# Fitted Numerical Scheme for Second-Order Singularly Perturbed Differential-Difference Equations with Mixed Shifts

Authors: Meku Ayalew; Gashu Gadisa Kiltu; Gemechis File Duressa
Journal: Abstract and Applied Analysis
 (2021)

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## Abstract

This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size, where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10−03 up to 10−10, and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.


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## Body

## 1. Introduction

Singularly perturbed differential-difference equations (SPDDEs) occur frequently in the mathematical modeling of various physical and biological phenomena, for example, control theory, viscous elasticity, and population dynamics [1]. Recently, many researchers have started developing different numerical methods for solving differential equations. Reference [2] investigated that the nonlinear thermal radiation and dissipation with the Darcy-Forchheimer equation in the porous medium analysis by using the fifth-order Runge-Kutta method, [3] also discussed the cross-fluid flow containing gyrotactic microorganisms and nanoparticles on a horizontal and three-dimensional cylinder by using the Runge-Kutta Fehlberg fifth-order technique, [4] studied the three-dimensional convective heat transfer of magnetohydrodynamics nanofluid flow through a rotating cone by using the fifth-order Runge-Kutta method. Reference [5] discussed that the heat transfer hybrid nanofluid contains 1-butanol as the base fluid and MoS2–Fe3O4 hybrid nanoparticles by using the finite element method. In these papers, the influence of various parameters on velocity profile and temperature has been investigated. Reference [6] presented a finite difference numerical method for solving singularly perturbed delay differential equations, [7] introduced a Galerkin method for solving this problem, [8] also introduced a fitted tension spline method for solving such problem, [9] presented a fitted second-order numerical method for singularly perturbed problems, [10–14] also developed some numerical methods of different orders for solving singularly perturbed delay differential equations, and so on. However, the issue of convergence and accuracy still needs attention and improvement. In this paper, we present a stable, convergent, and more accurate exponentially fitted fourth-order numerical scheme for solving SPDDEs and investigate the influence of delay and advance parameters on the solution profile.


