In this study, we consider a linear nonhomogeneous differential equation with variable coefﬁcients and variable delays and present a novel matrix-collocation method based on Morgan–Voyce polynomials to obtain the approximate solutions under the initial conditions. The method reduces the equation with variable delays to a matrix equation with unknown Morgan– Voyce coefﬁcients. Thereby, the solution is obtained in terms of Morgan–Voyce polynomials. In addition, two test problems together with error analysis are performed to illustrate the accuracy and applicability of the method; the obtained results are scrutinized and interpreted by means of tables and ﬁgures. Keywords Morgan–Voyce polynomials Matrix method Collocation method Delay differential equation Variable delay Introduction including production of blood cells [1, 14, 23, 25, 27, 28, 34, 37]. In this paper, we consider nonhomogeneous differential In the case of bounded delays, many authors using equation with variable delays in the form [3, 5, 10, 12, standard techniques [3, 10, 20, 27, 34, 37] have studied the 23, 30, 37, 38]. asymptotic behavior of solutions, the asymptotic stability m in equations and the existence of positive periodic solutions y ðÞ t ¼ P ðtÞþ P ðtÞyðtÞþ P ðtÞyt s ðtÞ of delay equations. However, most of the mentioned type o 1 j j ð1Þ j¼2 delay equations have not analytical and numerical solu- tions; therefore, numerical methods are required to obtain under the initial condition yðaÞ¼ k, where the coefﬁcients approximate solutions. For this purpose, by means of the P ðtÞ and the delays s are continuous functions on the j j matrix method based on collocation points which have interval 0 a j b and the delays are nonnegative, been given by Sezer and coworkers [2, 6, 16, 17, 21, s ðÞ t 0 for t a. 26, 29, 36], we develop a novel matrix technique to ﬁnd the Delay differential equations of the type 1 arise in a approximate solution of Eq. 1 under the initial condition variety of applications including control systems, electro- yðaÞ¼ k in the truncated Morgan–Voyce series form dynamics, mixing liquids, neutron transportation, popula- tion models, physiological processes and conditions ytðÞ ﬃ y ðÞ t ¼ y b ðtÞ; a t b ð2Þ N n n n¼0 & Mustafa Ozel where y ; n ¼ 0; 1; ...; N are coefﬁcients to be determined; mustafa.ozel@deu.edu.tr n b ; n ¼ 0; 1; ...; N are the ﬁrst kind Morgan–Voyce poly- Department of Geophysical Engineering, Faculty of nomials deﬁned by [11, 32, 33] Engineering, Dokuz Eylul University, Tınaztepe Campus, Buca, 35160 Izmir, Turkey n þ j b ðÞ t ¼ t ; n 2 N; a t b n ð3Þ Department of Physics, Faculty of Science, Dokuz Eylul n j j¼0 University, Tınaztepe Campus, Buca, 35160 Izmir, Turkey Department of Mathematics, Faculty of Art and Science, Celal Bayar University, Manisa, Turkey 123 146 Mathematical Sciences (2018) 12:145–155 and Here, the set of polynomialsfg b ðÞ t has the following 2 3 0 1 2 N properties [11, 15, 19, 32] 6 7 0 1 2 N 6 7 6 7 1. The polynomials b ðÞ t deﬁned by 3 are recursively n 6 2 3 N þ 1 7 6 7 6 7 given by the relation 0 1 N 1 6 7 6 7 6 4 N þ 2 7 M ¼ 6 7 b ðÞ t ¼ðÞ t þ 2 b ðÞ t b ðÞ t ; n 2 n n1 n2 6 7 0 N 2 6 7 6 7 . . . . 