy

p p p

x y

2

1 * * *11 12 13

2 * * *21 22 23

2 * * *2 1 2 3

* * *1 2 3

1 2

0 0

0 0 000 0 0 00

0 0 000 0 0

0 0 0 000 0

T

xu u u

u u ux

T T Tk

p p p

x x

Stokes flow

Thermal Stokes flow

18


Other examples

* * *11 12 13

* * *21 22 23

* * *31 32 33

0 0

0 0

0 0

u u u

L u u u

u u u

22 2(1 ) (1 )

( )2 2ij ij

i j

D D vL

x x

2

3 3

(1 )

2i ii

DL L

x

22

33

(1 )

2

DL

Thick plate

2 22

2 * *1 1 2 11 12

* *2 22 21 22

21 2 2

( ) ( )0

0( ) ( )

x x x u u

u u

x x x

Solid deformation

19


Remark

Particular solutions for engineering

problems

Particular solutions for product operator

1 21 2( ) ( ) ( )

L

Hörmander operator decomposition techn

ique

20


Particular solutions for

( )

L

L

L


Particular solutions for product operator

1 21 2( ) ( ) ( )

L

Method of particular solutions (Partial fraction decomposition)

21

2009 Method of Particular SolutionsPartial fraction

decomposition (Theorem)

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(Proof 1)

,( ) ( , , ) ( , , )m

m

lm m lA x y z a x y z

1 21 2( ) ( ) ( ) ( , , ) ( , , )A x y z a x y z

1 21 2 ,( ) ( ) ( ) ( , , ) ( ) ( , , )m

m

lm m lA x y z A x y z

,1

( , , ) ( ) ( , , )i

mm l ii

A x y z A x y z

,m

i i

l

i m

23



(Proof 2)

1

,1 0 1

1 ( )m

im l i

m l i

C

1

,1 0 1

( , , ) ( ) ( , , )m

im l i

m l i

A x y z C A x y z

,

1

( ) ( , , ) ( , , )i

mi m li

A x y z A x y z

1

, ,0 0

( , , ) ( , , )m

mm l m lm l

A x y z C A x y z

1,

1 01

1

( ) ( )

m

i m

m ll

m li i m

C

24


Example

(1)

2 2( 4)( 9) ( , ) m nx y x y

(2) (1) (1) (2) (1)2L 2L 2M 2H 2H35

324 11664 2704 1053 123201

2 (2)2L

m nx y (1)2L

m nx y (1)2M( 4) m nx y

2 (2)2H( 9) m nx y (1)

2H( 9) m nx y

2 2 2 2

1 1 1 1 1 35

( 4)( 9) 324 11664 2704( 4) 1053( 9) 123201( 9)

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Example (2)

2 2 4 ( , )0 0 0( 2 cos 2 ) ( )m n m nr r r x y

2 2 ( , )1 2( )( ) ( )m n m nr x y

0

0

1 0

2 0

r e

r e

i

i

( , ) ( , )( , ) 1 2

2 2 2 21 2 2 1

( ) ( )( )

m n m nm n r r

r

2 ( , )1 1( ) ( )m n m nr x y

2 ( , )2 2( ) ( )m n m nr x y

2 2 2 2 2 2 2 21 2 1 2 1 2 1 2

1 1 1

( )( )

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Remark

( , , , )( ) ( , , ) ( ) ( ) ( )L L l m n l m nl m nP x y z x y z orT x T y T z

( , , , )0 ( , , ) ( ) ( ) ( )L L l m n l m n

l m nP x y z x y z orT x T y T z

1 21 2( ) ( ) ( ) ( ) ( ) ( )l m n

l m nP x y z orT x T y T z


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Particular solutions of Chebyshev polynomials (why orthogonal polynomials)Fourier series: exponential convergence but Gibb’s phenomena

Lagrange Polynomials: Runge phenomena

Chebyshev Polynomials (one of the orthogonal polynomials): exponential convergence

28


Chebyshev interpolation (1)

no matrix inverse

1 2

1 2 1 2

1 2

1 1 1 2 2 21 2 1 2

0 0 1 1 2 2

2 2( , ) ( , )

