using multiquadric method in the numerical solution of ...

© European Journal of Scientific Research, Vol 10, No 2, 2005

USING MULTIQUADRIC METHOD IN THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS WITH A SINGULARITY POINT AND PARTIAL DIFFERENTIAL EQUATIONS IN ONE AND TWO DIMENSIONS A. Aminataei * and Maithili Sharan ** * Department of Mathematics , K . N . Toosi University of Technology , P . O . Box 16315 – 1618, Tehran, Iran . ** Centre for Atmospheric Sciences , Indian Institute of Technology : Delhi , Hauz Khas, New Delhi -110016, India . ABSTRACT In this paper, the aim is to solve the differential equations on the line of Green’s function method with Multiquadric ( MQ ) approximation scheme . The scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering . For this purpose, it is proposed to solve the differential equations with a singularity at a point represented by a Dirac delta function . It is shown that, the MQ method works excellently, and the numerical solution is giving a very closed approximation to analytic one . Key Words: Multiquadric radial basis functions; Singularity point; Domain decomposition method for linear equation . 1. INTRODUCTION a . In the present study, MQ method is used to solve the following ordinary differential equations ( ODEs ) and partial differential equations ( PDEs ) . (i)

uxx + ( 2 S )2 sin ( 2 S x ) = 0

in : = ( 0,1 )

with B.C.’s : u ( 0 ) = u ( 1 ) = 0 and

uexact = u ( x ) = sin ( 2 S x ) .

( ii ) d2 u/dx2 = - Q . G ( x – a ) using Green’s function in which G ( x – a ) = 120 / ( 1 + v2 ) and v = 120 ( x - .5) with B.C.’s : u ( 0 ) = u ( O ) = 0 and ­ x(O  a), uexact = u ( x ) = ( Q / O ) . ® ¯a(O  x),

0 0.

(4)

for j = 1 , 2 , … , N

(5)

Rmax2 and Rmin2 are two input parameters chosen so that the ratio Rmax2/Rmin2 # 10 to 106 . Madych [ 11 ] proved that under circumstances very large values of a shape parameter are desirable . The adhoc formula in equation ( 4 ) is a way to have at least one very large value of a shape parameter without incurring the onset of severe ill-conditioning problems . Spatial partial derivatives of any function are formed by differentiating the spatial basis functions. Consider a 2 D problem . The first and second derivatives are given by simple differentiation: N

u/x |x = xi =

¦

aj . ( xi –xj )/hij

(6)

j 1

N

u/y | y = yi =

¦

aj . ( yi –yj )/hij

(7)

j 1

2u/x2 |x = xi =

N

¦

aj {1/hij [ 1 - ( xi – xj )2 / hij2 ] }

(8)

j 1

2u/y2 |y=yi =

N

¦

aj { 1/hij [ 1- ( yi-yj )2/hij2 ] }

(9)

j 1

N

2u/xy|x=xi & y=yi = - ¦ aj [ 1/hij3 (xi – xj ) (yi – yj ) ] j 1

22

( 10 )

© European Journal of Scientific Research, Vol 10, No 2, 2005

where hij = [ ( xi – xj )2 + ( yi – yj )2 + Rj2 ]1/2

( 11 )

Equations ( 4 ) and ( 6 ) through ( 11 ) give us a framework to approximate ODE and PDEs using the MQ basis function expansions . 2. NUMERICAL SOLUTION OF PROBLEMS AND DISCUSSION OF RESULTS a . The problem ( i ) , is solved for given values of Rmin and Rmax and the difference between numerical and analytical solution is negligibly small . The sum , is the sum of mean square errors between the n data points of the solutions . The results are shown in Tables I and II .

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b . The problem ( ii ) is solved for different number of points ( 9, 15, 19, 29, 39 and 61 points ) with the singularity point a = .5 . ( ii ) in dimensionless form is written as : d2u/dX2 = -Q’ . G ( X – a* ) where X = x/ O , Qc = Q . O and a* = a/ O . Hence, its solution is ­X (1 - a * ) , u ( X ) = Qc . ® ¯a * ( 1 - X ) ,

0Xa* a *  X  1.