## 2. Statement of the Problem of the Exponentially Fitted Method

Consider the governing equation [7, 14, 15]:
(1)εy"x+ϕxy′x+ψxyx−δ+φxyx+ϑxyx+η=rx,x∈0,1,subject to the boundary conditions,
(2)yx=αx,−δ≤x≤0,yx=βx,1≤x≤1+ηwhere ε is a perturbation parameter (0<ε<<1), δ is a delay parameter, η is the advance parameter with 0<δ,η=oε, and ϕx,ψx,φx,ϑx,rx,αx and βx are smooth functions on 0,1. Depending on the sign of ψx+φx+ϑx, different cases of boundary layers are reported in [12].From the Taylor series expansion in the neighborhood of the pointx, we obtain
(3)yx−δ≈yx−δy′x+οδ2,(4)yx+η≈yx+ηy′x+oη2.Replacing Equations (3) and (4) into Equation (1) gives an asymptotically equivalent SP two-point boundary value problem:
(5)Lyx≡εy″x+pxy′x+qxyx=rx,under the boundary conditions,
(6)y0=α0,y1=β0,where px=ϕx−δψx+ηϑx and qx=ψx+φx+ϑx.The transformation from Equation (1) with Equation (5) is accepted, because the conditions 0<δ,η<<1 are sufficiently small [16].Using the uniform mesh technique over the domain, we havexi=x0+ih,i=0,1,⋯,N.By using the Taylor series expansion, we obtain(7)yi+1=yi+hyi′+h22!yi″+h33!yi3+h44!yi4+oh5,(8)yi−1=y1−hyi′+h22!yi″−h33!yi3+h44!yi4+oh5.Subtracting Equation (8) from Equation (7) gives the approximation δc1yi, for the first derivative of yi as
(9)δc1yi=yi+1−yi−12h+T1,where T1=−h2/6yi3.Similarly, adding Equations (7) and (8) provides the approximation δc2yi, for the second derivative of yi as
(10)δc2yi=yi+1−2yi+yi−1h2+T2,where T2=−h2/12yi4.Substituting Equations (7) and (8) into Equation (9) yields
(11)δc1yi=yi′+h26yi3+T3,where T3=h4/120yi5+T1=h4/120yi5−h2/6yi3.Again, substituting Equations (7) and (8) into Equation (10) gives
(12)δc2yi=yi″+h212yi4+T4,where T4=h4/360yi6−h2/12yi4.Applyingδc2 to yi′ in Equation (9), we get
(13)yi3=δc2yi′−T12.Using Equation (13) into Equation (11), we obtain
(14)δc1yi=yi′+h26δc2yi′+T5,where T5=13h4/360yi5−h2/6yi3.Applyingδc2 to yi″ in Equation (10), we get a four-order finite difference scheme for Equation (5):
(15)yi4=δc2yi″−T22.Substituting Equation (15) into Equation (12), we get
(16)δc2yi=yi″+h212δc2yi″+T6,where T6=7h4/720yi6−h2/12yi4.From Equations (14) and (16), we get
(17)yi′=δc1yi−T51+h2/6δc2,yi″=δc2yi−T61+h2/12δc2.After evaluating Equation (5) at nodal point xi and using Equation (17), we obtain
(18)εδc2yi−T61+h2/12δc2+piδc1yi−T51+h2/6δc2+qiyi=ri.Simplifying Equation (18), we have
(19)εδc2yi+εh26δc4yi−ε1+h26δc2T6+piδc1yi+h2pi12δc3yi+qiyi+h2qi4δc2yi+h4qi72δc4yi=ri1+h24δc2+h472δc4+piT51+h212δc2.Rearranging Equation (5) and successively differentiating, evaluating at xi, and substituting into Equation (19), we get
(20)ε−h262pi′+h2qi12+h2pi212ε−h4qi72ε2pi′+qi−pi2εδc2yi−ε1+h26δc2T6+−h2pi12εpi′+qi+h4qi72εpipi′+qiε−pi″−2qi′+h26pipi′+qiε−pi″−2qi′+piδc1yi+h4qi72εpiqi′ε−qi″−h2piqi′12ε+qi+h26piqi′ε−qi″yi=ri+h24δc2ri+h472δc2ri″+piT51+h212δc2−h26ri″+h2pi12εri′−h4qi72εri″−h4piqi72ε2ri′.Introducing the fitting factorσ into Equation (20), we have
(21)σε1−h26ε2pi′+h2qi12ε+h2pi212ε2−h4qi72ε22pi′+qi−pi2εδc2yi−1+h26δc2T6+−h2pi12εpi′+qi+h4qi72εpipi′+qiε−pi″−2qi′+h26pipi′+qiε−pi″−2qi′+piδc1yi+h4qi72εpiqi′ε−qi″−h2piqi′12ε+qi+h26piqi′ε−qi″yi=ri+h24δc2ri+h472δc2ri″+piT51+h212δc2−h26ri″+h2pi12εri′−h4qi72εri″−h4piqi72ε2ri′.Using the central difference approximation forδc2yi and δc1yi, multiplying both sides of Equation (21) by h, and evaluating the limit as h⟶0, we obtain
(22)σ12ρ12+ρ2pi2limh⟶0yi−1−2yi+yi+1+pi2limh⟶0yi+1−yi−1=0,where ρ=h/ε.From the theory of singular perturbations and O’Malley [17], we have two cases for px>0 andpx<0.Case 1.
Forpx<0 (right-end boundary layer), we have
(23)limh⟶0yi−1−2yi+yi+1=α0−y00e−p0−1/ε+iρep0ρ+e−p0ρ−2,limh⟶0yi+1−yi−1=α0−y00e−p0−1/ε+iρe−p0ρ−ep0ρ.Thus, from Equation (22), we get
(24)σ0=6ρp012+ρ2p20cothp0ρ2.Case 2.
Forpx>0 (left-end boundary layer), we have
(25)limh⟶0yi−1−2yi+yi+1=β0−y01e−p1−1/ε+iρe−p1ρ+ep1ρ−2,(26)limh⟶0yi+1−yi−1=β0−y01e−p1−1/ε+iρe−p1ρ−ep1ρ.Thus, from Equation (22), we get
(27)σ1=6ρp112+ρ2p21cothp1ρ2.In general, for discretization, we take a variable fitting parameter as(28)σi=6ρpi12+ρ2pi2cothpiρ2.Now using Equations (9) and (10) into Equation (21) for δc1yi and δc2yi and making use of δc2ri=ri−1−2ri+ri+1/h2 and δc2ri″=ri−1″−2ri″+ri+1″/h2, we obtain
(29)σiεh2+qi12−162pi′+pi212ε−h2qi72ε2pi′+qi−pi2ε−12−hpi12εpi′+qi+h3qi72ε−pi″−2qi′+pipi′+qiε+h6−pi″−2qi′+pipi′+qiε+pihyi−1+−2σiεh2−162pi′+qi12+pi212ε−h2qi72εqi+2pi′−pi2ε+h4qi72ε−qi″+piqi′ε−h2piqi′12ε+qi+h26piqi′ε−qi″yi+σiεh2−162pi′+qi12+pi212ε−h2qi72ε2pi′+qi−pi2ε+12−hpi12εpi′+qi+h3qi72ε−pi″−2qi′+pipi′+qiε+h6−pi″−2qi′+pipi′+qiε+pihyi+1=ri+14ri−1−2ri+ri+1+h272ri−1″−2ri″+ri+1″−h26ri″+h2pi12εri′−h4qi72εri″−h4piqi72ε2ri′+Ti,whereTi=pih4/45yi5−σiεh4/240yi6+Oh5 is a local truncation error.Simplifying Equation (24), we get a tridiagonal system:
(30)LN≡Aiyi−1−Biyi+Ciyi+1=Di,fori=1,2,⋯,N−1,where
(31)Ai=σiεh2+qi12−162pi′+pi212ε−h2qi72ε2pi′+qi−pi2ε−12−hpi12εpi′+qi+h3qi72εpipi′+qiε−pi″−2qi′+h6pipi′+qiε−pi″−2qi′+pih,Bi=2σiεh2−162pi′+qi12+pi212ε−h2qi72ε2pi′+qi−pi2ε−h4qi72εpiqi′ε−qi″+h2piqi′12ε−qi−h26piqi′ε−qi″,Ci=σiεh2−162pi′+qi12+pi212ε−h2qi72ε2pi′+qi−pi2ε+12−hpi12εpi′+qi+h3qi72εpipi′+qiε−pi″−2qi′+h6pipi′+qiε−pi″−2qi′+pih,Di=ri+14ri−1−2ri+ri+1+h272ri−1″−2ri″+ri+1″−h26ri″+h2pi12εri′−h4qi72εri″−h4piqi72ε2ri′.We have used a discrete invariant imbedding algorithm to solve Equation (25).