M Mb a b a

m m m mm m b a b a

x x x x x xf x x f x x a T T

x x x x

1 21 2

1 2

1 21 1 2 2 1 1 2 2

1 2 1 1 2 2

0 01 2 , , , , 1 2

( , )4cos cos

M Ml l

m ml lM m M m M l M l

f x x m l m la

M M c c c c M M

1 2( , )a ax x

1 2( , )b bx x

( )cos

2 2b a b a

l

x x l x xx

M

29


Chebyshev

interpolation (2)

211 2

1 2

1 2

1 1 1 2 2 21 2

0 0 1 1 2 2

2 2( , )

mmM Mb a b a

m mm m b a b a

x x x x x xf x x a

x x x x

1 2

1 2 1 2 1 1 2 2

1 1 2 2

, ,

M M

m m m m m m m mm m m m

a a d d

,0

2 2mm

b a b am m m

mb a b a

x x x x x xT d

x x x x

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Chebyshev

interpolation (3)

no book keeping

by multiple loops

1 2

1 2

1 1

1 2

ˆ ˆˆ ˆ1 2 1 2

ˆ ˆ0 0

ˆ( , )M M

m mm m

m m

f x x a x x

1 1 2 21 21 2

1 2 1 21 2

1 1 2 2

ˆ ˆˆ ˆ1 1 1 2 2 2

ˆ ˆˆ ˆ 1 1 1 1 1 2 2 2 2 2

2 ! 2 !ˆ

ˆ ˆ ˆ ˆ! ! ! !

m m m mm mM Mb a b a

m m m mm mm m m m b a b a

x x m x x ma a

x x m m m x x m m m

ˆˆˆ

ˆ 0

2 !2

ˆ ˆ! !

m m mmmmb ab a

mmb a b a

x x mx x xx

x x x x m m m

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2009 Method of Particular SolutionsChebyshev

interpolation (4) Start

End

Input q , 1ax ,

1bx ,2ax ,

2bx

For all 0 m M , calculate

1 21 2

1 2

1 21 1 2 2 1 1 2 2

1 2 1 1 2 2

0 01 2 , , , , 1 2

( , )4cos cos

M Ml l

m ml lM m M m M l M l

q x x m l m la

M M c c c c M M

For all 0 m M , calculate

1 2

1 2 1 2 1 1 2 2

1 1 2 2

, ,

M M

m m m m m m m mm m m m

a a d d

For all ˆ0 m M , calculate

1 1 2 21 21 2

1 2 1 21 2

1 1 2 2

ˆ ˆˆ ˆ1 1 1 2 2 2

ˆ ˆˆ ˆ 1 1 1 1 1 2 2 2 2 2

2 ! 2 !ˆ

ˆ ˆ ˆ ˆ! ! ! !

m m m mm mM Mb a b a

m m m mm mm m m m b a b a

x x m x x ma a

x x m m m x x m m m

1 2

1

2

1 2

1 2 1 2

1 1 2 2

1 1

2 2

1 1

2 2

1 1 1 111

1

2 2 2 222

2

1 2

1 2 , ,

for 0 to

{

for 0 to

{

0

for 0 to

{

for 0 to

{

( ) cos

2 2

( ) cos

2 2

( , )4

m m

b a b al

b a b al

l lm m m m

M m M m M

m M

m M

a

l M

l M

x x x xlx

M

x x x xlx

M

q x xa a

M M c c c

1 2

1 2 1 1 2 2

1 1 2 2

0 0 , , 1 2

cos cos

}

}

}

}

M M

l l l M l

m l m l

c M M

32


( ) (? ) ( ) ( )i j kT x T y T zL


( )? i j kx y zL

33



34

2009 Method of Particular SolutionsParticular solutions of of Chebyshev polyn

omials (Generating Theorem)

35



36



37


Particular solutions of Chebyshev polynomials (poly-Helmholtz)

proof

Generating Theorem

Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

38


Particular solutions of Chebyshev polynomials (polyharmonic)

39


Particular solutions of Chebyshev polynomials (polyharmonic)

40


Method of fundamental solutions

1

( ) 0iAi h

i

u

1( )

1 0 1

( ) ( )i

i

i

A KA j

h ijk ki j k

u G

x x s

( ) ( ) ( )L LG x x Fig. 2: Geometry configuration of the MFS.

