# A Family of Trigonometrically Fitted Enright Second Derivative Methods for Stiff and Oscillatory Initial Value Problems

Authors: F. F. Ngwane; S. N. Jator
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

A family of Enright’s second derivative formulas with trigonometric basis functions is derived using multistep collocation method. The continuous schemes obtained are used to generate complementary methods. The stability properties of the methods are discussed. The methods which can be applied in predictor-corrector form are implemented in block
form as simultaneous numerical integrators over nonoverlapping intervals. Numerical results obtained using the proposed block form reveal that the new methods are efficient and highly competitive with existing methods in the literature.


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## Body

## 1. Introduction

Many real life processes in areas such as chemical kinetics, biological sciences, circuit theory, economics, and reactions in physical systems can be transformed into systems of ordinary differential equations (ODE) which are generally formulated as initial value problems (IVPs). Some classes of IVPs are stiff and/or highly oscillatory as described by the following model problem:(1)y′=Ay,ya=y0,x∈a,b,where y(x)∈Rm and A is m×m real matrix with at least one eigenvalue with a very negative real part and/or very large imaginary part, respectively (see Fatunla [1]). Many conventional methods cannot solve these types of problems effectively.Stiff systems have been solved by several authors including Lambert [2, 3], Gear [4, 5], Hairer [6], and Hairer and Wanner [7]. Different methods including the Backward Differentiation Formula (BDF) have been used to solve stiff systems. Second derivative methods with polynomial basis functions were proposed to overcome the Dahlquist [8] barrier theorem whereby the conventional linear multistep method was modified by incorporating the second derivative term in the derivation process in order to increase the order of the method, while preserving good stability properties (see Gear [9], Gragg and Stetter [10], and Butcher [11]).Many classical numerical methods including Runge-Kutta methods, higher derivative multistep schemes, and block methods have been constructed for solving oscillatory initial value problems (see Butcher [11, 12], Brugnano and Trigiante [13, 14], Ozawa [15], Nguyen et al. [16], Berghe and van Daele [17], Vigo-Aguiar and Ramos [18], and Calvo et al. [19]). Many methods for solving oscillatory IVPs require knowledge of the system under consideration in advance.Obrechkoff [20] proposed a general multiderivative method for solving systems of ordinary differential equations. Special cases of Obrechkoff method have been developed by many others including Cash [21] and Enright [22]. The methods by Enright [22] have order p=k+2 for a k step method.In this paper, we propose a numerical integration formula which more effectively copes with stiff and/or oscillatory IVPs. We will construct a continuous form of the second derivative multistep method (CSDMM) using a multistep collocation technique such that Enright’s second derivative methods (ESDM) will be recovered from the derived continuous methods. The aim of this paper is to derive a family of Enright’s second derivative formulas with trigonometric basis functions using multistep collocation method. Many methods for solving IVPs are implemented in a step-by-step fashion in which, on the partitionπN, an approximation is obtained at xn+1 only after an approximation at xn has been computed, where πN:a=x0<x1<⋯<xn<xn+1<⋯<xN=b, xn+1=xn+h, n=1,…,N, h=(b-a)/N, h is the step size, N is a positive integer, and n is the grid index. We implement ESDM in block form.In Section2, we present a derivation of the family of Enright methods. Error analysis and stability are discussed in Section 3. The implementation of the ESDM and numerical examples to show the accuracy and efficiency of the ESDM are given in Section 4. Finally, we conclude in Section 5.


## 2. Derivation of the Family of Methods

We consider the first-order differential equation(2)y′=fx,y,ya=y0,x∈a,b,where f is assumed to satisfy the conditions to guarantee the existence of a unique solution of the initial-value problem.


### 2.1. CSDMM

In what follows, we state the CSDMM which has the ability to produce the ESDM:(3)Ux=αn+k-1xyn+k-1+h∑j=0kβjxfn+j+h2γn+kxgn+k,where αn+k-1(x), βj(x), and γn+k(x) are continuous coefficients. We assume that yn+j=U(xn+j) is the numerical approximation to the analytical solution y(xn+j), yn+j′=U′(xn+j) is the numerical approximation to the analytical solution y′(xn+j), fn+j=U′(xn+j) is an approximation to y′(xn+j), and gn+j=U′′(xn+j) is an approximation to y′′(xn+j), where fn+j=f(xn+j,yn+j), gn+j=(df(x,y(x))/dx)yn+jxn+j, j=0,1,2,…,k.We now define the following vectors and matrix used in the following theorem:(4)V=yn+k-1,fn,fn+1,fn+2,…,fn+k,gn+kT,Px=P1x,P2x,…,Pk+3xT,Wk+3,k+3=P1xn+k-1P2xn+k-1⋯Pk+3xn+k-1P1′xnP2′xn⋯Pk+3′xnP1′xn+1P2′xn+1⋯Pk+3′xn+1P1′xn+2P2′xn+2⋯Pk+3′xn+2⋮⋮⋱⋮P1′xn+kP2′xn+k⋯Pk+3′xn+kP1′′xn+kP2′′xn+k⋯Pk+3′′xn+k,where Pi(x)=xi-1, i=1,2,…,k+1, Pk+2x=sin⁡wx, and Pk+3x=cos⁡wx.Remark 1.
In the derivation of the ESDM,  the basesP(x)≡Pi(x)T with Pi(x)=xi-1, i=1,2,…,k+1, Pk+2(x)=sin(wx), and Pk+3(x)=cos(wx) are chosen because they are simple to analyze. Other possible bases (see Nguyen et al. [16] and Nguyen et al. [23]) include the following: (1)
{sin⁡(wx),cos⁡(wx),xsin(wx),xcos(wx),…,xnsin⁡(wx),xncos⁡(wx)};(2)
{sinx,cosx,…,sin⁡(nwx),cos⁡(nwx)};(3)
{sin⁡(w1x),cos⁡(w1x),…,sin⁡(wnx),cos⁡(wnx)};(4)
{x,…,wxn}∪{sin⁡(wx),cos⁡(wx),…,sin⁡(mwx),cos⁡(mwx)};(5)
{x,…,xn,exp⁡(±wx),xexp(±wx),…,xmexp⁡(±wx)};(6)
{x,…,wxn-1,wxn}.Theorem 2.
LetU(x) satisfy U(xn+j)=yn+j, U′(xn+j)=fn+j, and U′′(xn+j)=gn+j and let W be invertible; then method (3) is equivalent to(5)Ux=VTW-1TPx.The proof of the above theorem can be found in Jator et al. [24].Through interpolation ofU(x) at the point xn+k-1, collocation of U′(x) at the points xn+j, j=0,1,2,…,k, and collocation of U′′(x) at the point xn+k, we get the system(6)Uxn+k-1=yn+k-1,U′xn+j=fn+jj=0,1,2,…,k,U′′xn+k=gn+k.To solve this system we require that method (3) be defined by the assumed basis functions (7)αn+k-1=∑i=0k+3αiPix;hβjx=∑i=0k+3hβi,jPix,j=0,1,…,k;h2γn+k=∑i=0k+3h2γiPix,where the constants αi, βi,j, and γi are to be determined.