## 3. Stability Analysis

The continuous minimum principle, continuous maximum principle, and stability of the solution of Equations (5) and (6) are presented in [13]. We present the stability of the scheme in Equation (25) for both cases.Case 1.
Whenqx<0, i.e., ψx+φx+ϑx<0, for x∈0,1.Lemma 1 (discrete minimum principle).
Ifwi is any mesh function such that wo≥0 and LNwi≤0, then wi≥0 for all xi∈0,1.Proof.
Suppose∃k∈ℤ+ such that wk<0 and wk=min0≤i≤Nwi.
Then, from Equation (25), we have
(32)LNwk≡Akwk−1−Bkwk+Ckwk+1=σkεh2+pk212ε+qk12−pk′3+h2qk72εpk2ε−2pk′−qkwk−1−wk+σkεh2+pk212ε+qk12−pk′3+h2qk72εpk2ε−2pk′−qkwk+1−wk+pkh+hpkpk′+qk12ε−hpk″6−hqk′3+h3qk72εpkpk′+qkε−pk″−2qk′wk+1−wk−1+Fkwk,where Fk=h2/6pkqk′/ε−qk″−h2pkqk′/12ε+qk+h4qk/72εpkqk′/ε−qk″.
For sufficiently smallh (i.e., h⟶0) and for suitable values of pk, we obtain LNwk>0. Since, wk<0 (by assumption), ε,σk>0 and Fk⟶qk<0..
However, this is a contradiction. Hence,wi≥0 for all xi∈0,1.☐Theorem 2.
The operatorLN in Equation (25) is stable for ψx+φx+ϑx<0 if wi is any mesh function, then wi≤λmaxw0,maxxi∈0,1Lwi, for some constant λ≥1.Proof. (see [13]).
This proves the stability of the scheme for the case of boundary layer behaviour.☐Case 2.
Whenqx>0, i.e., ψx+φx+ϑx>0, for x∈0,1.Lemma 3 (discrete maximum principle).
Ifwi is any mesh function such that w0≥0 and LNwi≥0, then wi≥0 for all xi∈0,1.Proof.
Suppose∃k∈ℤ+ such that wk<0 and wk=max0≤i≤Nwi.
Then, from Equation (25), we have
(33)LNwk≡Akwk−1−Bkwk+Ckwk+1=σkεh2+pk212ε+qk12−pk′3+h2qk72εpk2ε−2pk′−qkwk−1−wk+σkεh2+pk212ε+qk12−pk′3+h2qk72εpk2ε−2pk′−qkwk+1−wk+pkh+hpkpk′+qk12ε−hpk″6−hqk′3+h3qk72εpkpk′+qkε−pk″−2qk′wk+1−wk−1+Fkwk.
For sufficiently smallh and for suitable values of pk, we obtain LNwk<0. Since, wk<0 (by assumption), ε,σk>0 and Fk⟶qk>0..
However, this is a contradiction. Hence,wi≥0 for all xi∈0,1.☐Theorem 4.
The operatorLN in Equation (25) is stable for ψx+φx+ϑx>0, if wi is any mesh function, then wi≤κmaxw0,maxxi∈0,1Lwi for some constant κ≥1.Proof.
The proof is analogous to Theorem2.
This proves the stability of the scheme for the case of oscillatory behaviour.☐