6 7 . . . . with b ðÞ t ¼ 1 and b ðÞ t ¼ t þ 1. 6 . . . . 7 o n 6 7 4 5 2N 2. The polynomials y ¼ b ðÞ t ; n ¼ 0; 1; ... are solutions n 00 0 of the differential equation 00 0 Besides, the relation between the matrix and its derivative ttðÞ þ 4 y þ 2ðÞ t þ 1 y nnðÞ þ 1 y ¼ 0: X ðÞ t can be written in the form [13, 18, 22, 24] 3. The ﬁrst four Morgan–Voyce polynomials of the ﬁrst X ðÞ t ¼ XtðÞT ð6Þ kind are obtained from 3 as where b ðÞ t ¼ 1; b ðÞ t ¼ t þ 1; b ðÞ t ¼ t þ 3t þ 1; o 1 2 2 3 3 2 01 0 ... 0 b ðÞ t ¼ t þ 5t þ 6t þ 1; ... 6 7 00 2 ... 0 6 7 6 7 . . . . 6 7 . . . . T ¼ : . . . . 6 7 Fundamental matrix relations 6 7 4 5 00 0 ... N 00 0 ... 0 In this section, we compose the matrix relations of Eq. 1 and its solution Eq. 2. For this aim, we ﬁrst write the Then, by means of the matrix relations Eqs. 4, 5, and 6,we matrix form of the ﬁnite Morgan–Voyce series Eq. 2 as obtain ytðÞ ﬃ y ðÞ t ¼ btðÞY ð4Þ ytðÞ ﬃ y ðtÞ¼ btðÞY ð7Þ so that ¼ XtðÞMY btðÞ ¼½ b ðÞ t ; b ðÞ t ; .. .; b ðÞ t o 1 N and 0 0 0 Y ¼½ y ; y ; ...; y ; o 1 N y ðÞ t ﬃ y ðtÞ¼ X ðÞ t MY ð8Þ then, by using the Morgan–Voyce polynomials Eq. 3,we ¼ XtðÞTMY obtain the matrix form btðÞ as follows By putting t ! t s ðÞ t in Eq. 7, we gain the recurrence btðÞ ¼ XtðÞM ð5Þ relation [13, 18, 22, 24] where yt s ðtÞ ﬃ y ðtÞ¼ Xt s ðtÞ MY j N j ð9Þ 2 N ¼ XtðÞL s ðtÞ MY XtðÞ ¼ 1; t; t ; ...; t so that 2 3 0 1 2 N 0 1 2 N s ðÞ t s ðÞ t s ðÞ t s ðÞ t j j j j 6 7 0 0 0 0 6 7 6 7 6 7 1 2 N 0 1 N1 6 7 0 s ðÞ t s ðÞ t s ðÞ t j j j 6 7 1 1 1 6 7 6 7 6 2 N 7 0 N2 L s ðÞ t ¼ 6 7 00 s ðÞ t s ðÞ t j j 6 7 2 2 6 7 6 7 . . . . 6 7 . . . . 6 7 . . . . 6 7 4 N 5 000 s ðÞ t 123 Mathematical Sciences (2018) 12:145–155 147 Note that the matrix Xt s ðtÞ can be written as U ¼ XðaÞM ¼½ u ; u ; ...; u 00 01 0N Xt s ðtÞ ¼ XtðÞL s ðtÞ j j Consequently, in order to get the approximate solution of Eq. 1 subject to yðaÞ¼ k, we replace the row matrix in By substituting the relations Eqs. 7, 8, and 9 into Eq. 1,we Eq. 13 by the last row(or any row) of the augmented matrix have the matrix equation in Eq. 12; then, we obtain the result matrix "# ~ ~ ~ ~ W; P , WY ¼ P ð14Þ o o XtðÞT P ðÞ t XtðÞ P ðÞ t XtðÞL s ðÞ t MY 1 j j j¼2 ~ ~ ~ If rank W ¼ rank W; P ¼ N þ 1, then we can write, ¼ P ðÞ t ð10Þ ~ ~ Y ¼ W P . Thus the matrix, Y (thereby the Morgan– and by placing the collocation points deﬁned by Voyce coefﬁcients y ; y ; ...; y ) is uniquely determined; o 1 N thus Eq. 1 has a unique solution. b a t ¼ a þ i; i ¼ 0; 1; ...; N: in Eq. 10, the compact form of the obtained matrix equa- Error analysis tions system "# In this section, an error analysis will be presented for the XT P X P ðÞ t XL s MY ¼ P ð11Þ 1 j j o Morgan–Voyce polynomial solution in Eq. 16 with the j¼2 residual error function [4, 8, 9, 13, 21, 22, 28, 31, 36]. where In addition, we will improve the Morgan–Voyce poly- 2 3 2 3 2 N nomial solution y ðÞ t with the aid of the residual error XtðÞ 1 t t t o o o o function. 