### 2.2. ESDM

The general second derivative formula for solving (2) using the k-step second derivative linear multistep method is of the form(8)∑j=0kαjyn+j=h∑j=0kβjfn+j+h2∑j=0kγjgn+j,where yn+j≈y(xn+jh), fn+j=f(xn+jh,y(xn+jh)) and fj=f(xj,yj), gn+j=df(x,y(x))/dx|yn+jxn+j; xn is a discrete point at node n; and αj, βj, and γj are parameters to be determined. It is worth noting that Enright’s method is a special case of (7). We solve (6) to get the coefficients αi, βi,j, and γi in (7) which are then used to obtain the continuous multistep method of Enright in the form(9)Ux=yn+k-1+h∑j=0kβjxfn+j+h2γn+kxgn+k.Evaluating (9) at x=xn+k and setting yn+k=U(xn+kh) yield the following Enright’s second derivative multistep method:(10)yn+k=yn+k-1+h∑j=0kβjfn+j+h2γn+kgn+k,whereas the (k-1) complementary methods(11)yn+i=yn+k-1+h∑j=0kβ^j,ifn+j+h2γ^n+k,ign+kare obtained by evaluating (9) at x=xn+i, i=0,1,2,…,k-2, with k≥2.We note that, in order to avoid the cancellations which might occur whenh is small, the use of the power series expansions of βj, γn+k, β^j,i, and γ^n+k,i is preferable (see Simos [25]).Case
k
=
1. This case has only the main method given by (10) with the coefficients defined by(12)β0=-csc⁡u/22-u+sin⁡u2u=13+u290+u42520+u675600+u82395008+691u1054486432000+Oh11,β1=-csc⁡u/22ucos⁡u-sin⁡u2u=23-u290-u42520-u675600-u82395008-691u1054486432000+Oh11,γn+1=-2+ucot⁡u/2u2=-16-u2360-u415120-u6604800-u823950080-691u10653837184000+Oh11.Case
k
=
2. The coefficients of the main method (10) and the complimentary method (11) are, respectively, defined by(13)β0=hcsc⁡u/22ucos⁡u/2-2sin⁡u/224uucos⁡u-sin⁡u=-148-u2360-13u457600-89u66048000-143203u8167650560000-126473u102724321600000+Oh11,β1=csc⁡u/22-2-3u2+2-u2cos⁡2u+4usin⁡u+2usin⁡2u8uucos⁡u-sin⁡u=512+u2720+13u450400+121u66048000+52133u841912640000+761473u1010897286400000+Oh11,β2=-hcsc⁡u/222-4+3u2cos⁡u+2+u2cos⁡2u+4usin⁡u8uucos⁡u-sin⁡u=2948+u2720-13u4403200-u6189000-5939u815240960000-255581u1010897286400000+Oh11,γn+2=-ucos⁡u/2-2sin⁡u/2sin⁡u/2uucos⁡u-sin⁡u=-18-u2240-13u467200-19u62016000-12979u827941760000-83437u103632428800000+Oh11,(14)β^0,0=-csc⁡u/22-2-u2+4+3u2cos⁡u-2cos⁡2u-2usin⁡2u8uucos⁡u-sin⁡u=-1748-u272-251u4403200-169u66048000-213203u8167650560000-161023u102724321600000+Oh11,β^1,0=csc⁡u/222+u2+-2+3u2cos⁡2u+4usin⁡u-6usin⁡2u8uucos⁡u-sin⁡u=-1112+17u2720+53u450400+281u66048000+87133u841912640000+1037873u1010897286400000+Oh11,β^2,0=-csc⁡u/224+-4+u2cos⁡u+u2cos⁡2u-2usin⁡2u8uucos⁡u-sin⁡u=1348-7u2720-173u4403200-u654000-135329u8167650560000-393781u1010897286400000+Oh11,γ^n+2,0=-ucos⁡u/2-2sin⁡u/2sin⁡u/2uucos⁡u-sin⁡u=-18-u2240-13u467200-19u62016000-12979u827941760000-83437u103632428800000+Oh11.Case
k
=
3. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined by(15)β0=csc⁡u/44-42-5u2-8-6+u2cos⁡u+-6+u2cos⁡2u+32usin⁡u-4usin⁡2u96uu+2ucos⁡u-3sin⁡u=71080+163u2151200+1529u413608000+1203457u6125737920000+5143273u87005398400000+364730953u106865290432000000+Oh11,β1=-csc⁡u/44-102-16u2-3-34+7u2cos⁡u+6cos⁡2u-6cos⁡3u+u2cos⁡3u+88usin⁡u-2usin⁡2u-4usin⁡3u96uu+2ucos⁡u-3sin⁡u=-120-11u28400-173u4756000-153049u66985440000-4786027u82724321600000-422813u103259872000000+Oh11,β2=csc⁡u/44-30-23u2-24cos⁡u+78-21u2cos⁡2u-24cos⁡3u+8u2cos⁡3u+34usin⁡u+52usin⁡2u-22usin⁡3u96uu+2ucos⁡u-3sin⁡u=1940-u2224-29u4302400+4187u62794176000+41551u899066240000+6204403u10152562009600000+Oh11,β3=-csc⁡u/4430-78+23u2cos⁡u+233+8u2cos⁡2u-18cos⁡3u-5u2cos⁡3u+68usin⁡u-22usin⁡2u96uu+2ucos⁡u-3sin⁡u=307540+71u215120+289u41360800+12391u61143072000+2957783u84903778880000+24651509u10686529043200000+Oh11,γn+3=csc⁡u/27ucos⁡u/2+5ucos⁡3u/2+12sin⁡u/2-sin⁡3u/212uu+2ucos⁡u-3sin⁡u=-19180-97u225200-491u42268000-285163u620956320000-7286749u88172964800000-67916227u101144215072000000+Oh11,(16)β^0,0=csc⁡u/24-12+2u2+9-4u2cos⁡u+26+7u2cos⁡2u⁡-9cos⁡3u+34usin⁡u-20usin⁡2u-6usin⁡3u48uu+2ucos⁡u-3sin⁡u=-43135-43u24725-193u4850500-2561u61964655000+1107763u83064861800000+8005337u10214540326000000+Oh11,β^1,0=-csc⁡u/24-15-4u2+34+5u2cos⁡u+15cos⁡2u-12cos⁡3u+7u2cos⁡3u-5usin⁡u+13usin⁡2u-19usin⁡3u24uu+2ucos⁡u-3sin⁡u=-75+29u21050+139u4189000+3683u6436590000-516289u8681080400000-473779u105297292000000+Oh11,β^2,0=csc⁡u/24-12+2u2+15cos⁡u+62+5u2cos⁡2u-15cos⁡3u+4u2cos⁡3u+44usin⁡u-28usin⁡2u-20usin⁡3u48uu+2ucos⁡u-3sin⁡u=-15-u235-17u418900-961u643659000-1367u86191640000+104537u104767562800000+Oh11,β^3,0=-csc⁡u/243+u2cos⁡u+-3+4u2cos⁡2u+u2cos⁡3u-usin⁡u-usin⁡2u-3usin⁡3u24uu+2ucos⁡u-3sin⁡u=-11135+19u21890+19u448600+1063u671442000+756881u81225944720000+2591419u1085816130400000+Oh11,γ^n+3,0=-2cot⁡u/22u+ucos⁡u-3sin⁡u3uu+2ucos⁡u-3sin⁡u=245-u21575-31u4283500-5927u6654885000-95237u8145945800000-3218641u1071513442000000+Oh11,(17)β^0,1=csc⁡u/24-30-u2+86+u2cos⁡u+-18+5u2cos⁡2u+16usin⁡u-20usin⁡2u96uu+2ucos⁡u-3sin⁡u=231080+467u2151200+3841u413608000+2702753u6125737920000+74202119u849037788800000+707967737u106865290432000000+Oh11,β^1,1=-csc⁡u/24-54-8u2+96+39u2cos⁡u-42cos⁡2u+5u2cos⁡3u-4usin⁡u-22usin⁡2u-8usin⁡3u96uu+2ucos⁡u-3sin⁡u=-920-59u28400-517u4756000-371321u66985440000-10263683u82724321600000-32697403u10127135008000000+Oh11,β^2,1=csc⁡u/2454+5u2-48cos⁡u+-6+39u2cos⁡2u-8u2cos⁡3u+50usin⁡u-76usin⁡2u+10usin⁡3u96uu+2ucos⁡u-3sin⁡u=-2940+u23360+59u4302400+53563u62794176000+1581469u81089728640000+1395817u1013869273600000+Oh11,β^3,1=-csc⁡u/2478+-96+5u2cos⁡u+29+4u2cos⁡2u-u2cos⁡3u-20usin⁡u-2usin⁡2u96uu+2ucos⁡u-3sin⁡u=83540+11u23024+281u41360800+14279u61143072000+3937807u84903778880000+36676261u10686529043200000+Oh11,γ^n+3,1=csc⁡u/2-13ucos⁡u/2+ucos⁡3u/2+24sin⁡u/212uu+2ucos⁡u-3sin⁡u=-11180-113u225200-739u42268000-474827u620956320000-12620021u88172964800000-119414483u101144215072000000+Oh11.Case
k
=
4. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined by(18)β0=Nβ0384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-175760-251u2483840-2003u433868800-354659u662589542400-51104113u8102521670451200-600902219u1014353033863168000+Oh11,β1=Nβ1384u6ucos⁡2u+16sin⁡u-11sin⁡2u=145+29u230240+181u41058400+70811u63911846400+2645263u81601901100800+9724577u1069004970496000+Oh11,β2=Nβ2192u6ucos⁡2u+16sin⁡u-11sin⁡2u=-41480+11u25760-169u42822400-70207u65215795200-12382199u88543472537600-14328617u10108735105024000+Oh11,β3=Nβ3384u6ucos⁡2u+16sin⁡u-11sin⁡2u=4790-241u230240-187u4529200-72019u63911846400-3489379u83203802201600-64028929u10897064616448000+Oh11,β4=Nβ4384u6ucos⁡2u+16sin⁡u-11sin⁡2u=31335760+2719u2483840+10207u433868800+1216471u662589542400+142053797u8102521670451200+1494030511u1014353033863168000+Oh11,γn+4=Nγn+42u6ucos⁡2u+16sin⁡u-11sin⁡2u=-332-3u2896-37u4188160-529u638635520-195109u8189854945280-6348281u1079739077017600+Oh11with Nβi, i=0,1,…,4, and Nγn+4 defined in part A of Appendix 1 of the supplementary material (see Supplementary Material available online at http://dx.doi.org/10.1155/2015/343295). Consider  (19)β^0,0=Nβ^0,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-201640-23u22560-1019u43763200-20569u62318131200-1712483u83797098905600-6213083u10177197948928000+Oh11,β^1,0=Nβ^1,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-75+39u21120+359u4352800+4481u6144883200+260119u8177989011200+34002959u10299021538816000+Oh11,β^2,0=Nβ^2,064u6ucos⁡2u+16sin⁡u-11sin⁡2u=-99160-219u24480-417u4313600-6317u6193177600-40941u835158323200-12088171u10132898461696000+Oh11,β^3,0=Nβ^3,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-910+29u21120+29u458800-809u6144883200-151769u8118659340800-3275539u1033224615424000+Oh11,β^4,0=Nβ^4,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=149640-51u217920+1013u411289600+37621u62318131200+16324541u811391296716800+530557669u104784344621056000+Oh11,γ^n+4,0=Nγ^n+4,02u6ucos⁡2u+16sin⁡u-11sin⁡2u=-332-3u2896-37u4188160-529u638635520-195109u8189854945280-6348281u1079739077017600+Oh11with Nβ^i,0, i=0,1,…,4, and Nγ^n+4,0 defined in part B of Appendix 1 of the supplementary material. Consider(20)β^0,1=Nβ^0,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=190+u2756+u410800+u6199584+691u82971987200+u10102643200+Oh11,β^1,1=Nβ^1,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1745-u2189-u42700-u649896-691u8742996800-u1025660800+Oh11,β^2,1=Nβ^2,124u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1915+u2126+u41800+u633264+691u8495331200+u1017107200+Oh11,β^3,1=Nβ^3,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1745-u2189-u42700-u649896-691u8742996800-u1025660800+Oh11,β^4,1=Nβ^4,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=190+u2756+u410800+u6199584+691u82971987200+u10102643200+Oh11,γ^n+4,1=0with Nβ^i,1, i=0,1,…,4, defined in part C of Appendix 1 of the supplementary material. Consider(21)β^0,2=Nβ^0,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-111920-169u2161280-12281u4101606400-2158889u6187768627200-7853429u87886282342400-272561573u103312238583808000+Oh11,β^1,2=Nβ^1,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=7135+31u210080+3701u49525600+1346483u635206617600+16159921u84805703302400+752918951u102691193849344000+Oh11,β^2,2=Nβ^2,2192u6ucos⁡2u+16sin⁡u-11sin⁡2u=-83160-17u213440-841u42822400-534077u615647385600-9061597u82847824179200-974213257u103588258465792000+Oh11,β^3,2=Nβ^3,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1930-59u210080-589u41587600-278329u611735539200-5135051u83203802201600-28211849u10244653986304000+Oh11,β^4,2=Nβ^4,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=183117280+821u2161280+122327u4304819200+17519503u6563305881600+743666159u8307565011353600+8152441741u1043059101589504000+Oh11,γ^n+4,2=Nγ^n+4,22u6ucos⁡2u+16sin⁡u-11sin⁡2u=-11288-3u2896-1447u45080320-218147u69388431360-9544999u85126083522560-11813881u1079739077017600+Oh11with Nβ^i,2, i=0,1,…,4, and Nγ^n+4,2 defined in part D of Appendix 1 of the supplementary material.