## 4. Convergence Analysis

Writing the scheme in Equation (25) in matrix form, we obtain
(34)MY=V,where M=mij,i,j=1,2,⋯,N−1, is a tridiagonal matrix of order N−1.Multiplying both sides of Equation (25) by −h2, we have
(35)mii+1=−σiε−h262pi′+h2pi212ε+h2qi12−h4qi72ε2pi′+qi−pi2ε−h2pi+h26pipi′+qiε−pi″−2qi′−h2pipi′+qi12ε+h4qi72εpipi′+qiε−pi″−2q′,mii=2σiε−h262pi′+h2pi212ε+h2qi12−h4qi72ε2pi′+qi−pi2ε−h2h26piqi′ε−qi″−h2piqi12ε+qi+h4qi72εpiqi′ε−qi″,mii−1=−σiε−h262p′+h2pi212ε+h2qi12−h4qi72ε2pi′+qi−pi2ε+h2pi+h6pipi′+qiε−pi″−2qi′−h2pipi′+q12ε+h4qi72εpipi′+qiε−pi″−2q′,and V=vi is a column vector, where
(36)v1=−h2D1−A1α0,vi=−h2Difori=2,3,⋯,N−2,vN−1=−h2DN−1−CN−1β0,with a local truncation error:
(37)Tih=h615K+Oh7,where K=pi/3yi5−σiε/16yi6.Equation (34) can also be written in error form as
(38)MY¯−Th=V,whereY¯=y¯1,y¯2,⋯,y¯N−1t stands for the exact solution and Th=T1h,T2h,⋯,TN−1ht is a local truncation error.From Equations (34) and (38), we obtain
(39)MY¯−Y=Th,which implies
(40)ME=Th,where E=Y¯−Y=e1,e2,⋯,eN−1t.LetSi be the sum of elements of the ith row of a matrix M, then we get
(41)Si=σiε+h−pi2+h2−σ62pi′+σpi212ε+σqi12−qi+h36pi″+2qi′−h3pipi′+qi2ε+Oh4,fori=1,Si=h2−qi+Oh4,fori=2,3,⋯,N−2,Si=σiε+hpi2+h2−σ62pi′+σpi212ε+σqi12+112pipi′+qiε−pi″−2qi′−qi+h3−pipi′+qi24ε+Oh4,fori=N−1.For a sufficiently smallh, the matrix M is irreducible and monotone [1]. Thus, M−1 exists and M−1≥0.Thus, Equation (40), gives
(42)E=M−1Th,(43)E≤M−1.Th.Letm¯k,i≥0 be the k,ith element of M−1. From the theory of matrices, we have
(44)∑i=1N−1m¯k,iSi=1,k=1,2,⋯,N−1.Therefore,(45)∑i=1N−1m¯k,i≤1min1≤i≤N−1Si=1h2Bi0,where Bi0=−qi.We defineM−1=max1≤k≤N−1∑i=1N−1m¯k,i and Th=max1≤i≤N−1Tih.From Equations (34), (42), (43), and (45), we get
(46)ei=∑k=1N−1m¯k,iTih,i=1,2,⋯,N−1,which implies
(47)ei≤∑k=1N−1m¯k,imax1≤i≤N−1Tih≤h6K15h2Bi0=h4K15Bi0,where i0 is some number between i and N.Therefore,E=Oh4. Hence, the present method is of fourth-order convergence.