2 N 6 7 6 7 XtðÞ 1 t t t 1 1 6 7 6 1 1 7 6 7 6 7 Firstly, we consider the operator Eq. 1, under the initial X ¼ ¼ ; 6 . 7 6 . . . . 7 . . . . . 4 5 4 5 condition yðaÞ¼ k, . . . . 2 N XtðÞ 1 t t t N N N N Ly½ ðÞt ¼ gðtÞ 2 3 2 3 L sðÞ t P ðt Þ o o j o 6 7 6 7 Ly½ ðtÞ ¼ y ðÞ t P ðtÞ P ðtÞyðtÞ P ðtÞyt s ðtÞ : 6 L sðÞ t 7 P ðt Þ o 1 j j o 1 j 1 6 7 6 7 6 7 j¼2 P ¼ ; L s ¼ 6 7 o j 6 7 . . 6 7 4 5 4 5 ð15Þ P ðt Þ L sðÞ t o N j N Here, y ðtÞ is the approximate solution of the problem and P ¼ diag P ðt Þ; P ðt Þ; ...; P ðt Þ ; j ¼ 1; 2; ...; m o j o j 1 j N satisﬁes the problem X ¼ diag½ Xðt Þ; Xðt Þ; ...; Xðt Þ ; o 1 N L½y ðtÞ ¼ gðtÞþ R ðtÞ; a t b; N N ð16Þ y ðtÞ¼ k: Morgan–Voyce matrix method Also, the residual function of the Morgan–Voyce polyno- mial approximation y ðÞ t is deﬁned as The fundamental matrix Eq. 11 of Eq. 1 can be expressed in the form R ðtÞ¼ L½y ðtÞ gðtÞ: ð17Þ N N WY ¼ P ,½ W; P ð12Þ o o If we know the exact solution ytðÞ, then the error function where is calculated as the difference between the approximate and "# the exact solutions deﬁned by W ¼ w ¼ XT P X P ðÞ t XL s M pq 1 j j e ðÞ t ¼ ytðÞ y ðÞ t : ð18Þ N N j¼2 By using the Eqs. 15, 16, 17 and 18, we get the error p; q ¼ 0; 1; ...; N: problem By using the relation Eq. 7, we obtain the corresponding L½e ðtÞ ¼ L½yðtÞ L½y ðtÞ ¼ R ðtÞ: N N N matrix form to the initial condition yðaÞ¼ k as ð19Þ e ðÞ a ¼ 0: UY ¼ k ,½ U; Y ð13Þ such that 123 148 Mathematical Sciences (2018) 12:145–155 By solving the error problem in Eq. 19 with the method Numerical examples presented in Sect. 3, we get the approximation e ðtÞ to N;M e ðtÞ as follows N Example 1 Consider the differential equation with vari- able delay t þ 1 e ðtÞ¼ a b ðtÞ; ðM NÞ: N;M n 0 2 2 2 2 y ðtÞ¼3t t þ t yðtÞ yðt t Þþ tyðt t 1Þ n¼0 ð20Þ 0 t 1 Consequently, by means of the polynomials y ðtÞ and e ðtÞ, ðM NÞ, we obtain the corrected Morgan–Voyce N;M polynomial solution y ðtÞ¼ y ðtÞþ e ðtÞ. Here, N;M N N;M subject to the initial condition yð0Þ¼1. The exact e ðtÞ¼ yðtÞ y ðtÞ, E ðtÞ¼ e ðtÞ e ðtÞ¼ N N N;M N N;M solution of this equation is yðtÞ¼ t 1. First of all, let us yðtÞ y ðtÞ and e ðtÞ denote the error function, the N;M N;M determine the collocation points by the formula t ¼ a þ corrected error function and the estimated error function, ba i; i ¼ 0; 1; ...; N for a ¼ 0, b ¼ 1, and m ¼ 3; N ¼ 2. respectively. Therefore the collocation points are obtained as If the exact solution of Eq. 1 can not been known, then t ¼ 0; t ¼ ; t ¼ 1. By Eq. 11, the fundamental matrix o 1 