OPERATOR SPLITTING METHODS FOR COMPUTATION OF EIGENVALUES OF REGULAR STURM-LIOUVILLE PROBLEMS ˙ Ismail G¨ UZEL ismailgzel@gmail.com Dokuz Eyl¨ ul University ˙ IZM˙ IR 13/06/2016 Liouville J. Sturm J.C.F OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Outline 1 Introduction 2 The Sequential Splitting Method for Cauchy Problem 3 The Symmetrical Weighted Sequential Splitting Method 4 Application The Symmetrical Weighted Sequential Splitting Method To Regular SLP 5 Asymptotic Behaviour for Eigenvalues of SLP 6 Numerical Results 7 References Thales BC 624-546 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Introduction We discuss the computation of higher eigenvalues of regular Sturm- Liouville problem (SLP) in canonical Liouville normal form − y ′′ ( t ) + q ( t ) y ( t ) = λy ( t ) (1) with Dirichlet boundary conditions y (0) = y (1) = 0 (2) for q ( t ) ∈ C [0 , 1] and t ∈ [0 , 1] . Pythagoras BC 570-495 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Concerning numerical solution of the Sturm-Liouville problems, finite difference methods are very popular. Generally speaking, finite difference methods (including asymptotic correction techniques, (Anderssen&De Hoog) 1 , (Andrew) 2 , extrap- olation, (Somali&Oger) 3 have the advantage of simplicity and pro- gramming ease. But it is inefficient for computation of higher eigenvalues. Asymptotic correction has proved most successful when the deriva- tives of q ( t ) are small. 1 Anderssen,R.S.,& De Hoog,F.R.(1984). On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems with general boundary conditions. BIT Numerical Mathematics,24(4),401-412. 2 Andrew,A.L.(1988)Correction of finite element eigenvalues for problems with natural or periodic boundary conditions. BIT Numerical Mathematics, 28(2), 254-269. 2 3 Somali,S.,&Oger,V.(2005).Improvement of eigenvalues of Sturm-Liouville problem with t-periodic boundary conditions. Journal of Computational and Applied mathematics, 180(2),433-441 Euclid BC 330-275 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The Sequential Splitting Method for Cauchy Problem The main idea of the splitting method is to lead the complex problem to the sequence of sub-problems with simpler structure. (Geiser) 4 dY ( t ) = ( A + B ) Y ( t ) t ∈ [0 , T ] with Y (0) = Y 0 , (3) dt where A, B ∈ R m × m are constant matrices, Y = ( y 1 , . . . , y m ) T is the solution vector, the initial condition Y 0 ∈ R m is a given constant vector. The solution is given as Y ( t ) = e t ( A + B ) Y 0 . ¨ 4 Geiser,J.(2011) Iterative splitting methods for differential equations. CRC Press. Omer 1048-1131 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The method solves two subproblems sequentially an subintervals [ t i , t i +1 ] , for i = 0 , 1 , . . . , N − 1 , dU ( t ) = A U ( t ) with U ( t i ) = Y sp,i (4) dt dV ( t ) = B V ( t ) with V ( t i ) = U ( t i +1 ) , (5) dt where Y sp, 0 = Y 0 and Y sp,i +1 = V ( t i +1 ) , t 0 = 0 and t N = T . Fibonacci 1170-1250 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The exact solutions of the equation (4) and (5) respectively are U ( t i +1 ) = e ( t i +1 − t i ) A Y sp,i and V ( t i +1 ) = e ( t i +1 − t i ) B U ( t i +1 ) = e ( t i +1 − t i ) B e ( t i +1 − t i ) A Y sp,i The approximate split solution at the point t i +1 is defined as Y sp,i +1 = V ( t i +1 ) . That is Y sp,i +1 = e hB e hA Y sp,i , where h = t i +1 − t i is the stepsize. Galileo 1564-1642 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The local splitting error of the sequential splitting method is obtained as Err local = ( e h ( A + B ) − e hB e hA ) Y sp,i = 1 2 h 2 ( BA − AB ) Y sp,i + O ( h 3 ) and then the global error of the method Err global = O ( h ) when AB � = BA . The splitting error is O ( h ) . So, it is called First-Order Splitting Method Descartes 1596-1650 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The Symmetrical Weighted Sequential Splitting Method We consider the Cauchy Problem (3) and define the splitting of the operator on the time interval [ t i , t i +1 ] as the following dU 1 ( t ) = A U 1 ( t ) with U 1 ( t i ) = Y sp,i dt dV 1 ( t ) = B V 1 ( t ) with V 1 ( t i ) = U 1 ( t i +1 ) dt and dU 2 ( t ) = B U 2 ( t ) with U 2 ( t i ) = Y sp,i dt dV 2 ( t ) = A V 2 ( t ) with V 2 ( t i ) = U 2 ( t i +1 ) , dt where Y sp, 0 = Y 0 . Fermat 1601-1665 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The