Variation in the number of points, by taking more points show that the mean square error (E), mean absolute error ( E1 ) and maximum absolute error ( MAE ) increases .It is shown in Table III. An increase in Q also increases the results, which itself increases the error . With a = .5 , there is a symmetry in the slope of the curve , and numerical result is consistent with the analytic one. Fig. 1 shows a comparison between numerical and analytic solutions with 9, 15, 19 and 29 points. There is a good agreement between the numerical solution and analytic one. Fig. 2 represents the comparison between numerical solutions for different length values of 5, 20, 40 and 100 . The slopes of the curves increases up-to the singularity point a and thereafter decreases.

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c . Problem ( iii ) is solved for different number of data points . When c # 1 or c < 1 , the method provides an excellent approximation . For c > 1 , ( iii ) is written in the form :

with the homogeneous boundaries and matching conditions in dimensionless form as: U( 0 ) = 0 = U( 1 ) ; U |X = 2a+ = U |X = 2a- ; dU/dX|X=2a+ = dU/dX|X=2a- ; U|X=a+ = U|X=a- ; U’2|X=a – U’1|X=a = -1 & U = u/Q . We have obtained the analytic solution as :

U( X ) = 1/( c sinh c ).

­sinh c ( 1 - a ) . sinh cX , ° ®sinh ca . sinh c ( 1 - X ) , °sinh ca . sinh c ( 1 - X ) , ¯

0Xa a d X < 2a 2a d X < 1.

(13)

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This dimensionless solution ( eqs. ( 13 )) is the same as the solution of ( iii ) in dimensionless form as given on page 2. i.e. ,

U( X ) = 1/( c sinh c ).

­sinh c ( 1 - a ) . sinh cX , ® ¯sinh ca . sinh c ( 1 - X ) ,

0Xa a d X < 1.

(14)

Hence, for c > 1, the numerical solution defers from the analytic solution after the point 2a where a is the singularity point . To overcome this, we have considered the domaindecomposition from 0< X < a , a d X < 2a to 2a d X < 1 ( i.e. , eqs. ( 12 )) . We have derived the analytic solution ( eqs. ( 13 )) and it is shown that it is the same as in the case where 0 < X < a and a d X < 1( i.e.,eqs.(14 )), and the same track in their and in the second region, there is a mismatching between the results . Further, for c > 1 , the domain decomposition is from 0 < X < b to b d X < 1 where b is considered as 2a and a is the singularity point . Therefore , ( iii ) is rewritten in the form :

­° P c d 2 U/dX 2 - U = - P c . G ( X - a ) , ® °¯ P c d 2 U/dX 2 - U = 0 ,

0 0 ) and Rj > 0 is the shape parameter of the i-th basis function given by ( 5 ) . Carlson and Foley [ 12 ] have used a constant value for different functions . They observed that the root mean squared errors of many of the bivariate functions ( 3 ) were reduced several orders of magnitude when the optimal shape parameter was used . Kansa [ 9 , 10 ] has found numerically that the following power law function increases accuracy up to 5 orders of magnitude for many monotonic functions . Later , Hagan and Kansa [ 13 ] studied the role of the parameter Rj2 in the multiquadric function using a simple two- parameter optimization procedure developed by Marquardt [ 14 ] . Golberg & Chen [ 32

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15, 16 , 17 ] used the method of cross-validation to estimate the optimal shape parameter of elliptic PDE problems in 2 and 3 D ’s and observed exponential convergence . Spatial partial derivatives of the function in ( 18 ) are obtained by differentiating the spatial basis functions . Substitution of ( 18 ) and its partial derivatives in ( 16 ) together with the boundary conditions ( 17 ) , yields a system of linear algebraic equations which can be solved for the coefficients { aj } . We illustrate the method for a 2 D problem . The derivatives at ( xi , yi ) are given by : N