### 2.3. Block Specification and Implementation of the Methods

We consider a general procedure for the block implementation of the methods in matrix form (see Fatunla [26]). First we define the following vectors:(22)Yμ+1=yn+1,yn+2,yn+3,…,yn+kT,Yμ=yn-k+1,yn-k+2,yn-k+3,…,ynT,Fμ+1=fn+1,fn+2,fn+3,…,fn+kT,Fμ=fn-k+1,fn-k+2,fn-k+3,…,fnT,Gμ+1=gn+1,gn+2,gn+3,…,gn+kT,where yn+j=y(xn+jh),fn+j=f(xn+jh,y(xn+jh)), and gn+j=df(x,y(x))/dx|yn+jxn+j. The integration on the entire block will be compactly written as(23)A1Yμ+1=A0Yμ+hB0Fμ+hB1Fμ+1+h2C1Gμ+1,μ=0,1,…,which forms a nonlinear equation because of the implicit nature, and hence we employ the Newton iteration for the evaluation of the approximate solutions. We use Newton’s approach for the implementation of implicit schemes to get the following solution of the block:(24)Yμ+1i+1=Yμ+1i-A1-hB1∂Fμ+1∂Y-h2C1∂Gμ+1∂Y-1·A1Yμ+1-A0Yμ-hB0Fμ-hB1Fμ+1-h2C1Gμ+1.The k×k matrices A0, A1, B0, B1, and C1 are defined as follows.Case
k
=
2. Consider(25)A0=0-100,A1=-10-11,B0=0β^0,00β0,B1=β^1,0β^2,0β1β2,C1=0γ^n+2,00γn+2with βi, β^i,j, γn+k, γ^n+k,j, j=0, i=1,2, defined in methods (13) and (14).Case
k
=
3
. Consider(26)A0=00000-1000,A1=1-100-100-11,B0=00β^0,100β^0,000β0,B1=β^1,1β^2,1β^3,1β^1,0β^2,0β^3,0β1β2β3,C1=00γ^n+3,100γ^n+3,000γn+3,with βi, β^i,j, γn+k, γ^n+k,j, j=0,1, i=1,2,3, defined in methods (15), (16), and (17).Case
k
=
4
. Consider(27)A0=00000000000-10000,A1=10-1001-1000-1000-11,B0=000β^0,1000β^0,2000β^0,0000β0,B1=β^1,1β^2,1β^3,1β^4,1β^1,2β^2,2β^3,2β^4,2β^1,0β^2,0β^3,0β^4,0β1β2β3β4,C1=000γ^n+4,1000γ^n+4,2000γ^n+4,0000γn+4with βi, β^i,j, γn+k, γ^n+k,j, i=1,2,…,k and j=0,1,2,…,k-2, defined in methods (18) through (21).


## 2.1. CSDMM

In what follows, we state the CSDMM which has the ability to produce the ESDM:(3)Ux=αn+k-1xyn+k-1+h∑j=0kβjxfn+j+h2γn+kxgn+k,where αn+k-1(x), βj(x), and γn+k(x) are continuous coefficients. We assume that yn+j=U(xn+j) is the numerical approximation to the analytical solution y(xn+j), yn+j′=U′(xn+j) is the numerical approximation to the analytical solution y′(xn+j), fn+j=U′(xn+j) is an approximation to y′(xn+j), and gn+j=U′′(xn+j) is an approximation to y′′(xn+j), where fn+j=f(xn+j,yn+j), gn+j=(df(x,y(x))/dx)yn+jxn+j, j=0,1,2,…,k.We now define the following vectors and matrix used in the following theorem:(4)V=yn+k-1,fn,fn+1,fn+2,…,fn+k,gn+kT,Px=P1x,P2x,…,Pk+3xT,Wk+3,k+3=P1xn+k-1P2xn+k-1⋯Pk+3xn+k-1P1′xnP2′xn⋯Pk+3′xnP1′xn+1P2′xn+1⋯Pk+3′xn+1P1′xn+2P2′xn+2⋯Pk+3′xn+2⋮⋮⋱⋮P1′xn+kP2′xn+k⋯Pk+3′xn+kP1′′xn+kP2′′xn+k⋯Pk+3′′xn+k,where Pi(x)=xi-1, i=1,2,…,k+1, Pk+2x=sin⁡wx, and Pk+3x=cos⁡wx.Remark 1.
In the derivation of the ESDM,  the basesP(x)≡Pi(x)T with Pi(x)=xi-1, i=1,2,…,k+1, Pk+2(x)=sin(wx), and Pk+3(x)=cos(wx) are chosen because they are simple to analyze. Other possible bases (see Nguyen et al. [16] and Nguyen et al. [23]) include the following: (1)
{sin⁡(wx),cos⁡(wx),xsin(wx),xcos(wx),…,xnsin⁡(wx),xncos⁡(wx)};(2)
{sinx,cosx,…,sin⁡(nwx),cos⁡(nwx)};(3)
{sin⁡(w1x),cos⁡(w1x),…,sin⁡(wnx),cos⁡(wnx)};(4)
{x,…,wxn}∪{sin⁡(wx),cos⁡(wx),…,sin⁡(mwx),cos⁡(mwx)};(5)
{x,…,xn,exp⁡(±wx),xexp(±wx),…,xmexp⁡(±wx)};(6)
{x,…,wxn-1,wxn}.Theorem 2.
LetU(x) satisfy U(xn+j)=yn+j, U′(xn+j)=fn+j, and U′′(xn+j)=gn+j and let W be invertible; then method (3) is equivalent to(5)Ux=VTW-1TPx.The proof of the above theorem can be found in Jator et al. [24].Through interpolation ofU(x) at the point xn+k-1, collocation of U′(x) at the points xn+j, j=0,1,2,…,k, and collocation of U′′(x) at the point xn+k, we get the system(6)Uxn+k-1=yn+k-1,U′xn+j=fn+jj=0,1,2,…,k,U′′xn+k=gn+k.To solve this system we require that method (3) be defined by the assumed basis functions (7)αn+k-1=∑i=0k+3αiPix;hβjx=∑i=0k+3hβi,jPix,j=0,1,…,k;h2γn+k=∑i=0k+3h2γiPix,where the constants αi, βi,j, and γi are to be determined.