approximate split solution at the point t i +1 = t i + h is defined as Y sp,i +1 = 1 2 { V 1 ( t i +1 ) + V 2 ( t i +1 ) } (6) = 1 2 { e hB e hA + e hA e hB } Y sp,i . Pascal 1623-1662 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The local spliting error of the symmetrical weighted splitting method is � e hB e hA + e hA e hB �� e h ( A + B ) − 1 � Err local = Y sp,i , 2 = O ( h 3 ) , and Err global = O ( h 2 ) , The splitting error is O ( h 2 ) if AB � = BA . So, it is called Second-Order Splitting Method Newton 1643-1727 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The diagram of splitting methods A t i + 1 t i + 1 B B B t i t i A A Leibniz 1646-1716 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Application The Symmetrical Weighted Sequential Splitting Method To Regular SLP Sturm-Liouville problem (1) and (2) are equivalent with the first order system by y ′ = z Y ′ ( t ) = A ( t ) Y ( t ) , 0 ≤ t ≤ 1 , (7) C 1 Y (0)+ C 2 Y (1) = 0 , (8) Bernoulli 1655-1705 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Application The Symmetrical Weighted Sequential Splitting Method To Regular SLP Sturm-Liouville problem (1) and (2) are equivalent with the first order system by y ′ = z Y ′ ( t ) = A ( t ) Y ( t ) , 0 ≤ t ≤ 1 , (7) C 1 Y (0)+ C 2 Y (1) = 0 , (8) where � y ( t ) � � � 0 1 Y ( t ) = , A ( t ) = , z ( t ) q ( t ) − λ 0 � 1 � � 0 � 0 0 C 1 = and C 2 = . 0 0 1 0 Bernoulli Bernoulli 1655-1705 1655-1705 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The matrix A ( t ) is splitted as a sum of M and q ( t ) N A ( t ) = M + q ( t ) N, where � 0 1 � � 0 0 � M = and N = . − λ 0 1 0 We consider the partition of the interval [0 , 1] h = 1 t i = ih , i = 0 , 1 , . . . , n , n . L’Hˆ opital 1661-1704 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The symmetrical weighted sequential splitting of the system on time interval [ t i , t i +1 ] is defined as in the following algorithm, U ′ 1 ( t ) = M U 1 ( t ) U 1 ( t i ) = Y sp,i V ′ 1 ( t ) = q ( t ) N V 1 ( t ) V 1 ( t i ) = U 1 ( t i +1 ) and U ′ 2 ( t ) = q ( t ) N U 2 ( t ) U 2 ( t i ) = Y sp,i V ′ 2 ( t ) = M V 2 ( t ) V 2 ( t i ) = U 2 ( t i +1 ) , for i = 0 , 1 , . . . , n − 1 and Y sp, 0 is a vector to be determined. Taylor 1685-1731 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The approximate split solution at the point t i +1 is defined as Y sp,i +1 = 1 2 { V 1 ( t i +1 ) + V 2 ( t i +1 ) } , = 1 e s i +1 N e hM + e hM e s i +1 N � � Y sp,i , 2 � t i +1 where s i +1 = q ( ξ ) dξ , i = 0 , 1 , . . . , n − 1 . t i Maclaurin 1698-1746 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Finally, we can write the approximate split solution of (7) at t n = 1 as Y sp,n = KY sp, 0 ≈ Y (1) , where K is 2 × 2 matrix � n − 1 � K = 1 [ e s n − i N e hM + e hM e s n − i N ] � . 2 n i =0 Cramer 1704-1752 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
It is apparent that M 2 j = ( − 1) j λ j I , (9) M 2 j +1 = ( − 1) j λ j M for j = 0 , 1 , . . . (10) Using (9) and (10), we have √ √ λt ) I 2 × 2 + 1 e tM = cos( √ sin( λt ) M λ √ √ � � 1 cos( λt ) λ sin( λt ) √ √ √ √ = . − λ sin( λt ) cos( λt ) Emilie 1706-1749 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
Since N is nilpotent matrix of index 2 ( N k = 0 , k ≥ 2 ), it is clear that e s n − i N = I + s n − i N . (11) We obtained that � n − 1 � K = 1 [2 e hM + s n − i [ b ( λ ) I + 2 a ( λ ) N ]] � . 2 n i =0 where √ √ b ( λ ) = sin( λh ) a ( λ ) = cos( λh ) and √ . λ Euler 1707-1783 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
The solution Y sp,n will be the solution of (7) and (8) C 1 Y sp, 0 + C 2 Y sp,n = 0 ( C 1 + C 2 K ) Y sp, 0 = 0 . For a non-trivial solution Y sp, 0 , the determinant of C 1 + C 2 K must be zero. It follows that Q ( λ ) = det( C 1 + C 2 K ) is the approximate characteristic function of SLP (7). Note that; Q ( λ ) is the (1 , 2) th entry of K . D’Alembert 1717-1783 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
If q ( t ) = 0 , then s i = 0 . Since nh = 1 , we have n − 1 K = 1 2 e hM = e M . � 2 n i =0 From det( C 1 + C 2 K ) = 0 , we get the characteristic equation of the original SLP √ 1 √ sin( λ ) = 0 λ and then the eigenvalues of SLP (1) and (2) are λ k = k 2 π 2 , k = 1 , 2 , . . . . Maria 1718-1799 OPERATOR SPLITTING METHOD FOR COMPUTATION OF EIGENVALUES OF REGULAR SLP
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