¦

( u/x ) | ( xi , yi ) =

aj [ (x – xj )/hij – ( x – x1 )/hi1 ] ,

J 2

N

( u/y ) |( xi , yi ) =

¦

aj [ ( y – yj )/hij – ( y – y1 )/hi1 ] ,

j 2

( 2 u/x2 ) |( xi , yi ) =

N

¦

aj [1/hij { 1 - ( x - xj )2/hij2 } - 1/hi1{ 1 - ( x - x1 )2/hi12 } ]

j 2

( 2 u/y2 ) |( xi ,yi ) =

N

¦

aj [ 1/hij { 1-( y – yj )2/hij2} – 1/hi1 { 1- ( y – y1 )2/hi12 } ]

( 21 )

j 2

where hij is given by ( 11 ). Substituting the expansion of u , ( 18 ) and its derivatives ( 21 ) , in the 2 D version of ( 16 ) , we find for each of the interior nodes , i . If a point i lies on the boundary * , it may satisfy a Dirichlet boundary condition . With the Dirichlet condition u = u1 , the matrix elements for the i-th equation , is found. For all the knots , a system of linear algebraic equations of the form PA=B

( 22 ) T

is obtained , where P = { Pij } is a square matrix of order N , and A = ( a1,a2,…,aN ) and B = ( b1,b2,…,bN )T are column vectors of length N , where A is the vector of unknown expansion coefficient, and B is the column vector of known source terms . Non-linear PDEs are solved by replacing ( 18 ) everywhere with a non-linear expansion of u in the governing set of PDEs . Dubal [ 18 ] solved for the expansion coefficients of the MQ expansion ( 18 ) , by Picard iteration , and Galperin et al [ 19 ] solved for the expansion coefficients by a combination of picard & Newton-Raphson schemes . Here, we have solved the linear PDE ( 22 ) , successfully using the Gauss elimination method with total pivoting [ 20 ] . The mean square error ( E ) is computed from N

E = 1/N .

¦ i 1

( u – u i )2

( 23 )

where u is the computed value of u using the MQ method and u is the known value of u at the point i . It may be known either analytically or computationally by means of some other methods such as finite differences , finite elements , etc . Ex : Poisson equation in a rectangle with right hand as a function of space coordinates . Consider a Poisson equation in a rectangular plate , ’ 2u = 2xy ( xy + x + y – 3 ) . exp ( x + y )

in

: = ( 0 , 1 )u( 0 , 1 )

( 24 ) 33

© European Journal of Scientific Research, Vol 10, No 2, 2005

with B.C.’s : u| w: = 0 . This problem is solved for given values of Rmin & Rmax and the MQ method converges well to the exact solution . It is solved for different number of points both in x as well as in y – directions ( 9×9 pts, 7×7 pts and 5×5 pts ) . and e . In ( v ) , as mentioned throughout this study , the aim is to solve the differential equation on the line of Green’s function method with MQ . For this purpose , we have solved the problem v and obtained its analytic solution . After that , it is proposed to solve the differential equation d2w/dx2 – c2 w = - G ( x – a ) ,

0< x , a < 1

( 25 )

with the homogeneous B.C.’s : w ( 0 ) = w ( 1 ) = 0 , with a singularity at appoint represented by a Dirac delta function . The solution of the above equation is w ( x , a ) = 1/(c sinh c) .

­sinh c ( 1 - a ) . sinh cx , ® ¯sinh ca . sinh c ( 1 - x ) ,

0xa a d x < 1.

(26)

To apply the Green’s function method : u ( x ) = -³

x

0

1

w( x , a ) . ea da - ³ w( x , a ) . ea da x

( 27 )

which is u ( x ) = 1/( 1 - c2 ) { ex - 1/sinh c [ sinh c ( 1-x ) – e sinh ( cx ) ] } .

( 28 )

Equation ( 28 ) is equal to the uexact of problem v given on page 2 . Hence , the problem v is solved with MQ method using the analytic solution ( 28 ) . The numerical MQ method and the analytic solution converges excellently and the results are given in Tables XI and XII for different values of c . It is shown that by decreasing c , the error increases , using variable step sizes in the x – direction with 20 points . The values of Rmin and Rmax are obtained using Marquardt [ 14 ] optimization procedure .