## 2.2. ESDM

The general second derivative formula for solving (2) using the k-step second derivative linear multistep method is of the form(8)∑j=0kαjyn+j=h∑j=0kβjfn+j+h2∑j=0kγjgn+j,where yn+j≈y(xn+jh), fn+j=f(xn+jh,y(xn+jh)) and fj=f(xj,yj), gn+j=df(x,y(x))/dx|yn+jxn+j; xn is a discrete point at node n; and αj, βj, and γj are parameters to be determined. It is worth noting that Enright’s method is a special case of (7). We solve (6) to get the coefficients αi, βi,j, and γi in (7) which are then used to obtain the continuous multistep method of Enright in the form(9)Ux=yn+k-1+h∑j=0kβjxfn+j+h2γn+kxgn+k.Evaluating (9) at x=xn+k and setting yn+k=U(xn+kh) yield the following Enright’s second derivative multistep method:(10)yn+k=yn+k-1+h∑j=0kβjfn+j+h2γn+kgn+k,whereas the (k-1) complementary methods(11)yn+i=yn+k-1+h∑j=0kβ^j,ifn+j+h2γ^n+k,ign+kare obtained by evaluating (9) at x=xn+i, i=0,1,2,…,k-2, with k≥2.We note that, in order to avoid the cancellations which might occur whenh is small, the use of the power series expansions of βj, γn+k, β^j,i, and γ^n+k,i is preferable (see Simos [25]).Case
k
=
1. This case has only the main method given by (10) with the coefficients defined by(12)β0=-csc⁡u/22-u+sin⁡u2u=13+u290+u42520+u675600+u82395008+691u1054486432000+Oh11,β1=-csc⁡u/22ucos⁡u-sin⁡u2u=23-u290-u42520-u675600-u82395008-691u1054486432000+Oh11,γn+1=-2+ucot⁡u/2u2=-16-u2360-u415120-u6604800-u823950080-691u10653837184000+Oh11.Case
k
=
2. The coefficients of the main method (10) and the complimentary method (11) are, respectively, defined by(13)β0=hcsc⁡u/22ucos⁡u/2-2sin⁡u/224uucos⁡u-sin⁡u=-148-u2360-13u457600-89u66048000-143203u8167650560000-126473u102724321600000+Oh11,β1=csc⁡u/22-2-3u2+2-u2cos⁡2u+4usin⁡u+2usin⁡2u8uucos⁡u-sin⁡u=512+u2720+13u450400+121u66048000+52133u841912640000+761473u1010897286400000+Oh11,β2=-hcsc⁡u/222-4+3u2cos⁡u+2+u2cos⁡2u+4usin⁡u8uucos⁡u-sin⁡u=2948+u2720-13u4403200-u6189000-5939u815240960000-255581u1010897286400000+Oh11,γn+2=-ucos⁡u/2-2sin⁡u/2sin⁡u/2uucos⁡u-sin⁡u=-18-u2240-13u467200-19u62016000-12979u827941760000-83437u103632428800000+Oh11,(14)β^0,0=-csc⁡u/22-2-u2+4+3u2cos⁡u-2cos⁡2u-2usin⁡2u8uucos⁡u-sin⁡u=-1748-u272-251u4403200-169u66048000-213203u8167650560000-161023u102724321600000+Oh11,β^1,0=csc⁡u/222+u2+-2+3u2cos⁡2u+4usin⁡u-6usin⁡2u8uucos⁡u-sin⁡u=-1112+17u2720+53u450400+281u66048000+87133u841912640000+1037873u1010897286400000+Oh11,β^2,0=-csc⁡u/224+-4+u2cos⁡u+u2cos⁡2u-2usin⁡2u8uucos⁡u-sin⁡u=1348-7u2720-173u4403200-u654000-135329u8167650560000-393781u1010897286400000+Oh11,γ^n+2,0=-ucos⁡u/2-2sin⁡u/2sin⁡u/2uucos⁡u-sin⁡u=-18-u2240-13u467200-19u62016000-12979u827941760000-83437u103632428800000+Oh11.Case
k
=
3. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined by(15)β0=csc⁡u/44-42-5u2-8-6+u2cos⁡u+-6+u2cos⁡2u+32usin⁡u-4usin⁡2u96uu+2ucos⁡u-3sin⁡u=71080+163u2151200+1529u413608000+1203457u6125737920000+5143273u87005398400000+364730953u106865290432000000+Oh11,β1=-csc⁡u/44-102-16u2-3-34+7u2cos⁡u+6cos⁡2u-6cos⁡3u+u2cos⁡3u+88usin⁡u-2usin⁡2u-4usin⁡3u96uu+2ucos⁡u-3sin⁡u=-120-11u28400-173u4756000-153049u66985440000-4786027u82724321600000-422813u103259872000000+Oh11,β2=csc⁡u/44-30-23u2-24cos⁡u+78-21u2cos⁡2u-24cos⁡3u+8u2cos⁡3u+34usin⁡u+52usin⁡2u-22usin⁡3u96uu+2ucos⁡u-3sin⁡u=1940-u2224-29u4302400+4187u62794176000+41551u899066240000+6204403u10152562009600000+Oh11,β3=-csc⁡u/4430-78+23u2cos⁡u+233+8u2cos⁡2u-18cos⁡3u-5u2cos⁡3u+68usin⁡u-22usin⁡2u96uu+2ucos⁡u-3sin⁡u=307540+71u215120+289u41360800+12391u61143072000+2957783u84903778880000+24651509u10686529043200000+Oh11,γn+3=csc⁡u/27ucos⁡u/2+5ucos⁡3u/2+12sin⁡u/2-sin⁡3u/212uu+2ucos⁡u-3sin⁡u=-19180-97u225200-491u42268000-285163u620956320000-7286749u88172964800000-67916227u101144215072000000+Oh11,(16)β^0,0=csc⁡u/24-12+2u2+9-4u2cos⁡u+26+7u2cos⁡2u⁡-9cos⁡3u+34usin⁡u-20usin⁡2u-6usin⁡3u48uu+2ucos⁡u-3sin⁡u=-43135-43u24725-193u4850500-2561u61964655000+1107763u83064861800000+8005337u10214540326000000+Oh11,β^1,0=-csc⁡u/24-15-4u2+34+5u2cos⁡u+15cos⁡2u-12cos⁡3u+7u2cos⁡3u-5usin⁡u+13usin⁡2u-19usin⁡3u24uu+2ucos⁡u-3sin⁡u=-75+29u21050+139u4189000+3683u6436590000-516289u8681080400000-473779u105297292000000+Oh11,β^2,0=csc⁡u/24-12+2u2+15cos⁡u+62+5u2cos⁡2u-15cos⁡3u+4u2cos⁡3u+44usin⁡u-28usin⁡2u-20usin⁡3u48uu+2ucos⁡u-3sin⁡u=-15-u235-17u418900-961u643659000-1367u86191640000+104537u104767562800000+Oh11,β^3,0=-csc⁡u/243+u2cos⁡u+-3+4u2cos⁡2u+u2cos⁡3u-usin⁡u-usin⁡2u-3usin⁡3u24uu+2ucos⁡u-3sin⁡u=-11135+19u21890+19u448600+1063u671442000+756881u81225944720000+2591419u1085816130400000+Oh11,γ^n+3,0=-2cot⁡u/22u+ucos⁡u-3sin⁡u3uu+2ucos⁡u-3sin⁡u=245-u21575-31u4283500-5927u6654885000-95237u8145945800000-3218641u1071513442000000+Oh11,(17)β^0,1=csc⁡u/24-30-u2+86+u2cos⁡u+-18+5u2cos⁡2u+16usin⁡u-20usin⁡2u96uu+2ucos⁡u-3sin⁡u=231080+467u2151200+3841u413608000+2702753u6125737920000+74202119u849037788800000+707967737u106865290432000000+Oh11,β^1,1=-csc⁡u/24-54-8u2+96+39u2cos⁡u-42cos⁡2u+5u2cos⁡3u-4usin⁡u-22usin⁡2u-8usin⁡3u96uu+2ucos⁡u-3sin⁡u=-920-59u28400-517u4756000-371321u66985440000-10263683u82724321600000-32697403u10127135008000000+Oh11,β^2,1=csc⁡u/2454+5u2-48cos⁡u+-6+39u2cos⁡2u-8u2cos⁡3u+50usin⁡u-76usin⁡2u+10usin⁡3u96uu+2ucos⁡u-3sin⁡u=-2940+u23360+59u4302400+53563u62794176000+1581469u81089728640000+1395817u1013869273600000+Oh11,β^3,1=-csc⁡u/2478+-96+5u2cos⁡u+29+4u2cos⁡2u-u2cos⁡3u-20usin⁡u-2usin⁡2u96uu+2ucos⁡u-3sin⁡u=83540+11u23024+281u41360800+14279u61143072000+3937807u84903778880000+36676261u10686529043200000+Oh11,γ^n+3,1=csc⁡u/2-13ucos⁡u/2+ucos⁡3u/2+24sin⁡u/212uu+2ucos⁡u-3sin⁡u=-11180-113u225200-739u42268000-474827u620956320000-12620021u88172964800000-119414483u101144215072000000+Oh11.Case
k
=
4. The coefficients of the main method (10) and the complimentary methods (11) are, respectively, defined by(18)β0=Nβ0384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-175760-251u2483840-2003u433868800-354659u662589542400-51104113u8102521670451200-600902219u1014353033863168000+Oh11,β1=Nβ1384u6ucos⁡2u+16sin⁡u-11sin⁡2u=145+29u230240+181u41058400+70811u63911846400+2645263u81601901100800+9724577u1069004970496000+Oh11,β2=Nβ2192u6ucos⁡2u+16sin⁡u-11sin⁡2u=-41480+11u25760-169u42822400-70207u65215795200-12382199u88543472537600-14328617u10108735105024000+Oh11,β3=Nβ3384u6ucos⁡2u+16sin⁡u-11sin⁡2u=4790-241u230240-187u4529200-72019u63911846400-3489379u83203802201600-64028929u10897064616448000+Oh11,β4=Nβ4384u6ucos⁡2u+16sin⁡u-11sin⁡2u=31335760+2719u2483840+10207u433868800+1216471u662589542400+142053797u8102521670451200+1494030511u1014353033863168000+Oh11,γn+4=Nγn+42u6ucos⁡2u+16sin⁡u-11sin⁡2u=-332-3u2896-37u4188160-529u638635520-195109u8189854945280-6348281u1079739077017600+Oh11with Nβi, i=0,1,…,4, and Nγn+4 defined in part A of Appendix 1 of the supplementary material (see Supplementary Material available online at http://dx.doi.org/10.1155/2015/343295). Consider  (19)β^0,0=Nβ^0,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-201640-23u22560-1019u43763200-20569u62318131200-1712483u83797098905600-6213083u10177197948928000+Oh11,β^1,0=Nβ^1,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-75+39u21120+359u4352800+4481u6144883200+260119u8177989011200+34002959u10299021538816000+Oh11,β^2,0=Nβ^2,064u6ucos⁡2u+16sin⁡u-11sin⁡2u=-99160-219u24480-417u4313600-6317u6193177600-40941u835158323200-12088171u10132898461696000+Oh11,β^3,0=Nβ^3,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=-910+29u21120+29u458800-809u6144883200-151769u8118659340800-3275539u1033224615424000+Oh11,β^4,0=Nβ^4,0128u6ucos⁡2u+16sin⁡u-11sin⁡2u=149640-51u217920+1013u411289600+37621u62318131200+16324541u811391296716800+530557669u104784344621056000+Oh11,γ^n+4,0=Nγ^n+4,02u6ucos⁡2u+16sin⁡u-11sin⁡2u=-332-3u2896-37u4188160-529u638635520-195109u8189854945280-6348281u1079739077017600+Oh11with Nβ^i,0, i=0,1,…,4, and Nγ^n+4,0 defined in part B of Appendix 1 of the supplementary material. Consider(20)β^0,1=Nβ^0,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=190+u2756+u410800+u6199584+691u82971987200+u10102643200+Oh11,β^1,1=Nβ^1,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1745-u2189-u42700-u649896-691u8742996800-u1025660800+Oh11,β^2,1=Nβ^2,124u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1915+u2126+u41800+u633264+691u8495331200+u1017107200+Oh11,β^3,1=Nβ^3,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1745-u2189-u42700-u649896-691u8742996800-u1025660800+Oh11,β^4,1=Nβ^4,148u6ucos⁡2u+16sin⁡u-11sin⁡2u=190+u2756+u410800+u6199584+691u82971987200+u10102643200+Oh11,γ^n+4,1=0with Nβ^i,1, i=0,1,…,4, defined in part C of Appendix 1 of the supplementary material. Consider(21)β^0,2=Nβ^0,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-111920-169u2161280-12281u4101606400-2158889u6187768627200-7853429u87886282342400-272561573u103312238583808000+Oh11,β^1,2=Nβ^1,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=7135+31u210080+3701u49525600+1346483u635206617600+16159921u84805703302400+752918951u102691193849344000+Oh11,β^2,2=Nβ^2,2192u6ucos⁡2u+16sin⁡u-11sin⁡2u=-83160-17u213440-841u42822400-534077u615647385600-9061597u82847824179200-974213257u103588258465792000+Oh11,β^3,2=Nβ^3,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=-1930-59u210080-589u41587600-278329u611735539200-5135051u83203802201600-28211849u10244653986304000+Oh11,β^4,2=Nβ^4,2384u6ucos⁡2u+16sin⁡u-11sin⁡2u=183117280+821u2161280+122327u4304819200+17519503u6563305881600+743666159u8307565011353600+8152441741u1043059101589504000+Oh11,γ^n+4,2=Nγ^n+4,22u6ucos⁡2u+16sin⁡u-11sin⁡2u=-11288-3u2896-1447u45080320-218147u69388431360-9544999u85126083522560-11813881u1079739077017600+Oh11with Nβ^i,2, i=0,1,…,4, and Nγ^n+4,2 defined in part D of Appendix 1 of the supplementary material.


