# An Analytical Computational Algorithm for Solving a System of Multipantograph DDEs Using Laplace Variational Iteration Algorithm

Authors: Mohamed S. M. Bahgat; A. M. Sebaq
Journal: Advances in Astronomy
 (2021)

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## Abstract

In this research, an approximation symbolic algorithm is suggested to obtain an approximate solution of multipantograph system of type delay differential equations (DDEs) using a combination of Laplace transform and variational iteration algorithm (VIA). The corresponding convergence results are acquired, and an efficient algorithm for choosing a feasible Lagrange multiplier is designed in the solving process. The application of the Laplace variational iteration algorithm (LVIA) for the problems is clarified. With graphics and tables, LVIA approximates to a high degree of accuracy with a few numbers of iterates. Also, computational results of the considered examples imply that LVIA is accurate, simple, and appropriate for solving a system of multipantograph delay differential equations (SMPDDEs).


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## Body

## 1. Introduction

Many physical phenomena are formulated by delay differential equations which are similar to ordinary differential equations, but their development at assured time instant depends on past values. Pantograph equations are one of the most prominent kinds of functional differential equations with proportional delay and the pantograph kind equations have been studied extensively because of the various implementations in which these equations arise. The name of the pantograph is begun from being crafted by Ockendon and Tayler on the gathering of current by the pantograph head of an electric loco; this equation has appeared in many scientific models which is exceedingly applicable in physics, mathematics, engineering, and biology as in astrophysics, population models, probability theory, quantum mechanics, number theory, nonlinear dynamic system, electronic system, cell growth, and so forth (for further, see [1–3] and the references therein). Numerous examinations were done on the estimated arrangement of the referenced condition in the one-dimensional case. As of late, a collocation method based on the Genocchi delay operational matrix and the operational matrix of fractional derivative for solving generalized fractional pantograph equations is given in [4]. Some papers like [5, 6] have presented the solution for a system of multipantograph delay differential equations (SMPDDEs) with higher order by two different algorithms. A collocation method based on the Genocchi operational matrix for solving generalized pantograph equations is given in [7]. Then, again, the mathematical resolvability of another form of differential issues can be found in [8–18] and the references therein. The pantograph equation,(1)y′t=αyt+ft,yβt,0<β<1,y0=y0,is one of the important types of DDE that emerge in numerous scientific models which appear in dynamical systems, population studies, electrodynamics, and number theory. In particular, it was used by [3] to research how the electric current is composed through the pantograph of an electric locomotive, from where it gets its name. In this research, we are building on our work in [5] to develop the Laplace variational iteration algorithm (LVIA) for SMPDDEs:(2)y1′t=α1,1y1t+α1,2y2t+⋯+α1,nynt+g1t,y1β1,1t,y2β1,2t,…,ynβ1,nty2′t=α2,1y1t+α2,2y2t+⋯+α2,nynt+g2t,y1β2,1t,y2β2,2t,…,ynβ2,nt⋮yn′t=αn,1y1t+αn,2y2t+⋯+αn,nynt+gnt,y1βn,1t,y2βn,2t,…,ynβn,nt,subject to the initial conditions:(3)yit0=yi,0,i=1,2,3,…,n,where 0=t0<t≤T, αi,j, yi,0 are finite constants, gi are analytical functions such that 0<βi,j<1,i,j=1,2,…,n, which verify all needed requirements for finding a unique solution, and yit,i=1,2,…,n, are functions that need to be found on the given interval. We were motivated to apply LVIA to find approximate solutions for SMPDDEs. By choosing a suitable value for the initial approximation, LVIA can easily be applied to the given problems. Moreover, the solution and its derivative are usable for each arbitrary point in the interval. LVIA provides a direct scheme for solving the problem with no physically unsuitable assumptions, discretization, linearization, transformation, or perturbation. Generally, in applications of LVIA to IVP (initial value problem) of SMPDDEs, one generally follows the following three proceedings:(a)
Establishment of the correction functional(b)
Identifying the Lagrange multipliers(c)
Defining the initial iterationLVIA is worth mentioning that the method is capable of decreasing the size of the computational work as compared to the classic methods while yet maintaining the elevation accuracy of the numerical results. The main feature of the variational iteration algorithm over decomposition restraint of Adomian is that the previous algorithm gives the solution of the problem without calculating Adomian’s polynomials which requires complex calculations. Accordingly, it is not affected by calculation round-off errors and one is not reputed with the requirement of large PC time and memory. Generally, by using LVIA, one iteration results in an accurate solution if the initial solution is closely chosen. The convergence of the method is a systematical debate in [8]. The remainder of this study is ordered as follows: in Section 2, we presented the basic idea of LVIA together with the analysis of the method. In Section 3, the symbolic approximate solutions for (2)-(3) can be provided using the extended LVIA. Depending on the above-mentioned, some of the numerical applications are given to explain the adequacy of LVIA in Section 3. The computations show that the approximate solutions can be achieved accurately and efficiently with a few iterations. Finally, Section 4 gives a brief conclusion.