2 the absolute errorsjj e ðt Þ ¼jj yðt Þ y ðt Þ ,(a t b) 2 N i i N i i equation of this problem is written as are not computed. However, the absolute errors "# jj e ðt Þ ¼jj yðt Þ y ðt Þ ,(a t b) can be estimated by 3 N i i N i i X XT P X P ðÞ t XL s MY ¼ P 1 j j o using the absolute error function e ðtÞ . N;M j¼2 where 2 3 2 3 2 3 2 3 0 00 0 000 10 0 6 7 5 1 1 6 7 6 7 6 7 6 7 P ¼ ; P ¼ 0 0 ; P ¼ 0 10 ; P ¼ 0 0 4 5 4 5 4 5 o 1 2 3 4 5 4 4 2 00 1 2 00 1 001 XtðÞ ¼ 1 tt 2 3 2 3 2 3 2 3 XðÞ 0 10 0 010 111 6 7 1 1 1 6 7 6 7 6 7 6 7 T ¼4002 5; M ¼ 4013 5; X ¼ 6 X 7 ¼ 4 1 5 4 2 5 2 4 000 001 11 1 XðÞ 1 2 3 2 3 XðÞ 0 00 10 000 00 00 6 7 1 1 1 6 7 6 7 X ¼6 0 X 0 7 ¼ 400 01 00 0 5 4 2 5 2 4 00 000 01 11 00 XðÞ 1 2 3 2 3 100 1 11 6 7 6 7 010 01 2 6 7 6 7 6 7 6 7 6 7 6 7 001 00 1 6 7 6 7 6 1 1 7 6 5 25 7 6 7 6 7 1 1 6 7 6 7 4 16 4 16 6 7 6 7 6 1 7 6 5 7 LðÞ s ðt Þ¼ ; LðÞ s ðt Þ¼ : 2 i 3 i 6 7 6 7 01 01 6 7 6 7 2 2 6 7 6 7 6 001 7 6 7 00 1 6 7 6 7 6 7 6 7 1 11 1 24 6 7 6 7 6 7 6 7 4 5 4 5 01 2 01 4 001 00 1 123 Mathematical Sciences (2018) 12:145–155 149 Table 1 Numerical results of t N ¼ 3; M ¼ 4 N ¼ 4; M ¼ 5 N ¼ 12; M ¼ 13 the exact, approximate and corrected solutions of Exact Appr. Corrected Appr. Corrected Appr. Corrected Example 2 for some N values 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.1 1.22140 1.39581 1.20219 1.20219 1.22571 1.22140 1.22140 0.2 1.49182 1.75739 1.46663 1.46663 1.49689 1.49182 1.49182 0.3 1.82212 2.11993 1.79678 1.79678 1.82697 1.82212 1.82212 0.4 2.22554 2.51864 2.20151 2.20151 2.23024 2.22554 2.22554 0.5 2.71828 2.98872 2.69510 2.69510 2.72299 2.71828 2.71828 0.6 3.32012 3.56539 3.29724 3.29724 3.32478 3.32012 3.32012 0.7 4.05520 4.28383 4.03301 4.03301 4.05954 4.05520 4.05520 0.8 4.95303 5.17927 4.93292 4.93292 4.95683 4.95303 4.95303 0.9 6.04965 6.28689 6.03286 6.03286 6.05299 6.04965 6.04965 1 7.38906 7.64191 7.37414 7.37414 7.39236 7.38906 7.38906 Fig. 1 Exact, approximate and 9 corrected solutions of Example 2 for N ¼ 3 Exact Approximation Corrected 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 2 Exact, approximate and 9 corrected solutions of Example 2 for N ¼ 4 Exact Approximation Corrected 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 3 The augmented matrix of the fundamental matrix equation 12 4 0 is computed as 6 7 ½ W; P ¼ 4 0:25 1:75 5:46875 1:25 5 10 2 1 y(t) y(t) 150 Mathematical Sciences (2018) 12:145–155 Table 2 Comparison of the t N ¼ 3; M ¼ 4 N ¼ 4; M ¼ 5 N ¼ 12; M ¼ 13 absolute errors and corrected absolute errors of Example 2 Appr. Corrected Appr. Corrected Appr. Corrected 0 7.99361e-15 6.21725e-15 3.10862e-15 1.02141e-14 2.02061e-14 4.97380e-14 0.1 1.74411e-01 1.92085e-02 1.92085e-02 4.30490e-03 5.60576e-11 2.89457e-12 0.2 2.65564e-01 2.51946e-02 2.51946e-02 5.06186e-03 5.52192e-11 2.88503e-12 0.3 2.97810e-01 2.53429e-02 2.53429e-02 4.85123e-03 5.51801e-11 2.87947e-12 0.4 