## 2.3. Block Specification and Implementation of the Methods

We consider a general procedure for the block implementation of the methods in matrix form (see Fatunla [26]). First we define the following vectors:(22)Yμ+1=yn+1,yn+2,yn+3,…,yn+kT,Yμ=yn-k+1,yn-k+2,yn-k+3,…,ynT,Fμ+1=fn+1,fn+2,fn+3,…,fn+kT,Fμ=fn-k+1,fn-k+2,fn-k+3,…,fnT,Gμ+1=gn+1,gn+2,gn+3,…,gn+kT,where yn+j=y(xn+jh),fn+j=f(xn+jh,y(xn+jh)), and gn+j=df(x,y(x))/dx|yn+jxn+j. The integration on the entire block will be compactly written as(23)A1Yμ+1=A0Yμ+hB0Fμ+hB1Fμ+1+h2C1Gμ+1,μ=0,1,…,which forms a nonlinear equation because of the implicit nature, and hence we employ the Newton iteration for the evaluation of the approximate solutions. We use Newton’s approach for the implementation of implicit schemes to get the following solution of the block:(24)Yμ+1i+1=Yμ+1i-A1-hB1∂Fμ+1∂Y-h2C1∂Gμ+1∂Y-1·A1Yμ+1-A0Yμ-hB0Fμ-hB1Fμ+1-h2C1Gμ+1.The k×k matrices A0, A1, B0, B1, and C1 are defined as follows.Case
k
=
2. Consider(25)A0=0-100,A1=-10-11,B0=0β^0,00β0,B1=β^1,0β^2,0β1β2,C1=0γ^n+2,00γn+2with βi, β^i,j, γn+k, γ^n+k,j, j=0, i=1,2, defined in methods (13) and (14).Case
k
=
3
. Consider(26)A0=00000-1000,A1=1-100-100-11,B0=00β^0,100β^0,000β0,B1=β^1,1β^2,1β^3,1β^1,0β^2,0β^3,0β1β2β3,C1=00γ^n+3,100γ^n+3,000γn+3,with βi, β^i,j, γn+k, γ^n+k,j, j=0,1, i=1,2,3, defined in methods (15), (16), and (17).Case
k
=
4
. Consider(27)A0=00000000000-10000,A1=10-1001-1000-1000-11,B0=000β^0,1000β^0,2000β^0,0000β0,B1=β^1,1β^2,1β^3,1β^4,1β^1,2β^2,2β^3,2β^4,2β^1,0β^2,0β^3,0β^4,0β1β2β3β4,C1=000γ^n+4,1000γ^n+4,2000γ^n+4,0000γn+4with βi, β^i,j, γn+k, γ^n+k,j, i=1,2,…,k and j=0,1,2,…,k-2, defined in methods (18) through (21).


## 3. Error Analysis and Stability

### 3.1. Local Truncation Error (LTE)

Suppose that method (10) is associated with a linear difference operator:(28)Lyxn:h=yx+kh-yx+k-1h-h∑j=0kβjy′x+jh-h2γn+ky′′x+kh,where y(x) is an arbitrary smooth function. Then L[y(xn;h)] is called the local truncation error at xn+k if y represents a solution of the IVP (2). By a Taylor series expansion of y(x+jh), y′(x+jh), and y′′(x+jh),j=0,1,2,…,k, we have (29)Lyxn;h=C0yx+C1hy′x+C2h22!y′′x+⋯+Cqhqq!yqx+⋯,where C0=∑j=0kαj, C1=1-∑j=0kβj, and C2=(1/2!)(-(k-1)2+k2)-∑j=0kjβj-γn+k,…,  Cq=(1/q!)(-(k-1)2+kq)-(1/q-1!)∑j=0kjq-1βj-(1/q-2!)kq-2γn+k.Method (10) is said to be of order p if C0=C1=⋯=Cp=0, Cp+1≠0 (see [3]).Theorem 3.
Thek-step method (10) ESDM has a local truncation error (LTE) of (30)Ck+3hk+3w2yk+1xn+yk+3xn+Ohk+4.Proof.
We consider a Taylor series expansion ofyn+j,yx+jh,yn+j′,y′x+jh,yn+j′′,y′′x+jh and assume that y(xn+j)=yn+j, y′(xn+j)=fn+j, y′′(xn+k)=gn+k. Then by substituting these into method (10) and simplifying we get that(31)LTE=yxn+k-yn+k=Ck+3hk+3w2yk+1xn+yk+3xn+Ohk+4,where the values of Ck+3 are given in Table 1.Table 1
The local truncation error for various cases.
Case(k)
Method
Order(p)
Error constant(Cp+1)
1
(12)
3
1/7
2
(13)
4
7/1440
(14)
4
23/1440
3
(15)
5
17/7200
(16)
5
−2/225
(17)
5
11/2400
4
(18)
6
41/30240
(19)
6
11/1120
(20)
6
−1/756
(21)
6
19/10080Define the local truncation error of (23) as follows:(32)LZx;h=Zμ+1-AZμ+hBF¯μ+hDF¯μ+1+h2CG¯μ+1,where(33)Zμ+1=yxn+1,yxn+2,yxn+3,…,yxn+kT,Zμ=yxn-k+1,yxn-k+2,yxn-k+3,…,yxnT,F¯μ+1=fxn+1,fxn+2,fxn+3,…,fxn+kT,F¯μ=fxn-k+1,fxn-k+2,fxn-k+3,…,fxnT,G¯μ+1=gxn+1,gxn+2,gxn+3,…,gxn+kT,LZx;h=L1Zx;h,L2Zx;h,L3Zx;h,…,LkZx;hTa linear difference operator. Assuming that Z(x) is sufficiently differentiable, we can expand the terms in (23) as a Taylor series about x to obtain the expression for the local truncation error (34)LZx;h=C0Zx+C1hZ′x+⋯+CqhqZqx+⋯,where Cq=(C1,q,C2,q,…,Ck,q)T, q=0,1,…, are constant coefficients (see Ehigie et al. [27]).Definition 4.
The block method (23) has algebraic order p≥1, provided there exists a constant Cp+1≠0 such that the local truncation error Eμ satisfies Eμ=Cp+1hp+1+O(hp+2), where · is the maximum norm.Remark 5.
(i) The local truncation error constants(Cp+1) of the block method (23) are presented in Table 1. (ii) From the local truncation error constant computation, it follows that the order (p) of the method (23) is p=k+2.