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Acknowledgment: The work presented in this paper is partially supported by a grant under the Computational Modeling Project of the Indian Institute of Technology : Delhi , India ( Project No. IITD/IRD/RPO 1023/3635 ) .

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REFERENCES [ 1 ] R.L. Hardy , Multiquadric equations of topography and other irregular surfaces , J. Geophys. Res., 76 , 1905 ( 1971 ) . [ 2 ] R.L. Hardy , Theory and applications of the multiquadric bi-harmonic method : 20 years of discovery , Computers Math. Applic. 19 ( 8/9 ), 163 ( 1990 ) . [ 3 ] R.Franke ,Scattered methods,Math.Comput.,38,181(1972).

data

interpolation:Tests

of

some

[ 4 ] M. Zerroukat , H. Power and C.S. Chen., A numerical method for heat transfer problems using collocation and radial basis functions., Int. J. for Numeri. Meth. in Engg. 42, 1263 ( 1998 ) . [ 5 ] N. Mai – Duy and T. Tran – Cong, Numerical solution of differential equations using multiquadric radial basis function networks, Neural Networks., 14 , 185 ( 2001 ) . [ 6 ] A. Aminataei and M.M. Mazarei , Numerical solution of elliptic partial differential equations using direct and indirect radial basis function networks,Euro–Asian J.of Applied Sciences.Press [ 7 ] C.A. Micchelli, Interpolation of scattered data : Distance matrices and conditionally positive definite functions, Constr. Approx., 2, 11 ( 1986 ) . [ 8 ] W.R. Madych and S.A. Nelson, Multivariable interpolation and conditionally positive definite functions II , Math. Comp., 54,211 ( 1990 ) . [ 9 ] E.J. Kansa, Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics – I. Surface approximations and partial derivative estimates, Computers Math. Applic. 19 ( 8/9 ), 127 ( 1990 ) . [ 10 ] E.J. Kansa, Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics – II. Solutions to hyperbolic, parabolic and elliptic partial differential equations., Computers Math. Applic., 19 ( 8/9 ) ,147 ( 1990 ) . [ 11 ] W.R. Madych, Miscellaneous error bounds for multiquadric and related interpolants., Computers Math. Applic. 24 ( 12 ), 121 ( 1992 ) . [ 12 ] R.E. Carlson and T.A. Foley , The parameter R2 in multiquadric interpolation, Computers Math. Applic., 21 ( 9 ), 29 ( 1991 ) . [ 13 ] R.E. Hagan and E.J. Kansa, Studies of the R parameter in the multiquadric function applied to groundwater pumping , J. Appl. Sci. & Comp., 1 ( 2 ) , 266 ( 1994 ) . [ 14 ] D.M. Marquardt, An algorithm for least – square estimation of non-linear parameters , J. Soc. Indust. Appl. Math., 11 , 431 ( 1963 ) . [ 15 ] M.A. Golberg and C.S. Chen, The theory of radial basis functions used in the dual reciprocity boundary element method, Appl. Math. Comp., 60, 125 ( 1994 ) . [ 16 ] M.A. Golberg and C.S. Chen, Improved multiquadric approximation for partial differential equations., Engg. Anal. with Bound. Elem., 18 , 9 ( 1996 ) . [ 17 ] M.A. Golberg and C.S. Chen., Discrete projection methods for integral equations, Comput. Mech. Boston, M.A., ( 1997 ) .

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[ 18 ] M.R. Dubal., Domain decomposition and local refinement for multiquadric approximations, I. second-order equations in 1D., Int. J. Appl. Sci. Comput., 1 ( 1 ), 146 ( 1994 ) . [ 19 ] E.A. Galperin, E.J. Kansa, A. Makroglou and S.A. Nelson, Numerical solutions of weakly singular volterra equations using global optimization and radial basis functions, Proc. Hermas 96, Ed. E.A. Lipitakis, Hellenic European Research on Mathematics and Informations 96, LEA, Athens., 3, 550 ( 1997 ) . [ 20 ] M. Sharan, E.J. Kansa and S. Gupta., Applications of the multiquadric method for the solution of elliptic partial differential equations, Appl. Math. Comput., 84, 275 ( 1997 ) .

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