41


2 2( 4)( 9) ( ) 0hu x

2 (2)2LG

(1)2LG (1)

2M( 4)G

2 (2)2H( 9) G (1)

2H( 9)G

(2) (1) (1) (2) (1)1 2L 2 2L 3 2M 4 2H 5 2H

1

( ) ( ( ) ( ) ( ) ( ) ( ))K

h k k k k k k k k k kk

u G r G r G r G r G r

x

Method of fundamental

solutions (example)

42


Example (2D modified Helmholtz)

( 900) 899(e e )x yu

e ex yu

RMSEs 2.16E-02 1.38E-06 4.18E-12 3.07E-13 2.41E-12

Table I: The RMSEs for Example 3

4l m 8l m 12l m 16l m 20l m

Numerical example I

43


Example (2D Laplace)

e ex yu e ex yu

RMSEs 4.77E-05 2.94E-10 1.92E-10 1.92E-10 1.76E-10

Table II: The RMSEs for Example 4

4l m 8l m 12l m 16l m 20l m

Numerical example I

44


Example (3D modified Helmholtz)

( 900) 899(e e e )x y zu

e e ex y zu

RMSEs 1.48E-01 5.45E-06 8.33E-12 4.15E-12 1.77E-11

Table III: The RMSEs for Example 5

4l m n 8l m n 12l m n 16l m n 20l m n

Numerical example I

45


Example (3D Laplace)

e e ex y zu

e e ex y zu

RMSEs 4.18E-05 2.65E-10 4.17E-11 4.15E-11 2.89E-10

Table IV: The RMSEs for Example 6

4l m n 8l m n 12l m n 16l m n 20l m n

Numerical example I

46


Example (2D polyharmonic)

e ex yu 4 e ex yu

2 3

2 31

T

n n n

B

RMSEs 4.75E-10 2.98E-12 2.98E-12 2.98E-12 2.98E-12

Table V: The RMSEs for Example 7

4l m 8l m 12l m 16l m 20l m

6 4 21 2 3 4

1

( ) ( ln ln ln ln )K

h k k k k k k k k k k kk

u r r r r r r r

x

Numerical example I

47


Example (2D product operator)

e ex yu 2 3

2 31

T

n n n

B

2 ( 900)( 100) 89001(e e )x yu

21 0 2 0 3 4

1

( ) ( (30 ) (10 ) ln ln )K

h k k k k k k k k kk

u K r K r r r r

x

RMSEs 7.29E-06 2.61E-10 3.29E-10 3.29E-10 3.23E-10

Table VI: The RMSEs for Example 8

4l m 8l m 12l m 16l m 20l m

Numerical example I

48


Example (Reissner plate)

Numerical example I

Fig. 1: Geometric configuration of the Ressiner plate model.

( ) ( )ij j iL u E q 2

2 2

2

3 3

22

33

(1 ) (1 )( )

2 2

(1 )

2

(1 )

2

D D vL

x x

DL L

x

DL

2

3

(1 )

1

Ex

E

49


3 2 22 2 2 2(1 )

( ) ( ) ( ) det( ) ( )4

adj adj ikij jk ij jk ik

DL L L L

L

24 2 2

2 23

2 23

2 2 2 2 233

2 [(1 ) (1 ) ]

(1 ) ( )

(1 ) ( )

( )[2 (1 ) ] /

adj

adj

adj

adj

Lx x

Lx

Lx

L

(A,B)1 2( ) ( ) A B

ij j iL P E x x

Example (Reissner plates: particular solutions)

50


(A,B) ( ) ( )adjj jk kP L E F

(A,B)1 2( ) ( ) A B

ij j iL P E x x

3 2 22 2 2 2

1 2

(1 )( )

4A BD

F x x

?F 3 2 2

2 2 2 2(1 )( ) ( )

4

DG

x s


51


3 2 22 2 2 2

1 2

(1 )( )

4A BD

F x x

( , )( , ) ( , )31 2

3 4 2 3 6 2 3 6 2

44 4

(1 ) (1 ) (1 )

A BA B A B FF FF

D D D

3 2 2 3 4 2 2 3 6 2 3 6 2 22 2

1 4 4 4

(1 ) (1 ) (1 ) (1 ) ( )( )

4D D D D

2 2 ( , )1 1 2

2 ( , )2 1 2

2 2 ( , )3 1 2( )

A B A B

A B A B

A B A B

F x x

F x x

F x x

1

2

3

?