## 2. Methodology Basic Ideas of Laplace Variational Iteration Algorithm (LVIA)

To clarify the basic connotation of LVIA, firstly, we rewrite (2)-(3) as the next form:(4)Y′t=αYt+Gt,Yβt,with the initial conditions:(5)Yt0=Y0,where Y′t=y1′ty2′t…yn′tT∈ℝn, Yt=y1ty2t…yntT∈ℝn, α,β∈ℝn×n, G=g1g2…gnT∈ℝn where gi=git,y1βi,1t,y2βi,2t,…,ynβi,nt, and Y0=y1,0y2,0…yn,0T∈ℝn. The idea of the VIA for (4) is to construct the next correction functional:(6)Yk+1t=Ykt+∫0tμt,ρYk′ρ−αYρ−Gρ,Yβρdρ,where μ is the Lagrange multiplier which has a critical part in this study; it is recognized ideally by variety hypothesis and Ykt,k≥0 is the n-th approximate solution order for the exact solution Yt, which will be gotten by utilizing starting approximation Y0, which is acquired from (5). So, when n tends to infinity, the approximate solution Ykt converges to Yt. The numerical procedure of LVIA delineates how LVIA is utilized to approximate the solution of SMPDDEs. The basic steps included are given as follows:(i)
The correction functional is acquired by taking the Laplace transform of (4):(7)Yk+1s=Yks+μssYks−Y0+ℒ−αYt−Gt,Yβt,(ii)
whereYs=ℒYx=∫0∞e−sxYxdx, ℒ is used to indicate Laplace transform.(iii)
To find the optimal value ofμs, take the variation with respect to Yk. Let (7) be stationary with respect to Yk and Y0=0; then, we obtain(8)δδYkYk+1s=δδYkYks+δδYkμssYks−Y0+ℒ−αYt−Gt,Yβt.(iv)
Considering the termsℒ−αYt−Gt,Yβt as constrained variations, then we get(9)δYk+1s=δYks+μssδYks.(v)
From (9), we derive the Lagrange multiplier as(10)μs=−1s.(vi)
By using the inverse of Laplace transformℒ−1 for (7), we can obtain the succeeding approximations which give(11)Yk+1t=Ykt+ℒ−1μssYks−Y0+ℒ−αYkt−Gt,Ykβt=ℒ−1Y0s+ℒ−11sℒαYt+Gt,Yβt,and initial approximation Y0t can be determined by(12)Y0t=ℒ−1Y0s+⋯+Yk−10sk=Y0+Y′0t+⋯+Yk−10tk−1k−1!.Beginning with an initial approximationY0t, we obtain the sequential approximations, and the exact solution can be acquired by using(13)Yt=limk⟶∞Ykt.Now, we display that the sequenceYktk=1∞ given by (9) with Y0t=Y0 converges to the exact solution of (4)-(5). To do this, we declare the following theorem.Theorem 1 (see [9]).
LetYt∈C1ℝn, t∈ℝ=0,T be the exact solution of (4) and Ynt∈C1ℝk be the solution of the sequence(14)Yk+1t=Ykt+ℒ−1μssYks−Y0+ℒ−αYkt−Gt,Ykβt=ℒ−1Y0s+ℒ−11sℒαYt+Gt,Yβt,with the initial approximation Y0t.IfExtkt=Yt−Ykt and d/dtExtkt2≤Extkt2, then the functional sequence Yktk=1∞ defined by the above sequence converges to Yt.Proof.
To prove this theorem, we follow the same proof of Theorem1 in [9].
The accompanying considerable definitions given underneath are needed for this work and furthermore are needed for the convergence analysis in the next corollary.Corollary 1.
