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Xqghu Frpelqhg Ordglqj +phfkdqlfdo dqg hqylurqphqwdo,/ lq Surjuhvv lq Gxudelolw| Dqdo|vlv ri Frpsrvlwh V|vwhpv/ Fdugrq/ Ixnxgd ) Uhlivqlghu+hgv1,/ Edonhpd/ Urwwhugdp +4<<9,/ ss1 4<80 5351 91 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 65/ 6440654 +4<<6,1 :1 Wkljshq/ O1/ J1 Z1 Khgvwurp/ E1 S1Erqqhu/ Mrxuqdo ri Dssolhg Phfkdqlfv/ 83/ 6940699 +4<;6,1 ;1 Wvfkrhjo/ Q1 Z1/ Wkh skhqrphqrorjlfdo wkhru| ri olqhdu ylvfrhodvwlf ehkdylrxu/ Vsulqjhu0Yhuodj +4<;<,1 <1 \rqj0Vrn R1/ F1 Slorw ) F1 Mrugdq/ Prghol}dwlrq ri Ylvfrhodvwlf Ehkdylru ri Fduerq2Hsr{| Xqlgluhfwlrqdo Frpsrvlwhv/ lq= Surfhhglqj ri wkh Hxurshdq Phfkdqlfv Froortxlxp 4;5 +4<;7,/ ss1 :60;61 431 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 65/ 655065: +4<<6,1 441 Hpul/ L1 ) Q1 Z1 Wvfkrhjo/ Ukhro Dfwd/ 66/ 930:3 +4<<7,1 451 Ndvfkwd/ M1 ) I1 U1 Vfkzdu}o/ Ukhro Dfwd/ 66/ 84:085< +4<<7,1 461 Ndvfkwd/ M1 ) I1 U1 Vfkzdu}o/ Ukhro Dfwd/ 66/ 8630874 +4<<7,1 471 Vfkdshu|/ U1 D1/ Dssur{lpdwlrq phwkrgv ri wudqvirup lqyhuvlrq iru ylvfrhodvwlf vwuhvv dqdo|vlv/ lq= Surfhhglqjv ri wkh 7wk X1V1 Qdwlrqdo 4; Frqjuhvv ri Dssolhg Phfkdqlfv/ DVPH/ 5 +4<95,/ ss1 43:8043;81 4< Wdeoh L1 G|qdplf fxuyhv iru wkh vshfwud jlyhq e| Htxdwlrq 531 sEM5a  EC@ 3  a  EC@ 3   fff eSfDDe D2fef f 3 f2 ff2f eDffD bbbDe f 3 f2 fff eeDe feebf.D fff 2.DDD. fD.Db. fff fHS2SS. fe.b. fff fDfb2 Db.feH f 3 f2 fff fe.e.. bHbbS2 f 3 f2 53 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 110 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 a Ψ τ1 Ψ τ1 ; s positive parameters N=5 Iljxuh 4= Plqlpxp ri wkh vxp ri uhodwlyh huuruv/ [/ dqg wkh fruhvsrqghqw uhwdugdwlrq wlph/  / iru wkh fdofxodwlrq ri wkh g|qdplf frpsoldqfhv/ dv ixqfwlrqv ri wkh orjdulwkplf glvwdqfh/ @/ ehwzhhq wkh uhwdugdwlrq wlphv1 -0.10 0.10 0.30 0.50 0.70 0.90 1.10 1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 τ1 ; s L(τ) , Jk/ln(2.5) ; GPa-1 Iljxuh 5= Qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@iru @'2Dyv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 531 54 0.00 0.50 1.00 1.50 0.001 0.01 0.1 1 10 100-12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error Iljxuh 6= Dssur{lpdwhg fuhhs frpsoldqfh iru @'2Dyv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 531 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02 1.E-01 110100 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 a Ψ τ1 Ψ τ1 ; s positive parameters N=5 10% Noise Iljxuh 7= Plqlpxp ri [dqg wkh fruhvsrqghqw uhwdugdwlrq wlph/  /iruwkh fdofxodwlrq ri wkh g|qdplf frpsoldqfhv zlwk fI qrlvh/ dv ixqfwlrqv ri wkh orjdulwkplf glvwdqfh/ @/ ehwzhhq wkh uhwdugdwlrq wlphv1 55 -0.10 0.10 0.30 0.50 0.70 0.90 1.10 1.E-05 1.E-03 1.E-01 1.E+01 1.E+03 1.E+05 τ ; s L(τ) , Jk/ln(2.5) ; GPa-1 Iljxuh 8= Qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@iru @'2Ddqg iru fI qrlvh g|qdplf gdwd +Y'f, yv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 531 0.00 0.50 1.00 1.50 0.001 0.01 0.1 1 10 100-12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error Iljxuh 9= Dssur{lpdwhg fuhhs frpsoldqfh iru @'2Ddqg iru fI qrlvh g|qdplf gdwd +Y'f, yv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 531 56 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E-07 1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17 τ ; s L(τ) , Jk/ln(10) ; GPa-1 Iljxuh := Grxeoh0orjdulwkplf sorw ri wkh qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@iru @'fyv wkh h{dfw uhwdugdwlrq vshfwuxp/ Ht1 541 1.E-01 1.E+00 1.E+01 1.E+02 1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17 -12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error Iljxuh ;= Grxeoh0orjdulwkplf sorw ri wkh dssur{lpdwhg fuhhs frpsoldqfh iru @'fyv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 541 57 1.E-10 1.E-08 1.E-06 1.E-04 1.E-02 1.E+00 1.E+02 1.E-10 1.E-04 1.E+02 1.E+08 1.E+14 1.E+20 τ ; s L(τ) , Jk/ln(10) ; GPa-1 Iljxuh <= Grxeoh0orjdulwkplf sorw ri wkh qrupdolvhg uhwdugdwlrq frpsrqhqwv a&*,?E@iru @'fyv h{dfw uhwdugdwlrq vshfwuxp/ Ht1 551 1.E-01 1.E+00 1.E+01 1.E+02 1.E-03 1.E+01 1.E+05 1.E+09 1.E+13 1.E+17 -12 -8 -4 0 4 8 12 t ; s J(t) ; GPa-1 error ; (%) exact approx. error Iljxuh 43= Grxeoh0orjdulwkplf sorw ri wkh dssur{lpdwhg fuhhs frpsoldqfh iru @'fyv h{dfw fuhhs frpsoldqfh dv fdofxodwhg iurp vshfwuxp jlyhq e| Ht1 55 58