2.93098e-01 2.40357e-02 2.40357e-02 4.69433e-03 5.49143e-11 2.87148e-12 0.5 2.70441e-01 2.31845e-02 2.31845e-02 4.71136e-03 5.42593e-11 2.86038e-12 0.6 2.45269e-01 2.28805e-02 2.28805e-02 4.66159e-03 5.29470e-11 2.80753e-12 0.7 2.28632e-01 2.21878e-02 2.21878e-02 4.33963e-03 5.04476e-11 2.68496e-12 0.8 2.26233e-01 2.01130e-02 2.01130e-02 3.79602e-03 4.58762e-11 2.45759e-12 0.9 2.37243e-01 1.67888e-02 1.67888e-02 3.34318e-03 3.76135e-11 2.08455e-12 1.0 2.52855e-01 1.49208e-02 1.49208e-02 3.29924e-03 2.63132e-11 1.53477e-12 Fig. 3 Comparison of the 0.35 absolute error with the corrected 0.3 absolute error for N ¼ 3 0.25 0.2 Absolute Error 0.15 Corrected Absolute Error 0.1 0.05 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 4 Comparison of the 0.05 absolute error with the corrected Absolute Error absolute error for N ¼ 4 0.04 Corrected Absolute Error 0.03 0.02 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 and the augmented matrix for initial condition is obtained Example 2 Consider the differential equation with vari- as able delay t 0 2t 2 2t ½ U; k¼½ 111 j 1 : 2y ðtÞ tyðtÞþ te yðt t Þ¼ 4e ; 0 t 1 By using the procedure in Sect. 3, we obtain the approxi- subject to the initial condition yð0Þ¼ 1. The exact solution 2t mate solution as of this equation is yðtÞ¼ e . The fundamental matrix equation is y ðtÞ¼ t 1 ½ XT P X P XLðÞ s ðÞ t MY ¼ P : 1 2 2 i o which is the exact solution. e( t) e( t) Mathematical Sciences (2018) 12:145–155 151 Table 3 Numerical results of t N ¼ 3; M ¼ 4 N ¼ 4; M ¼ 5 N ¼ 12; M ¼ 13 the exact, approximate and corrected solutions of Exact Appr. Corrected Appr. Corrected Appr. Corrected Example 3 for some N values 0 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 0.1 0.90484 0.90675 0.90500 0.90500 0.90485 0.90484 0.90484 0.2 0.81873 0.82159 0.81894 0.81894 0.81874 0.81873 0.81873 0.3 0.74082 0.74401 0.74103 0.74103 0.74083 0.74082 0.74082 0.4 0.67032 0.67350 0.67052 0.67052 0.67033 0.67032 0.67032 0.5 0.60653 0.60956 0.60672 0.60672 0.60654 0.60653 0.60653 0.6 0.54881 0.55168 0.54900 0.54900 0.54882 0.54881 0.54881 0.7 0.49659 0.49936 0.49677 0.49677 0.49660 0.49659 0.49659 0.8 0.44933 0.45209 0.44950 0.44950 0.44934 0.44933 0.44933 0.9 0.40657 0.40937 0.40673 0.40673 0.40658 0.40657 0.40657 1 0.36788 0.37069 0.36803 0.36803 0.36789 0.36788 0.36788 Fig. 5 Exact, approximate and 1 corrected solutions of Exact 0.9 Example 3 for N ¼ 3 Approximation 0.8 Corrected 0.7 0.6 0.5 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Fig. 6 Exact, approximate and corrected solutions of Exact 0.9 Example 3 for N ¼ 4 Approximation 0.8 Corrected 0.7 0.6 0.5 0.4 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 After the collocation points substituted into this matrix The absolute and corrected errors of Example 2 in equation, we solve the system and we obtain the solutions Table 2 are compared for N ¼ 3; 4 