Letyi,kt,i=1,2,…,m and k=1,2,…,n, be series for any iteration k; then the LVIA will attain approximately to the exact solution of (2)-(3). Now, we report some types of errors which are as follows:(a)
Residual errorResikt defined by Resikt=yi,k′t−αi,1y1,kt−αi,2y2,kt−⋯−αi,mym,kt−git,y1,kβi,1t,y2,kβi,2t,…,ym,kβi,mt(b)
Exact error (absolute error)Extik which is defined by Extik=yi,exactt−yi,kt(c)
Relative errorRelik, which is defined by Relik=yi,exactt−yi,kt/yi,exactt(d)
Consecutive errorConik, which is defined by Conik=yi,k+1t−yi,kt


## 3. Applications and Numerical Discussion

In this section, six numerical examples are given to illustrate the accuracy and the convergence of LVIA which is described in Section 2. In contrast, numerical results show that this method gives a good approximation to the exact solution for all possible values oft, while the accuracy is in continuous increase by using only a few approximations. For comparison purposes, the solution intervals of problems are chosen generally the same as those in the references. Throughout this research, all the symbolical and computations results used Maple 18.0 software program.Example 1.
Regarding the SMPDDEs [19],(15)y1′t−y1t+y2t−y1t2=f1t,y2′t+y1t+y2t+y2t2=f2t,t>0subject to the initial conditions:(16)y10=1,y20=1,where f1t=e−t−et/2,f2t=et+e−t/2, with the exact solutions y1t=et and y2t=e−t.
We apply LVIA for solving system (15) and (16); we begin with selecting the initial conditions of the approximations such as y1,0t=y10=1 and y2,0t=y20=1. Then, let us seek the approximate solutions y1,kt and y2,kt where k is a positive integer number greater than or equal to zero. By taking ℒ for system (15)-(16), we obtain(17)sy¯1s−y10=ℒ−y2t+y1t+y1t2+f1t,sy¯2s−y20=ℒ−y2t−y1t−y2t2+f2t.
The iteration formulas thus are(18)y¯1,k+1s=y¯1,ks+μssy¯1,ks−y1,00−ℒy1,kt−y2,kt+y1,kt2+f1t,y¯2,k+1s=y¯2,ks+μssy¯2,ks−y2,00−ℒ−y1,kt−y2,kt−y2,kt2+f2t,with the Lagrange multiplier μs=−1/s, and by taking ℒ−1, we obtain(19)y1,k+1t=y1,kt−ℒ−11ssy¯1,ks−y1,00−ℒy1,kt−y2,kt+y1,kt2+f1t=ℒ−11s+ℒ−11sℒy1,kt−y2,kt+y1,kt2+f1t,y2,k+1t=y2,kt−ℒ−11ssy¯2,ks−y2,00−ℒy1,kt−y2,kt−y2,kt2+f2t=ℒ−11s+ℒ−11sℒ−y1,kt−y2,kt−y2,kt2+f2t.
Therefore,(20)y1,k+1t=ℒ−11sℒy1,kt−y2,kt+y1,kt2+f1t+1,y2,k+1t=ℒ−11sℒ−y1,kt−y2,kt−y2,kt2+f2t+1,with the initial iteration y1,0t=y10=1,y2,0t=y20=1. Now, in order to obtain the first approximation y1,1t and y2,1t of the LVIA solution for system (15) and (16), we put k=0 through (20) to get(21)y1,1t=4−e−t−2et/2+t,y2,1t=2−2e−t/2−3t+et,by using the first approximation y1,1t and y2,1t; then, the second approximation y1,2 and y2,2t of the LVIA solution for system (15)-(16) can be written as follows:(22)y1,2t=18−8et/4−6et/2−2e−t/2+6t−et+94t2,y2,2t=14−8e−t/4−e−t+2et/2−6e−t/2−8t+74t2,and by proceeding with the comparable style, the third and the fourth iterations for the LVIA solution of (15)-(16) lead to the following outcomes:(23)y1,3t=176−24e−t/4−2e−t−56et/4−20et/2−8e−t/2+17t3/48+22t−64et/8−et+172t2,y2,3t=108−56e−t/4−e−t+24et/4+8et/2−20e−t/2−71t348−46t−64e−t/8+2et+3t2,y1,4t=2884−96e−t/4−1024et/16−400et/4−60et/2−20e−t/2+61t324+244t+721t41536−320e−t/8−960et/8−3et+79t22,y2,4t=2012−400e−t/4−3e−t−1024e−t/16+96et/8+20et/2−60e−t/2−49t312−392t+503t41536−960e−t/8+320et/8+47t22.⋮
To clarify the convergence of the solution of the proposed methodyi,kt to the exact solution yit,i=1,2, w.r.t. the number of iterations k of the solutions, we set numerical values graphically. Figure 1(a) shows the exact solution yit,i=1,2,t∈0,3, and yi,kt,k=3,5,7,9,i=1,2, respectively. These plots detect that LVIA is an effective and appropriate method to solve SMPDDEs with fewer calculations and numbers of iterations. Figure 1(b) illustrates the efficiency of the method whether it has an exact solution or not by using the residual error values which shows that the residual error Resikt decreases as k increases. Table 1 shows the CPU times of the present method for Example 1 to get 10th-order approximate symbolic solutions with variable t. The accuracy of the method can be spotted with the numerical results and by comparing the absolute errors of LVIA for k=4,5,6 and the Laplace decomposition algorithm (LDA) [19] is given in Table 2. It is obvious that, from this table, the approximate solutions are proven to be similar to the exact solution for all likely values of t in [0,1].Figure 1
(a) Comparison betweenyi,kt,k=3,5,7,9,i=1,2 solutions of the LVIA and the exact solution for Example 1. (b) The residual error Resikt,i=1,2 of LVIA solutions of (13) at k=4,6,8, respectively, for Example 1.
