NAG Fortran Library Routine Document D05BYF.1

Transcription

1 NAG Fortran Library Routine Document Note: before using tis routine, lease read te Users Note for your imlementation to ceck te interretation of bold italicised terms and oter imlementation-deendent details. 1 Purose comutes te fractional quadrature weigts associated wit te Backward Differentiation Formulae (BDF) of orders 4, 5 and 6. Tese weigts can ten be used in te solution of weakly singular equations of Abel tye. 2 Secification SUBROUTINE (IORDER, IQ, LENFW, WT, SW, LDSW, WORK, LWK, IFAIL) INTEGER IORDER, IQ, LENFW, LDSW, LWK, IFAIL real WT(LENFW), SW(LDSW,2IORDER-1), WORK(LWK) 3 Descrition comutes te weigts W n;j and! i for a family of quadrature rules related to a BDF metod for aroximating te integral: Z 1 t ðsþ ffi ds 2 2 X W n;j ðjþþ X n! n j ðjþ; 0 t T; ð1þ t s 0 j¼2 1 wit t ¼ n ðn 0Þ, for some given. In (1), is te order of te BDF metod used and W n;j,! i are te fractional starting and te fractional convolution weigts resectively. Te algoritm for te generation of! i is based on Newton s iteration. Fast Fourier transform (FFT) tecniques are used for comuting tese weigts and subsequently W n;j (see Baker and Deraksan (1987) and Henrici (1979) for ractical details and Lubic (1986) for teoretical details). Some secial functions can be reresented as te fractional integrals of simler functions and fractional quadratures can be emloyed for teir comutation (see Lubic (1986)). A descrition of ow tese weigts can be used in te solution of weakly singular equations of Abel tye is given in Section 8. 4 References Baker C T H and Deraksan M S (1987) Comutational aroximations to some ower series Aroximation Teory (ed L Collatz, G Meinardus and G Nürnberger) Henrici P (1979) Fast Fourier metods in comutational comlex analysis SIAM Rev Lubic C (1986) Discretized fractional calculus SIAM J. Mat. Anal Parameters 1: IORDER INTEGER Inut On entry: te order of te BDF metod to be used,. Constraint: 4 IORDER 6. 2: IQ INTEGER Inut On entry: determines te number of weigts to be comuted. fractional convolution weigts are comuted. Constraint: IQ 0. By setting IQ to a value, 2 IQþ1.1

2 NAG Fortran Library Manual 3: LENFW INTEGER Inut On entry: te lengt of te array WT. Constraint: LENFW 2 IQþ2. 4: WT(LENFW) real array Outut On exit: te first 2 IQþ1 elements of WT contains te fractional convolution weigts! i, for i ¼ 0; 1;...; 2 IQþ1 1. Te remainder of te array is used as worksace. 5: SW(LDSW,2IORDER-1) real array Outut On exit: SWðn; j þ 1Þ contains te fractional starting weigts W n 1;j, for n ¼ 1; 2;..., ð2 IQþ1 þ 2 IORDER 1Þ; j ¼ 0; 1;...; 2 IORDER 2. 6: LDSW INTEGER Inut On entry: te first dimension of te array SW as declared in te (sub)rogram from wic is called. Constraint: LDSW 2 IQþ1 þ 2 IORDER 1. 7: WORK(LWK) real array Worksace 8: LWK INTEGER Inut On entry: te dimension of te array WORK as declared in te (sub)rogram from wic is called. Constraint: LWK 2 IQþ3. 9: IFAIL INTEGER Inut/Outut On entry: IFAIL must be set to 0, 1 or 1. Users wo are unfamiliar wit tis arameter sould refer to Cater P01 for details. On exit: IFAIL ¼ 0 unless te routine detects an error (see Section 6). For environments were it migt be inaroriate to alt rogram execution wen an error is detected, te value 1 or 1 is recommended. If te outut of error messages is undesirable, ten te value 1 is recommended. Oterwise, for users not familiar wit tis arameter te recommended value is 0. Wen te value 1 or 1 is used it is essential to test te value of IFAIL on exit. 6 Error Indicators and Warnings If on entry IFAIL ¼ 0or 1, exlanatory error messages are outut on te current error message unit (as defined by X04AAF). Errors or warnings detected by te routine: IFAIL ¼ 1 On entry, IORDER < 4 or IORDER > 6, or IQ < 0, or LENFW < 2 IQþ2, or LDSW < 2 IQþ1 þ 2 IORDER 1, or LWK < 2 IQþ3. 7 Accuracy None..2