in Figs. 3 and 4, in the form 4 of Example 2 for N ¼ 3; 4; 12 in the interval respectively. ½ 0; 1 Table 1. Example 3 Consider the delay differential equation having Now, we will give the exact, approximation, and cor- variable delays lnðÞ t þ 1 and t rected solutions of Example 2 for N ¼ 3; 4 in Figs. 1 and 2, respectively. y(t) y(t) 152 Mathematical Sciences (2018) 12:145–155 Table 4 Comparison of the t N ¼ 3; M ¼ 4 N ¼ 4; M ¼ 5 N ¼ 12; M ¼ 13 absolute errors and corrected absolute errors of Example 3 Appr. Corrected Appr. Corrected Appr. Corrected 0 0 0 2.22045e-16 2.22045e-16 4.44089e-16 4.44089e-16 0.1 1.91653e-03 1.57584e-04 1.57584e-04 1.15985e-05 1.16573e-13 2.98650e-14 0.2 2.86131e-03 2.06131e-04 2.06131e-04 1.35662e-05 1.13909e-13 2.95319e-14 0.3 3.19084e-03 2.07981e-04 2.07981e-04 1.30496e-05 1.12577e-13 2.89768e-14 0.4 3.17959e-03 1.97746e-04 1.97746e-04 1.25881e-05 1.09690e-13 2.83107e-14 0.5 3.02786e-03 1.90112e-04 1.90112e-04 1.24366e-05 1.05804e-13 2.74225e-14 0.6 2.86876e-03 1.86905e-04 1.86905e-04 1.21434e-05 1.01363e-13 2.64233e-14 0.7 2.77470e-03 1.83479e-04 1.83479e-04 1.14580e-05 9.62008e-14 2.50355e-14 0.8 2.76306e-03 1.74494e-04 1.74494e-04 1.06297e-05 9.02611e-14 2.35367e-14 0.9 2.80153e-03 1.59153e-04 1.59153e-04 1.01563e-05 8.32667e-14 2.20934e-14 1 2.81276e-03 1.45931e-04 1.45931e-04 1.00352e-05 7.82152e-14 1.96509e-14 -3 Fig. 7 Comparison of the x 10 absolute error with the corrected 3.5 absolute error for N ¼ 3 2.5 1.5 Absolute Error Corrected Absolute Error 0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -4 Fig. 8 Comparison of the x 10 absolute error with the corrected absolute error for N ¼ 4 Absolute Error Corrected Absolute Error 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 t t 2 ½ XT P X P XLðÞ s ðÞ t P XLðÞ s ðÞ t MY ¼ P : y ðtÞ¼ t þ t 1 e tyðÞ t lnðÞ t þ 1 e yt t 1 2 2 i 3 3 i o þ yðtÞ; 0 t 1 subject to the initial condition yð0Þ¼ 1. The exact solution After the collocation points substituted into this matrix of this equation is yðtÞ¼ e . The fundamental matrix equation, we can solve the system and we obtain the equation is e(t) e(t) Mathematical Sciences (2018) 12:145–155 153 Table 5 Numerical results of t Exact Present Abs. error HTL Abs. error the Exact, Present and HTL solutions of Example 4 for N ¼ 0.0 1.0000000 1.0000000 5.55112e-15 1.000000067 6.70000e-08 6 value 0.2 0.8187308 0.8187339 3.10453e-06 0.807836989 1.08938e-02 0.4 0.6703200 0.6703221 2.03288e-06 0.649394632 2.09254e-02 0.6 0.5488116 0.5488127 1.09160e-06 0.529079726 1.97319e-02 0.8 0.4493290 0.4493294 4.48652e-07 0.436772222 1.25567e-02 1.0 0.3678794 0.3678793 1.14014e-07 0.361610346 6.26909e-03 1.2 0.3011942 0.3011936 5.72331e-07 0.298780813 2.41340e-03 1.4 0.2465970 0.2465960 9.17880e-07 0.249314192 2.71723e-03 1.6 0.2018965 0.2018953 1.22873e-06 0.212885438 1.09889e-02 1.8 0.1652989 0.1652975 1.43489e-06 0.173619567 