(a)(b)Table 1
CPU times (per second) of LVIA for Example1.
k12345678910CPU0.1430.2540.4150.5440.7440.9361.1581.4421.7202.078Table 2
Comparison of the absolute errors (Extik,i=1,2) for Example 1.
tExact solutiony1ty1,kt LDA [19]y1,kt (presented method)k=4k=6k=4k=50.21.22140275816021.210×10−51.210×10−51.576×10−52.276×10−60.41.49182469764134.238×10−43.170×10−65.057×10−49.470×10−50.61.82211880039053.499×10−35.583×10−53.981×10−38.488×10−50.82.22554092849251.594×10−24.460×10−41.736×10−34.721×10−41.002.71828182845915.236×10−22.259×10−35.484×10−31.810×10−4Exact solutiony2ty2,kt LDA [19]y2,kt (presented metdod)k=4k=6k=4k=50.20.81873075307805.219×10−57.807×10−81.725×10−58.753×10−60.40.67032004603561.668×10−31.310×10−55.739×10−41.407×10−50.60.54881163609401.266×10−22.227×10−44.271×10−35.764×10−50.80.44932896411725.338×10−21.668×10−31.765×10−22.080×10−41.00.36787944117141.632×10−17.956×10−35.284×10−23.014×10−4Example 2.
Consider SMPDDEs [20]:(24)y1′t−sinty1t2−costy2t2=f1t,y2′t+costy1t2−sinty2t2=f2t,t≥0subject to the initial conditions:(25)y10=0,y20=1,where f1t=cost−cost/2,f2t=−sint−sint/2 with the exact solutions y1t=sint and y2t=cost. By the same procedures of Example 1, we apply the LVIA approach for solving (24)-(25), according to (11). Table 3 shows the CPU times of the LVIA for Example 2 to get 10th-order approximate symbolic solutions with variable t. To clarify the convergence of the solution of the proposed method yi,kt to the exact solutions yit,i=1,2, with respect to the iteration number of the solution, we set numerical values graphically. Figure 2 shows the exact solution yit,i=1,2,t∈0,5, and some iterated approximations yi,kt,k=1,2,3,4,i=1,2, respectively. This graph and the results obtained in Table 4 detected that the LVIA is a very accurate and effective method to solve SMPDDEs with fewer computational and iteration steps.Table 3
CPU times (per second) of LVIA for Example2.
k12345678910CPU0.1530.3110.5710.90901.5232.7222.9973.2433.5083.792Figure 2
Plots of the LVIA solutionsyi,kt,i=1,2, k=1,2,3,4, respectively, and the exact solution yit, i=1,2, of Example 2 on 0,5, where yit is represented by a straight line and yi,kt is represented by a dashed line.Table 4
Error analysis.
tAbsolute errorExt110Relative errorRel110Consecutive errorCon110Residual errorRes110tError analysis ofy1t for Example 2 on 0,10.21.34×10−326.77×10−321.34×10−327.32×10−310.42.29×10−295.87×10−292.29×10−295.99×10−280.61.36×10−272.401×10−271.35×10−272.13×10−260.81.39×10−261.94×10−261.39×10−269.50×10−261.06.61×10−267.86×10−266.62×10−261.87×10−24Error analysis ofy2t for Example 4 on 0,10.24.66×10−334.75×10−334.66×10−332.79×10−310.41.80×10−291.95×10−291.80×10−295.33×10−280.62.11×10−272.56×10−272.11×10−275.00×10−260.82.76×10−268.27×10−265.76×10−268.14×10−261.06.83×10−251.26×10−248.83×10−257.39×10−24Example 3.