## 5. Numerical Examples and Results

To show the applicability of the method, three model examples have been considered. The exact solution of Equations (1) and (2) with constant coefficients is
(48)yx=c1em1x+c2em2x+rc3,where
(49)m1=δψ−ϕ−ηϑ+sqrtδψ−ϕ−ηϑ2−4εc32ε,,m2=δψ−ϕ−ηϑ−sqrtδψ−ϕ−ηϑ2−4εc32ε,c3=ψ+φ+ϑ,c1=−r+βc3+em2r−αc3c3em1−em2,c2=r−βc3+em1−r+αc3c3em1−em2.For the variable coefficients, the maximum absolute errors are computed using the double mesh principle [13].Example 5.
Consider the model equation (7),
(50)εy″x−y′x−2yx−δ+yx−2yx+η=0,with boundary conditions,
(51)yx=1,−δ≤x≤0,y1=−1,1≤x≤η.Example 6.
Consider the model equation (7),
(52)εy″x+0.5y′x−3yx−δ−2yx+2yx+η=1,with boundary conditions,
(53)yx=1,−δ≤x≤0,y1=0,1≤x≤η.Example 7.
Consider the model equation (14),
(54)εy″x−1+ex2y′x−xyx−δ+x2yx−1−e−xyx+η=1,with boundary conditions,
(55)yx=1,−δ≤x≤0,y1=−1,1≤x≤η.The following graphs (Figures1 and 2) show the effect of delay and advance parameters on the solution profile.Figure 1
Numerical solution of Example5 for ε=0.1 and N=20.
(a)(b)Figure 2
Numerical solution of Example6 for ε=0.1 and N=20.
(a)(b)The following graphs (Figure3) show the pointwise absolute errors for different values of mesh size h.Figure 3
Pointwise absolute errors of Examples6 and 7, respectively, for ε=0.1 and δ,η=0.5ε.
(a)(b)