3 8 Furter Comments Fractional quadrature weigts can be used for solving weakly singular integral equations of Abel tye. In tis section, we roose te following algoritm wic users may find useful in solving a linear weakly singular integral equation of te form yðtþ ¼fðtÞþ 1 Z t Kðt; sþyðsþ ffi ds; 0 t T; ð2þ 0 t s using. In (2), Kðt; sþ and fðtþ are given and te solution yðtþ is sougt on a uniform mes of size suc tat T ¼ N. Discretization of (2) yields y n ¼ fðnþþ 2 2 X W n;j Kðn; jþy j þ X n! n j Kðn; jþy j ; ð3þ j¼2 1 were y n yðnþ. We roose te following algoritm for comuting y n from (3) after a call to : (a) (b) (c) Set N ¼ 2 IQþ1 þ 2 IORDER 2 and ¼ T=N. Equation (3) requires 2 IORDER 2 starting values, y j, for j ¼ 1; 2;...; 2 IORDER 2, wit y 0 ¼ fð0þ. Tese starting values can be comuted by solving te system y n ¼ fðnþþ X 2IORDER 2 Comute te inomogeneous terms SWðn þ 1;jþ 1ÞKðn; jþy j ; n ¼ 1; 2;...; 2 IORDER 2: (d) n ¼ fðnþþ X 2IORDER 2 SWðn þ 1;jþ 1ÞKðn; jþy j ; n ¼ 2 IORDER 1; 2 IORDER;...;N: Start te iteration for n ¼ 2 IORDER 1; 2 IORDER;...;N to comute y n from: ð1 WTð1ÞKðn; nþþyn ¼ n þ X n 1 WTðn j þ 1ÞKðn; jþy j : j¼2iorder 1 Note tat for nonlinear weakly singular equations, te solution of a nonlinear algebraic system is required at ste (b) and a single nonlinear equation at ste (d). 9 Examle Te following examle generates te first 16 fractional convolution and 23 fractional starting weigts generated by te fourt-order BDF metod. 9.1 Program Text Note: te listing of te examle rogram resented below uses bold italicised terms to denote recision-deendent details. Please read te Users Note for your imlementation to ceck te interretation of tese terms. As exlained in te Essential Introduction to tis manual, te results roduced may not be identical for all imlementations. Examle Program Text Mark 16 Release. NAG Coyrigt Parameters.. INTEGER NOUT PARAMETER (NOUT=6) INTEGER IORDER, IQ, ITPMT, ITIQ, LENFW, LDSW, LWK PARAMETER (IORDER=4,IQ=3,ITPMT=2IORDER-1,ITIQ=2(IQ+1), + LENFW=2ITIQ,LDSW=ITIQ+ITPMT,LWK=4ITIQ).. Local Scalars.. INTEGER I, IFAIL, J.. Local Arrays.. real SW(LDSW,ITPMT), WORK(LWK), WT(LENFW).. External Subroutines.. EXTERNAL.3

4 NAG Fortran Library Manual.. Executable Statements.. WRITE (NOUT,) Examle Program Results WRITE (NOUT,) IFAIL = 0 CALL (IORDER,IQ,LENFW,WT,SW,LDSW,WORK,LWK,IFAIL) WRITE (NOUT,) Fractional convolution weigts WRITE (NOUT,) DO 20 I = 1, ITIQ WRITE (NOUT,99999) I - 1, WT(I) 20 CONTINUE WRITE (NOUT,) WRITE (NOUT,) Fractional starting weigts WRITE (NOUT,) DO 40 I = 1, LDSW WRITE (NOUT,99999) I - 1, (SW(I,J),J=1,ITPMT) 40 CONTINUE STOP FORMAT (1X,I5,7F9.4) END 9.2 Program Data None. 9.3 Program Results Examle Program Results Fractional convolution weigts Fractional starting weigts

5 (last)