### 3.2. Stability

Definition 6.
The block method (23) is zero-stable, provided the roots of the first characteristic polynomial have modulus less than or equal to one and those of modulus one are simple (see [2]).Remark 7.
Observe that, from the first characteristic polynomialρk(R) of the block method (23) specified by ρk(R)= det∑i=01AiRi=0, we obtain -Rk-1(1+R)=0. Thus the roots Rj, j=1,2,…,k of ρk(R) satisfy |Rj|≤1, j=1,2,…,k and, for those roots with |Rj|=1, the roots are simple.Definition 8.
The block method (23) is consistent if it has order p>1 (see [26]).Remark 9.
The block method (23) is consistent as it has order p>1 and zero-stable; hence it is convergent since zero-stability + consistency = convergence.Proposition 10.
The block method (23) applied to the test equations y′=λy and y′′=λ2y yields(35)Yμ+1=Mq;uYμ,Mq;u=A1-qB1-q2C1-1A0+qB0,q=λh2,u=wh.Proof.
We begin by applying (23) to the test equations y′=λy and y′′=λ2y which are expressed as f(x,y)=λy and g(x,y)=λ2y, respectively; letting q=hλ and u=wh, we obtain a linear equation which is used to solve for Yμ+1 with (23) as a consequence.Remark 11.
The rational functionM(q;u) is called the stability function which determines the stability of the method.Definition 12.
A region of stability is a region in theq-u plane, in which |M(q;u)|≤1.Corollary 13.
Method (23) has M(q;u) specified in Appendix 2 of the supplementary material.Remark 14.
In theq-u plane the ESDM (23) is stable for q≤0, and u∈[-2π,2π], since |M(q;u)|≤1, q≤0.Remark 15.
Figures1, 3, 5, and 7 are plots of the stability region of M(q;u) for case k=1,2,…,4, respectively. We note from these figures that the stability region of M(q;u) for k=1,2,…,4 includes the entire left side of the complex plane. Figures 2, 4, 6, and 8 show the respective zeros and poles of M(q;u).Figure 1
k
=
1.Figure 2
k
=
1: M(q;u) has zeros (□) and no poles (+) in C-.Figure 3
k
=
2.Figure 4
k
=
2: M(q;u) has zeros (□) and no poles (+) in C-.Figure 5
k
=
3.Figure 6
k
=
3: M(q;u) has zeros (□) and poles (+) in C-.Figure 7
k
=
4.Figure 8
k
=
4: M(q;u) has zeros (□) and poles (+) in C-.


### 3.3. Implementation

The ESDM (10) is implemented in the spirit of Ngwane et al. in [28, 29] to solve (2) without requiring starting values and predictors. For instance, if we let n=0 in (10), then y1,y2,…,yk are obtained on the subinterval [x0,xk], as y0 is known from the IVP. If n=1, then yk+1,yk+2,…,y2k are obtained on the subinterval [xk,x2k], as yk is known from the previous computation and so on, until we reach the final subinterval [xN-1,xN]. Note that, for linear problems, we solve (2) directly using the feature Solve[] in Mathematica 8.0, while for nonlinear problems we use Newton’s method enhanced by the feature FindRoot[].


## 3.1. Local Truncation Error (LTE)

Suppose that method (10) is associated with a linear difference operator:(28)Lyxn:h=yx+kh-yx+k-1h-h∑j=0kβjy′x+jh-h2γn+ky′′x+kh,where y(x) is an arbitrary smooth function. Then L[y(xn;h)] is called the local truncation error at xn+k if y represents a solution of the IVP (2). By a Taylor series expansion of y(x+jh), y′(x+jh), and y′′(x+jh),j=0,1,2,…,k, we have (29)Lyxn;h=C0yx+C1hy′x+C2h22!y′′x+⋯+Cqhqq!yqx+⋯,where C0=∑j=0kαj, C1=1-∑j=0kβj, and C2=(1/2!)(-(k-1)2+k2)-∑j=0kjβj-γn+k,…,  Cq=(1/q!)(-(k-1)2+kq)-(1/q-1!)∑j=0kjq-1βj-(1/q-2!)kq-2γn+k.Method (10) is said to be of order p if C0=C1=⋯=Cp=0, Cp+1≠0 (see [3]).Theorem 3.
Thek-step method (10) ESDM has a local truncation error (LTE) of (30)Ck+3hk+3w2yk+1xn+yk+3xn+Ohk+4.Proof.
We consider a Taylor series expansion ofyn+j,yx+jh,yn+j′,y′x+jh,yn+j′′,y′′x+jh and assume that y(xn+j)=yn+j, y′(xn+j)=fn+j, y′′(xn+k)=gn+k. Then by substituting these into method (10) and simplifying we get that(31)LTE=yxn+k-yn+k=Ck+3hk+3w2yk+1xn+yk+3xn+Ohk+4,where the values of Ck+3 are given in Table 1.Table 1
The local truncation error for various cases.
Case(k)
Method
Order(p)
Error constant(Cp+1)
1
(12)
3
1/7
2
(13)
4
7/1440
(14)
4
23/1440
3
(15)
5
17/7200
(16)
5
−2/225
(17)
5
11/2400
4
(18)
6
41/30240
(19)
6
11/1120
(20)
6
−1/756
(21)
6
19/10080Define the local truncation error of (23) as follows:(32)LZx;h=Zμ+1-AZμ+hBF¯μ+hDF¯μ+1+h2CG¯μ+1,where(33)Zμ+1=yxn+1,yxn+2,yxn+3,…,yxn+kT,Zμ=yxn-k+1,yxn-k+2,yxn-k+3,…,yxnT,F¯μ+1=fxn+1,fxn+2,fxn+3,…,fxn+kT,F¯μ=fxn-k+1,fxn-k+2,fxn-k+3,…,fxnT,G¯μ+1=gxn+1,gxn+2,gxn+3,…,gxn+kT,LZx;h=L1Zx;h,L2Zx;h,L3Zx;h,…,LkZx;hTa linear difference operator. Assuming that Z(x) is sufficiently differentiable, we can expand the terms in (23) as a Taylor series about x to obtain the expression for the local truncation error (34)LZx;h=C0Zx+C1hZ′x+⋯+CqhqZqx+⋯,where Cq=(C1,q,C2,q,…,Ck,q)T, q=0,1,…, are constant coefficients (see Ehigie et al. [27]).Definition 4.
The block method (23) has algebraic order p≥1, provided there exists a constant Cp+1≠0 such that the local truncation error Eμ satisfies Eμ=Cp+1hp+1+O(hp+2), where · is the maximum norm.Remark 5.
(i) The local truncation error constants(Cp+1) of the block method (23) are presented in Table 1. (ii) From the local truncation error constant computation, it follows that the order (p) of the method (23) is p=k+2.