Consider the two-dimensional nonlinear pantograph equations [19]:(26)y1′t+y1t+e−tcost2y2t2+2e−3/4tcost2sint4y1t4=0,y2′t−ety1t22+y2t22=0,0≤t≤1,the initial conditions in which the system subjected to(27)y10=1,y20=0.
The exact solutions arey1t=e−tcost and y2t=sint. By the same procedures of Example 1, we apply LVIA for solving (26)-(27), according to (11). To clarify the convergence of yi,kt to yit,i=1,2, with respect to the iteration number of the solution, we present numerical results of Example 3 graphically where Figures 3(a) and 3(b) show the exact solution and the approximate solution of LVIA for yi,kt,i=1,2,k=1,2,3, respectively. These plots clarify that LVIA is a good accurate method for solving such SMPDDEs with fewer calculations and numbers of iterations. Also, Figure 3(c) displays the absolute errors obtained by LVIA for k=1,2,3 of (26)-(27). From Figure 4(c), the approximate solutions at k=1,2,3 are identical to the exact solution for all values of t in [0,1]. These plots clarify that LVIA is an efficient method for solving such a system with a few iterations. We note that the error decreases while the value k increases.Figure 3
(a) Comparison betweeny1,kt,k=1,2,3 solutions of the LVIA and y1t for Example 3. (b) Comparison between y2,kt,k=1,2,3 of LVIA and the exact solution y2t for Example 3. (c) Comparison of Extik for k=1,2,3, respectively, for yi,kt,i=1,2 of Example 3.
(a)(b)(c)Figure 4
Plots of the LVIA solutionsyi,kt,i=1,2, k=2,4,6,8,10, respectively, and the exact solution yit, i=1,2, of Example 4 on 0,2, where yit is represented by a straight line and yi,kt is represented by a dashed line.Example 4.
Consider the 2-dimensional nonlinear SMPDDEs [21]:(28)y1′t−y1′t−1−4y2t=0,y2′t−y1t+y1t−1=0,0≤t≤2,subject to the initial functions:(29)y1t=e−2t,y2t=12e−2t−1−e−2t,−1≤t≤0.
The exact solutions arey1t=e−2t and 1/2e−2t−1−e−2t. By using the given initial functions and also using the same procedures of Example 1, we apply the LVIA approach for solving (28)-(29), according to (11). Without loss of generality, we will test the accuracy of LVIA for Example 4 using error analysis of yit, i=1,2, t∈0,1 for (28)-(29) with a step size of 0.2 as well as comparison among the absolute errors, relative errors, consecutive errors, and residual errors of 10th-order approximate LVIA solutions shown in Table 5. Also, to illustrate the convergence of yi,kt to yit,i=1,2, with respect to the order of k for the solutions, we ready numerical results of Example 4 graphically where Figure 4 shows the exact solutions yit,i=1,2, and approximate solutions yi,kt,i=1,2, and k=2,4,6,8,10, respectively. These plots clarify that LVIA is an efficient method for solving such a system with a few iterations. We note that the error decreases while the value k increases. The results show that the LVIA provides us with the precise approximate solutions of (28)-(29). Moreover, we can control the error also by evaluating more iterations.Table 5
Error analysis.
tAbsolute errorExt110Relative errorRel110Consecutive errorCon110Residual errorRes110tError analysis ofy1t for Example 4 on 0,10.21.02×10−121.52×10−128.00×10−134.28×10−110.42.02×10−94.49×10−91.15×10−92.94×10−80.61.69×10−75.61×10−76.03×10−89.13×10−70.83.88×10−61.92×10−55.44×10−73.19×10−61.04.40×10−53.24×10−43.38×10−68.45×10−5Error analysis ofy2t for Example 4 on 0,10.23.25×10−121.52×10−123.23×10−121.77×10−100.46.44×10−94.49×10−96.38×10−91.74×10−70.65.40×10−75.61×10−75.32×10−79.66×10−60.81.24×10−51.92×10−51.21×10−51.65×10−41.01.40×10−43.24×10−41.37×10−41.48×10−3Example 5.