## 6. Discussion and Conclusion

This study introduces an exponentially fitted fourth-order numerical method for solving singularly perturbed differential-difference equations. The results observed from the tables demonstrate that the present method approximates the solution very well and depicts the betterment over some existing numerical methods reported in the literature. The stability and convergence of the scheme have been established. The solutions presented in Table1 confirm that the numerical rate of convergence as well as theoretical error estimates indicates that the present method is of fourth-order convergence.Table 1
Rate of convergence forε=0.1,η=0.5ε.
δN163264128Example50.004.06554.01644.00414.00100.054.05664.01444.00364.00090.094.05004.01264.00314.0008Example60.004.02564.00644.00164.00040.054.03494.00874.00224.00050.094.04354.01084.00274.0007To demonstrate the effect of delay and advance parameters on the left and right boundary layers of the solution, the graphs for different values of delay parameterδ and advance parameter η are plotted in Figures 1 and 2. Accordingly, based on the sign of px, one can see that, from Figure 1, as δ increases, the width of the right boundary layer decreases for a fixed value of η, but as η increases, the width of the right boundary layer increases for a fixed value of δ while the width of the left boundary layer decreases when δ or η increases (Figure 2). Furthermore, as h decreases, the absolute error also decreases (see Tables 2–6 and Figure 3). In general, the present method is stable, convergent, and more accurate.Table 2
Maximum absolute error of Example5 for ε=0.1.
N⟶832128512Present methodη=0.5εδ↓0.004.3229e−031.5775e−056.1456e−082.4006e−100.053.8440e−031.3769e−055.4036e−082.1092e−100.093.4760e−031.2460e−054.8494e−081.8940e−10Result in [7]0.000.0313775380.0018002410.0001120717.0036e−060.050.0297480100.0017000260.0001054186.5860e−060.090.0282942850.0016110539.9793e−056.2344e−06Present methodδ=0.5εη↓0.003.3862e−031.2139e−054.7199e−081.8429e−100.053.8440e−031.3769e−055.4036e−082.1092e−100.094.2256e−031.5339e−055.9891e−082.3403e−10Result in [7]0.000.0279105290.0015876519.8361e−056.1442e−060.050.0297480100.0017000260.0001054186.5860e−060.090.0310685000.0017812070.0001108006.9223e−06Table 3
Maximum absolute error of Example6 for ε=0.1.
N⟶832128512Present methodη=0.5εδ↓0.002.9005e−031.0342e−054.0567e−081.5841e−100.053.5885e−031.2831e−054.9745e−081.9433e−100.094.1815e−031.4979e−055.8027e−082.2664e−10Result in [7]0.000.0253475100.0014253278.9204e−055.5742e−060.050.0275338260.0015677109.7155e−056.0690e−060.090.0286697700.0016455500.0001021866.3826e−06Present methodδ=0.5εη↓0.001.7013e−031.1139e−054.3477e−081.6984e−100.053.5885e−031.2831e−054.9745e−081.9433e−100.093.9801e−031.4251e−055.5183e−082.1551e−10Result in [7]0.000.0261746180.0014783419.2083e−055.7527e−060.050.0275338260.0015677109.7155e−056.0690e−060.090.0283482720.0016231130.000100576.2854e−06Table 4
Maximum absolute error of Example7 for ε=0.1.
N⟶832128512η=0.5εδ↓0.009.1099e−021.1121e−026.3825e−044.0044e−050.059.0471e−021.0955e−026.3063e−043.9502e−050.098.9962e−021.0822e−026.2443e−043.9063e−05δ=0.5εη↓0.009.6047e−021.1165e−026.4582e−043.9245e−050.059.6212e−021.1248e−026.4941e−043.9502e−050.099.6342e−021.1313e−026.5223e−043.9705e−05Table 5
Numerical results of Example7 for ε=0.01,N=100.
xδ=0.0,η=0.0δ=0.005,η=0.001δ=0.005=ηδ=0.005,η=0.009Method in [14]Present methodMethod in [14]Present methodMethod in [14]Present methodMethod in [14]Present method0.200.88325720.88346590.88323090.88344050.88325490.88346380.88327800.88348710.400.75188080.75209750.75177850.75199620.75186530.75208230.75195130.75216840.600.62654520.62663450.62633620.62642660.62650160.62659160.62666670.62675630.800.52047430.52037490.52015980.52006130.52039440.52029590.52062920.52053020.900.47669970.47649810.47634060.47613980.47660200.47640130.47686380.47666260.920.46872170.46849940.46835460.46813300.46862070.46839930.46888710.46866520.940.46100070.46075960.46062590.46038540.46089640.46065610.46116730.46092650.960.45332630.45327560.45294800.45289390.45322020.45316890.45349310.4534435Table 6
Maximum absolute error forδ,η=0.5ε.
εh2−52−62−72−82−9Example52−37.8680e−064.8845e−073.0476e−081.9040e−091.1899e−102−44.8096e−052.9675e−061.8546e−071.1578e−087.2378e−102−53.4554e−041.9965e−051.2232e−067.6501e−084.7753e−092−62.1829e−031.5228e−048.8556e−065.4340e−073.3824e−082−79.4645e−031.0205e−037.0761e−054.1300e−062.5365e−072−82.8510e−024.4918e−034.9169e−043.4000e−051.9883e−06Example62−37.8654e−068.8695e−073.0790e−083.4972e−091.2029e−102−43.6536e−052.2720e−061.4204e−078.8711e−095.5450e−102−51.8558e−041.1830e−057.3360e−074.5832e−082.8636e−092−61.2043e−036.8432e−054.2579e−062.6466e−071.6561e−082−76.7569e−034.7969e−042.7698e−051.6967e−061.0622e−072−82.8776e−022.9911e−032.0743e−041.2083e−057.4178e−07Example72−38.3545e−032.0138e−034.9861e−041.2495e−043.1217e−052−41.7199e−024.3785e−031.0417e−032.5713e−046.4294e−052−52.5179e−028.8894e−032.2383e−035.2902e−041.3037e−042−63.1540e−021.2943e−024.5167e−031.1313e−032.6648e−042−74.4783e−021.6224e−026.5594e−032.2763e−035.6865e−042−87.8783e−022.3176e−028.2240e−033.3015e−031.1426e−03



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*Source: 4573847-2021-07-01.xml*