## 3.2. Stability

Definition 6.
The block method (23) is zero-stable, provided the roots of the first characteristic polynomial have modulus less than or equal to one and those of modulus one are simple (see [2]).Remark 7.
Observe that, from the first characteristic polynomialρk(R) of the block method (23) specified by ρk(R)= det∑i=01AiRi=0, we obtain -Rk-1(1+R)=0. Thus the roots Rj, j=1,2,…,k of ρk(R) satisfy |Rj|≤1, j=1,2,…,k and, for those roots with |Rj|=1, the roots are simple.Definition 8.
The block method (23) is consistent if it has order p>1 (see [26]).Remark 9.
The block method (23) is consistent as it has order p>1 and zero-stable; hence it is convergent since zero-stability + consistency = convergence.Proposition 10.
The block method (23) applied to the test equations y′=λy and y′′=λ2y yields(35)Yμ+1=Mq;uYμ,Mq;u=A1-qB1-q2C1-1A0+qB0,q=λh2,u=wh.Proof.
We begin by applying (23) to the test equations y′=λy and y′′=λ2y which are expressed as f(x,y)=λy and g(x,y)=λ2y, respectively; letting q=hλ and u=wh, we obtain a linear equation which is used to solve for Yμ+1 with (23) as a consequence.Remark 11.
The rational functionM(q;u) is called the stability function which determines the stability of the method.Definition 12.
A region of stability is a region in theq-u plane, in which |M(q;u)|≤1.Corollary 13.
Method (23) has M(q;u) specified in Appendix 2 of the supplementary material.Remark 14.
In theq-u plane the ESDM (23) is stable for q≤0, and u∈[-2π,2π], since |M(q;u)|≤1, q≤0.Remark 15.
Figures1, 3, 5, and 7 are plots of the stability region of M(q;u) for case k=1,2,…,4, respectively. We note from these figures that the stability region of M(q;u) for k=1,2,…,4 includes the entire left side of the complex plane. Figures 2, 4, 6, and 8 show the respective zeros and poles of M(q;u).Figure 1
k
=
1.Figure 2
k
=
1: M(q;u) has zeros (□) and no poles (+) in C-.Figure 3
k
=
2.Figure 4
k
=
2: M(q;u) has zeros (□) and no poles (+) in C-.Figure 5
k
=
3.Figure 6
k
=
3: M(q;u) has zeros (□) and poles (+) in C-.Figure 7
k
=
4.Figure 8
k
=
4: M(q;u) has zeros (□) and poles (+) in C-.


## 3.3. Implementation

The ESDM (10) is implemented in the spirit of Ngwane et al. in [28, 29] to solve (2) without requiring starting values and predictors. For instance, if we let n=0 in (10), then y1,y2,…,yk are obtained on the subinterval [x0,xk], as y0 is known from the IVP. If n=1, then yk+1,yk+2,…,y2k are obtained on the subinterval [xk,x2k], as yk is known from the previous computation and so on, until we reach the final subinterval [xN-1,xN]. Note that, for linear problems, we solve (2) directly using the feature Solve[] in Mathematica 8.0, while for nonlinear problems we use Newton’s method enhanced by the feature FindRoot[].


## 4. Numerical Illustration

In this section we consider some standard problems: stiff, oscillatory, linear, and nonlinear systems that appear in the literature to experimentally illustrate the accuracy and efficiency of the ESDM (10) which is implemented in block form. The ESDM for k=1,2,3, and 4 as early stated are denoted by EM1, EM2, EM3, and EM4, respectively. Our numerical examples test this family of methods. We include examples of second-order IVPs and it would be pertinent to mention here that there are methods specifically designed for this type of problems. In this paper, all the numerical experiments are carried out with fixed h and ω, assuming that ω is known. This allows us to compute the coefficients of the ESDM once for all integration. Some of the methods of orders 4 and 6 in the literature have been compared to EM2 and EM4, respectively. We find the approximate solution on the partition πN, and we give the errors at the endpoints calculated as Error =yN-y(xN). We denote the Max|yN-y(xN)| by Err, the number of steps by N, and the number of function evaluations by NFEs. We will write an error of the form Err =q×10-r as q(-r).Example 16.
We consider the following inhomogeneous IVP by Simos [25]:(36)y′′=-100y+99sin⁡x,y0=1,y′0=11,x∈0,1000,where the analytic solution is given by y(x)=cos(10x)+sin(10x)+sin(x).EM2 is fourth-order and hence comparable to the exponentially fitted method by Simos [25] which is also of fourth-order. PC1 and PC2 denote the predictor-corrector mode for k=1 and k=2, respectively. The efficiency curves in Figure 9 show the computational efficiency of the two methods (Simos and EM2) by considering the NFEs over N integration steps for each method. Hence for this example, EM2 performs better than Simos. We see from Table 2 that ESDM is efficient for each case.Table 2
Results withω=10, for Example 16.
N
ESDM
Simos [25]
EM1Err
EM2Err
EM3Err
EM4Err
PC1Err
PC2Err
SimosErr
1000
1.2 (−4)
3.9 (−3)
2.1 (−3)
5.8 (−1)
5.11
4.24
1.4 (−1)
2000
3.7 (−2)
7.7 (−3)
3.9 (−5)
1.7 (−4)
2.49
8.42
3.5 (−2)
4000
4.9 (−4)
2.3 (−3)
2.3 (−4)
8.4 (−5)
2.76 (−2)
1.835 (1)
1.1 (−3)
8000
2.3 (−5)
3.9 (−5)
1.9 (−6)
3.4 (−7)
2.83 (−2)
3.75 (1)
8.4 (−5)
16000
6.8 (−6)
1.4 (−6)
3.4 (−8)
2.1 (−10)
4.33 (−3)
7.47 (1)
5.5 (−6)
32000
1.0 (−6)
5.3 (−8)
2.6 (−12)
3.1 (−11)
3.79 (−4)
1.51 (2)
3.5 (−7)Figure 9
Efficiency curves for Example16.Example 17.
We consider the nonlinear Duffing equation which was also solved by Ixaru and Vanden Berghe [30]:(37)y′′+y+y3=Bcos⁡Ωx,y0=C0,y′0=0,x∈0,300.The analytic solution is given by y(x)=C1cos(Ωx)+C2cos(3Ωx)+C3cos(5Ωx)+C4cos(7Ωx), where Ω=1.01, B=0.002, C0=0.200426728069, C1=0.200179477536, C2=0.246946143×10-3, C3=0.304016×10-6, C4=0.374×10-9, and ω=1.01.We compare the errors produced by our EM2 with the fourth-order methods by Ixaru and Vanden Berghe [30]. We see from Table 3 that the results produced by ESDM on Example 17 are very good. In fact, EM2 produces results that are better than Simos’ method used in [30], as it produces better error magnitude while using less number of steps and fewer number of function evaluations. This example once more shows us that the ESDM produces good results and in particular EM2 is very competitive to the method used by Ixaru and Vanden Berghe [30].Table 3
Results withω=1.01, for Example 17.
N
ESDM
Ixaru and Vanden Berghe [30]
EM1Err
EM2Err
EM3Err
EM4Err
SimosErr
Ixaru and Vanden Err
300
1.1 (−4)
2.8 (−4)
4.7 (−1)
5.7 (−4)
1.7 (−3)
1.1 (−3)
600
1.8 (−5)
2.3 (−5)
1.9 (−5)
4.5 (−6)
1.9 (−4)
5.4 (−5)
1200
2.7 (−6)
1.3 (−6)
3.3 (−7)
2.9 (−7)
1.4 (−5)
1.9 (−6)
2000
6.2 (−7)
1.6 (−7)
1.6 (−8)
1.2 (−8)
—
—
2400
13.7 (−7)
5.8 (−8)
1.1 (−8)
3.9 (−9)
8.7 (−7)
6.2 (−8)
3000
1.9 (−7)
1.2 (−8)
3.9 (−9)
1.1 (−9)
—
—
4800
4.8 (−8)
7.8 (−10)
4.0 (−10)
4.1 (−11)
—
—Example 18 (a nearly sinusoidal problem).
We consider the following IVP on the range0≤t≤10 (see the study by Nguyen et al. [16]): (38)y1′=-2y1+y2+sin⁡t,y10=2,y2′=-β+2y1+β+1y2+sin⁡t-cos⁡t,y20=3.We choose β=-3 and β=-1000 in order to illustrate the phenomenon of stiffness. Given the initial conditions y1(0)=2 and y2(0)=3, the exact solution is β-independent and is given by(39)Exact:y1t=2exp⁡-t+sin⁡t,y2t=2exp⁡-t+cos⁡t.We choose this example to demonstrate the performance of ESDM on stiff problems. We compute the solutions to Example18 with β=-3,-1000. We compare EM4 of order six to the method by Nguyen et al. [16] which is also of order six. For both β=-3 and β=-1000, EM4 clearly obtains better absolute errors compared to Nguyen et al. [16]. This efficiency is achieved using fewer number of steps and less number of function evaluations than Nguyen et al. [16].Example 19.
Consider the given two-body problem which was solved by Ozawa [15]: (40)y1′′=-y1r3,y2′′=-y2r3,r=y12+y22,y10=1-e,y1′0=0,y20=0,y2′0=1+e1-e,x∈0,50π,ω=1,where e, 0≤e<1, is an eccentricity. The exact solution of this problem is y1x=cosk-e, y2x=1-e2sin⁡(k), where k is the solution of Kepler’s equation k=x+esin⁡(k).Table5 contains the results obtained using the ESDM for k=1,2,3,4. These results are compared with the explicit singly diagonally implicit Runge-Kutta (ESDIRK) and the functionally fitted ESDIRK (FESDIRK) methods given in Ozawa [15]. In terms of accuracy, Table 5 clearly shows that ESDM performs better than those in Ozawa [15].