?

?

F

F

F


52



Start

End

Input q , 1x , 2x ,

1ax , 1bx ,

2ax ,2bx

For all ˆ0 m M , calculate 1 2ˆ ˆˆm ma


21 2

11 2

1

21 2

11 2

1

1 2

1

ˆ[ ] ˆ ˆ2 4 2

21 2 1 2ˆ ˆ( , )

10 1 2

ˆ[ ] ˆ ˆ2 2 2

21 2 1 2ˆ ˆ( , )

20 1 2

1 2 1ˆ ˆ( , )3

ˆ ˆ( 1) ( 1)! ! !

ˆ ˆ!( 2 4)!( 2 )!

ˆ ˆ( 1) ! ! !

ˆ ˆ!( 2 2)!( 2 )!

ˆ ˆ( )! !

mm l m ll

um mu

l

mm l m ll

um mu

l

m mu

l m m x xF

l m l m l

l m m x xF

l m l m l

l l m mF

ΛΛ

ΛΛ

Λ 1 2

1 1 2 2

1

1 2

1 2

ˆ ˆ[ ] [ ] ˆ ˆ2 2

2 22 1 2

2 2 20 0 1 2 1 1 2 2

!

ˆ ˆ! !( 2 )!( 2 )!

m mm l m l

u

l ll l

x x

l l m l m l

Λ


1 2 1 2 1 2

1 1 11 2

1

ˆ ˆ ˆ ˆ ˆ ˆ( , ) ( , ) ( , )1 2 3ˆ ˆ( , )

3 4 2 3 6 2 3 6 2

4 4 4

(1 ) (1 ) (1 )

m m m m m mu u um m

u

F F FF

D D D

Λ Λ ΛΛ

Calculate

1 2

1 2

1 1 1

1 2

ˆ ˆ( , )ˆ ˆ1 1 2

ˆ ˆ0 0

ˆ( , )M M

m mm m u

m m

u x x a F

Λ

53


*( ) ( )ij jk ikL u x s

* ( )adjjk jku L G

3 2 22 2 2 2(1 )

( ) ( )4

DG

x s

2 2 2 20

3 2 6

4 ( ) (4 ) log( , )

2 (1 )

r K r r rG

D

x s

Example (Reissner plates: fundamental solutions)

54


Example (Reissner plates: numerical results)

21 2 2 2( ) ( )sin sinh sinhij j iL u DE x x x x

22 2 2 2 2 2

1 22 2 32 2 2 2 2 2 2

1 2

6 8 1 4 2 cosh 2 1 sinhcos

1 48 6 1 cosh 9 9 6 2 sinh

48 1

x x x x x xx

x x x x x x xu

22 2 2 2 2

12 2 3

2 2 2 2 2

2 2

6 8 2 1 cosh 1 4 2 sinhsin

1 48 3 3 6 2 cosh 3sinh

48 1

x x x x xx

x x x x xu

22 2 2 2 2 2

1 22 2 32 2 2 2 2 2 2

3 2

6 2 8 1 4 2 cosh 2 1 sinhsin

1 48 6 1 cosh 9 9 6 2 sinh

48 1

x x x x x xx

x x x x x x xu

55


Example (Reissner plates: numerical results)

Clamped Free

RMSEs 3.12E-08 1.02E-13 9.77E-14 3.07E-13 3.40E-10

Table II: The RMSEs for Example 24M 8M 20M 12M 16M

RMSEs 8.15E-12 3.44E-15 3.44E-15 3.43E-15 1.85E-15

Table I: The RMSEs for Example 14M 8M 20M 12M 16M Clamped

RMSEs 5.58E-12 9.32E-14 9.34E-14 3.49E-11 5.54E-07

Table III: The RMSEs for Example 34M 8M 16M 20M 12M Peanut domain

56


1. MFS+Chebyshev => spectral convergence

2. Hörmander operator decomposition technique

3. Partial fraction decomposition

4. polyHelmholtz & Polyharmonic particular solutions

5. MFS for the product operator

Conclusion

57


Thank you

58


Particular solutions of spline (APS)