8.32068e-03 2.0 0.1353353 0.1353338 1.44145e-06 0.078902496 5.64328e-02 Fig. 9 Comparison of the exact, 1.2 present and HTL solutions of Exact Example 4 for N ¼ 6 value Present 1.0 HTL 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Fig. 10 RMSE values for N ¼ 1 1; 2; .. .; 100 for Examples 2, 3 Example 2 -2 and 4 Example 3 Example 4 -4 -6 -8 -10 -12 -14 -16 0 102030405060708090 100 y(t) RESM 154 Mathematical Sciences (2018) 12:145–155 creativecommons.org/licenses/by/4.0/), which permits unrestricted solutions in the form 4 of Example 3 for N ¼ 3; 4; 12 in use, distribution, and reproduction in any medium, provided you give the interval½ 0; 1 Table 3. appropriate credit to the original author(s) and the source, provide a All exact, approximation, and corrected solutions of link to the Creative Commons license, and indicate if changes were Example 3 for N ¼ 3; 4 are given in Fig. 5 and 6, respec- made. tively. The absolute and corrected errors of Example 3 in Table 4 are compared for N ¼ 3; 4 in Figs. 7 and 8, References respectively. 1. Abd-Elhameed, W.M., Youssri, Y.H., Doha, E.H.: A novel operational matrix method based on shifted Legendre polyno- Example 4 Consider the following delay differential 2 mials for solving second-order boundary value problems equation [7] having variable delay lnðÞ t þ 1 involving singular, singularly perturbed and Bratu-type equa- 0 2 t 2 tions. Math. Sci. 9, 93–102 (2015) y ðtÞ¼ t þ 1 e yt ln t þ 1 yðtÞ; 0 t 2 2. Akyu¨z, A., Sezer, M.: A Chebyshev collocation method for the solution of linear integro-differential equations. Int. J. Comput. subject to the initial condition yð0Þ¼ 1. The exact solution Math. 72, 491–507 (1999) of this equation is yðtÞ¼ e . 3. Ardjouni, A., Djoudi, A.: Fixed points and stability in linear neutral differential equations with variable delays. Nonlinear Similarly, we can solve this problem by present method Anal. Theory Methods Appl. 74, 2062–2070 (2011) and we obtain the solutions in the form 4 of Example 4 for 4. Bahs¸ı, M.M., C ¸ evik, M., Sezer, M.: Orthoexponential polynomial N ¼ 6 in the interval½ 0; 2 . Then, we also compare the solutions of delay pantograph differential equations with residual present solutions and Hybrid Taylor-Lucas Method [7] error estimation. Appl. Math. Comput. 271, 11–21 (2015) 5. Bahs¸ı, M.Mustafa, Bahs¸ı, Ays¸e Kurt, C ¸ evik, Mehmet, Sezer, (HTL) solutions in Table 5. The present method has been Mehmet: Improved Jacobi matrix method for the numerical shown to be suitable as graphically for nonhomogeneous solution of Fredholm integro-differential-difference equations. differential equation with variable delays in Fig. 9. Math. Sci. 10, 83–93 (2016) In order to better deﬁne the solution space of the prob- 6. 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Published: Jun 5, 2018
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