### 4.1. A Predictor-Corrector Mode Implementation of ESDM

We can implement our ESDM in a predictor-corrector (PC) mode. The predictor fork=1,2,3,4 is given below while the corrector for each case is given by the main method (10). PC1 and PC2 denote the PC mode for k=1 and 2, respectively:k=1: (41)yn+1=yn+hsin⁡uufn+2h2sin⁡u/22u2gn,k=2:(42)yn+2=yn+1-hcsc⁡u/22-u+sin⁡u2ufn+hcsc⁡u/22-ucos⁡u+sin⁡u2ufn+1+h21-3cos⁡u/22+ucot⁡u/2+sin⁡u/22u2·gn+1,k=3,4:(43)yn+k=yn+k-1+h∑i=0k-1β¯ifn+i+h2γ¯n+k-1gn+k-1.The coefficients β¯i, i=0,1,…,k-1, and γ¯n+k-1 for k=3 are given below while those for k=4 are given in Appendix 3 of the supplementary material. We observe that the PC implementation performs poorly relative to the block implementation; see Table 2. Consider(44)β¯0=-csc⁡u/222-3u2-2cos⁡u+-2+u2cos⁡u+2cos2u+4usin⁡u8uucos⁡u-sin⁡u;β¯1=csc⁡u/222-5u2-2cos⁡u+-2+u2cos⁡2u+2cos⁡3u+4usin⁡u+2usin⁡2u8uucos⁡u-sin⁡u;β¯2=-csc⁡u/222-5u2cos⁡u+-2+3u2cos⁡2u+2-cos⁡2u+cos⁡3u+2usin⁡u8uucos⁡u-sin⁡u;γ¯n+2=csc⁡u/22-2+6cos⁡u-6cos⁡2u+2cos⁡3u-6usin⁡u+3usin⁡2u8uucos⁡u-sin⁡u.


### 4.2. Estimating the Frequency

Though we are mainly interested in problems whereω is taken as the exact frequency of the analytical solution and ω is known in advance, it is important to note that the exact frequency may be unknown for some problems. A preliminary testing indicates that a good estimate of the frequency can be obtained by demanding that the LTE in Theorem 3 equals zero and solving for the frequency. That is, solve for ω given that Ck+3hk+3ω2y(k+1)(xn)+y(k+3)(xn)=0, where y(j), j=k+1,k+3, denotes derivatives. We used this procedure to estimate ω for the problem given in Example 16 and obtained ω≈±9.999996, which approximately gives the known frequency ω=10. Hence, this procedure is interesting and will be seriously considered in our future research.If a problem has multiple frequencies, thenω is approximatively calculated so that it is an indicative frequency (see Nguyen et al. [16]). We note that estimating the frequency when it is unknown as well as finding the frequency for problems for which the frequency varies over time is very challenging. This challenge and the choice of the frequency in trigonometrically fitted methods have grown in interest. Existing references on how to estimate the frequency and the choice of the frequency include Vanden Berghe et al. [31] and Ramos and Vigo-Aguiar [32].


## 4.1. A Predictor-Corrector Mode Implementation of ESDM

We can implement our ESDM in a predictor-corrector (PC) mode. The predictor fork=1,2,3,4 is given below while the corrector for each case is given by the main method (10). PC1 and PC2 denote the PC mode for k=1 and 2, respectively:k=1: (41)yn+1=yn+hsin⁡uufn+2h2sin⁡u/22u2gn,k=2:(42)yn+2=yn+1-hcsc⁡u/22-u+sin⁡u2ufn+hcsc⁡u/22-ucos⁡u+sin⁡u2ufn+1+h21-3cos⁡u/22+ucot⁡u/2+sin⁡u/22u2·gn+1,k=3,4:(43)yn+k=yn+k-1+h∑i=0k-1β¯ifn+i+h2γ¯n+k-1gn+k-1.The coefficients β¯i, i=0,1,…,k-1, and γ¯n+k-1 for k=3 are given below while those for k=4 are given in Appendix 3 of the supplementary material. We observe that the PC implementation performs poorly relative to the block implementation; see Table 2. Consider(44)β¯0=-csc⁡u/222-3u2-2cos⁡u+-2+u2cos⁡u+2cos2u+4usin⁡u8uucos⁡u-sin⁡u;β¯1=csc⁡u/222-5u2-2cos⁡u+-2+u2cos⁡2u+2cos⁡3u+4usin⁡u+2usin⁡2u8uucos⁡u-sin⁡u;β¯2=-csc⁡u/222-5u2cos⁡u+-2+3u2cos⁡2u+2-cos⁡2u+cos⁡3u+2usin⁡u8uucos⁡u-sin⁡u;γ¯n+2=csc⁡u/22-2+6cos⁡u-6cos⁡2u+2cos⁡3u-6usin⁡u+3usin⁡2u8uucos⁡u-sin⁡u.


## 4.2. Estimating the Frequency

Though we are mainly interested in problems whereω is taken as the exact frequency of the analytical solution and ω is known in advance, it is important to note that the exact frequency may be unknown for some problems. A preliminary testing indicates that a good estimate of the frequency can be obtained by demanding that the LTE in Theorem 3 equals zero and solving for the frequency. That is, solve for ω given that Ck+3hk+3ω2y(k+1)(xn)+y(k+3)(xn)=0, where y(j), j=k+1,k+3, denotes derivatives. We used this procedure to estimate ω for the problem given in Example 16 and obtained ω≈±9.999996, which approximately gives the known frequency ω=10. Hence, this procedure is interesting and will be seriously considered in our future research.If a problem has multiple frequencies, thenω is approximatively calculated so that it is an indicative frequency (see Nguyen et al. [16]). We note that estimating the frequency when it is unknown as well as finding the frequency for problems for which the frequency varies over time is very challenging. This challenge and the choice of the frequency in trigonometrically fitted methods have grown in interest. Existing references on how to estimate the frequency and the choice of the frequency include Vanden Berghe et al. [31] and Ramos and Vigo-Aguiar [32].


## 5. Conclusion

In this paper, we have proposed a family of Enright methods using trigonometric bases for solving stiff and oscillatory IVPs. The ESDM is zero-stable and produced good results on stiff IVPs. This method has the advantages of being self-starting and having good accuracy properties. ESDM has order(k+2) similar to that in Enright [22]. We have presented representative numerical examples that are linear, nonlinear, stiff, and highly oscillatory. The need that the frequency be known in advance might be a shortcoming, yet these examples show that the ESDM is not only promising but more accurate and efficient than those in Nguyen et al. [16], Simos [25], Ixaru and Vanden Berghe [30], and Ozawa [15]. Details of the numerical results are displayed in Tables 2, 3, 4, and 5, and the efficiency curves are presented in Figures 9, 10, 11, 12, and 13. Our future research will incorporate a technique for accurately estimating the frequency as suggested in Section 4.2 as well as implementing the method in a variable step mode.Table 4
Results withω=1, for Example 18.
ESDM
Nguyen et al. [16]
Forβ=-3 we have
EM1
EM2
EM3
EM4
Nguyen
N
6
6
6
6
—
Err
6.6 (−5)
3.8 (−5)
7.1 (−5)
1.2 (−4)
—
N
10
10
10
10
10
Err
1.9 (−5)
1.3 (−6)
4.1 (−6)
8.1 (−7)
5.4 (−6)
N
27
27
27
27
19
Err
1.2 (−6)
8.2 (−8)
3.1 (−8)
4.3 (−9)
8.3 (−8)
N
32
32
32
32
23
Err
7.1 (−7)
6.3 (−8)
1.3 (−8)
1.9 (−9)
4.5 (−4)
Forβ=-1000 we have
EM1
EM2
EM3
EM4
Nguyen
N
6
6
6
6
13
Err
6.6 (−5)
3.8 (−5)
7.1 (−5)
1.3 (−5)
1.0 (−6)
N
16
16
16
16
16
Err
5.3 (−6)
6.2 (−7)
1.8 (−7)
2.5 (−8)
1.6 (−7)
N
20
20
24
24
21
Err
2.8 (−6)
3.2 (−7)
5.3 (−8)
6.5 (−9)
7.0 (−8)Table 5
Results withω=1, e=0.005, for Example 19.
ESDM
FESDIRK
ESDIRK
EM1
EM2
EM3
EM4
EM5
FESDIRK4 (3)
ESDIRK4 (3)
N
300
600
600
600
1200
381
884
Err
2.4 (−3)
3.0 (−4)
1.2 (−4)
2.6 (−1)
1.2 (−7)
1.4 (−3)
9.4 (−3)
N
600
1200
1200
1200
2000
680
1573
Err
1.8 (−3)
9.8 (−6)
4.5 (−6)
4.6 (−7)
2.4 (−9)
1.7 (−4)
6.2 (−4)
N
1200
2000
2000
2000
2400
1207
2796
Err
2.8 (−4)
7.5 (−7)
4.0 (−7)
1.3 (−8)
1.1 (−9)
1.8 (−5)
4.4 (−5)
N
2000
3200
3200
3200
3200
2144
4970
Err
6.2 (−5)
7.3 (−8)
3.9 (−8)
4.9 (−10)
1.6 (−10)
1.9 (−6)
3.4 (−6)
N
4000
4800
4800
4800
4800
3806
8833
Err
7.8 (−6)
9.6 (−9)
5.2 (−9)
2.9 (−11)
9.2 (−12)
1.9 (−7)
2.8 (−7)
N
8000
8000
8000
8000
8000
6762
15706
Err
9.7 (−7)
7.5 (−10)
4.0 (−10)
8.8 (−13)
3.6 (−13)
2.0 (−8)
2.5 (−8)Figure 10
Efficiency curves for Example17.Figure 11
Efficiency curves for Example18 with β=-3.Figure 12
Efficiency curves for Example18 with β=-1000.Figure 13
Efficiency curves for Example19.



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*Source: 343295-2015-06-08.xml*