( ) ( );u q L x x

59

2009 Method of Particular SolutionsParticular solutions of spline (APS)

( )

( ) ( )

j jP p

F f r

L

L

60

2009 Method of Particular SolutionsParticular solutions of spline (Definition)

61

2009 Method of Particular SolutionsParticular solutions of spline (2D Poly-Hel

mholtz Operator)

62


mholtz Operator)

Generating Theorem

proof

63


mholtz Operator)

64


mholtz Operator)

65


mholtz Operator)

Generating Theorem

proof

66

2009 Method of Particular SolutionsParticular solutions of spline (Limit Behav

ior)

67

2009 Method of Particular SolutionsParticular solutions of spline (Limit Behav

ior)

68

2009 Method of Particular SolutionsParticular solutions of spline (2D&3D Pol

yharmonic Operator)

Cheng (2000)

69


2 2 2 ( ) in ( ) ( ) ( )pD G k qu u u xx x x

Numerical example II

1 1

2 2

( ) ( ) on

( ) ( ) on

B u u

B u u

x x

x x

70

2009 Method of Particular SolutionsNumerical example II

1 2 1 2 2 2 1 1

1 1 1

( )( ) ( )( ) ( )( )

71

2009 Method of Particular SolutionsNumerical example II (BC)

( ) 1u K

1 1

2 2

( ) ( ) on

( ) ( ) on

Bu u

B u u

x x

x x

( )( )

x

Kn

22

2

( )( ) ( ) (1 )m

xx

Kn

2 2( ) ( )( ) (1 )v

x

x x x x

Kn t n t

( ) 1u K1 2

1 2( ) ( )

( )x xn n

K

2 2 2

1 2 32 21 21 2

( ) ( ) ( )( )m x xx x

K

3 3 3 3

1 2 3 43 2 2 31 1 2 1 2 2

( ) ( ) ( ) ( )( )v x x x x x x

K

2 21 21 Dn Dn

1 22 2(1 )Dn n 2 2

2 13 Dn Dn 2 2

1 2 1 21 (1 )Dn n Dn n

2 3 22 1 2 1 22 (1 ) 2(1 )Dn n Dn n nD

2 3 21 2 1 2 13 (1 ) 2(1 )Dn n Dn n nD

2 24 2 1 2 1(1 )Dn n Dn n

72

2009 Method of Particular SolutionsNumerical example II (BC)

73

2009 Method of Particular SolutionsNumerical example II (MFS)

1 1 1

2 2 2

( ) ( ) ( )

( ) ( ) ( )

on

on

h p

h p

B u u B u

B u u B u

x x x

x x x

1 2 1 1 2 21 1

( ; , , ) ( , ) ( , )L L

j j j jh j j j

j j

u G G

x s x s x s

1 0 1

2 0 2

( , ) ( )

( , ) ( )

j j

j j

G K r

G K r

x s

x s

21

22

21

22

2 ( )

2 ( )

( )( )

G

G

x s

x s

1 2 1 1 2 21 1

( ; , , ) ( , ) ( , )L L

j j j jh j j j

j j

u G G

x s x s x s

2 0 1 0 2

1 0 1 0 2

( , ) ( ) ( )

( , ) [ ( ) ( )]

G K r K r

G K r K r

x s

x s i

2 2 2 21 2( )( ) ( ) ( )u qD x x

74

2009 Method of Particular SolutionsNumerical example II (results)

75


1. MFS+Chebyshev => spectral convergence

2. Hörmander operator decomposition technique

3. Partial fraction decomposition

4. polyHelmholtz & Polyharmonic particular solutions

5. MFS for the product operator

6. MFS+APS => scattered data in right-hand sides

Conclusion

76


Thank you

2009 Method of Particular Solutions 1 A Study on Particular Solutions of Coupled PDE System...

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