Consider 3-dimensional SMPDDEs [22]:(30)y1′t−2y2t2−y3t=f1t,y2′t+2y3t22=f2t,t≥0,y3′t−y2t+y1t=f3t,subject to the initial conditions:(31)y10=−1,y20=0,y30=0,where f1t=−tcos1/2t,f2t=1−tsint and f3t=−tcost with the exact solutions y1t=−cost, y2t=tcost, and y3t=sint. Using the LVIA method for solving (30)-(31), at first we select the initial values of the approximations such as y1,0=−1 and y2,00=y3,00=0; then, let us seek the approximate solutions y1,k, y2,k, and y3,k. Taking ℒ for (30)-(31), we obtain(32)sy¯1s−y1,00=ℒ2y2t2+y3t+f1t,sy¯2s−y2,00=ℒ−2y3t22+f2t,sy¯3s−y3,00=ℒy2t−y1t+f3t.
The iteration formulas thus are(33)y¯1,k+1s=y¯1,ks+μssy¯1,ks−y1,00−ℒ2y2,kt2+y3,kt+f1t,y¯2,k+1s=y¯2,ks+μssy¯2s−y2,00−ℒ−2y3,kt22+f2t,y¯3,k+1s=y¯3,ks+μssy¯3,ks−y3,00−ℒy2,kt−y1,kt+f3t,with the Lagrange multiplier μs=−1/s, and by taking ℒ−1, we obtain(34)y1,k+1t=y1,kt−ℒ−11ssy¯1,ks−y1,00−ℒ2y2,kt2+y3,kt+f1t=ℒ−1−1s+ℒ−11sℒ2y2,kt2+y3,kt+f1t,y2,k+1t=y2,kt−ℒ−11ssy¯2,ks−y2,00+ℒ−2y3,kt22+f2t=ℒ−11sℒ−2y3,kt22+f2t,y3,k+1t=y3,kt−ℒ−11ssy¯3,ks−y3,00−ℒy2,kt−y1,kt+f3t=ℒ−11sℒy2,kt−y1,kt+f3t,with the initial iterations above. Now, in order to obtain the first approximation y1,1t, y2,1, and y3,1t of LVIA solution for (30)-(31), we put k=0 through (34) to get(35)y1,1t=−2tsint2−4cost2,y2,1t=tcost−sint+t,y3,1t=−tsint−cost+t+1,by using the first approximations y1,1t,y2,1t, and y3,1t and then the second approximations yi,2t,i=1,2,3, of LVIA solution for (30)-(31) which can be written in the form(36)y1,2t=2tcost22−4sint2cost2+t2+4cost2−5,y2,2t=12cost2sint2t2−7cost2sint2+12tsint2+16sint2+5tcost22−2cost2t2−14t3−4tcost2−t2+24cost2−92t−24,y3,2t=2cost22−4tcost2+12t2+16sint2−3t−2.
Using a similar approach, we get the 4th-order approximations yi,4t,i=1,2, of the LVIA solution for (30) and (31) leading to the following results:(37)y1,3t=8cost43sint4−8cost4sint4t−4sint4t2−4cost4sint4−16tsint4+512sint4−12cost42t2−t464−52cost42−80tcost4−94t2−192cost4−49t+243,y2,3t=−16cost43sint4+64cost/42sint/4t3−18cost4sint4t2+4sint4t3+24cost4sint4t−48sint4t2+1704cost4sint4−1984tsint/43+3072sint4+8tcost44−t5160+316t4+1792cost/439−336tcost42+80cost4t2−83t3+48cost42−768tcost4−3t2−9472cost/43−93t+261929,y3,3t=2cost2sint2t−4sint2t2−8tsint2+120sint2−12cost22t2−116t4−23t3+5cost22−40tcost2−54t2−48cost2−19t+43,y1,4t=142847+53788t9−297t22−t3+2cost8sint8t2−1152cost8sint8t−t615360−28cost82t2+512cost87sint8−768cost85sint8+32768cost/82sint/827−512tcost/839−192tcost88+384tcost86+64cost84t2−8cost8t3−t5128−13t4−32cost87sint8t2+48cost85sint8t2−20cost83sint8t2−896cost83sint8t−704cost83sint8+512sint8t2+1248sint8cost8−9216tsint8−112tcost84−176tcost82+192cost8t2+40448tcost/83−3648cost84−122880cost8−4210688sint/827−16320cost82,y2,4t=−160053t32−203t2−12345t364+163tcost/444−132cost43sint4t4−cost/4sint/4t6256−112cost4sint4t5+51cost/43sint/4t28−200cost/42sint/4t39+293cost/4sint/4t464−64cost42sint4t2+2276cos14tsin14tt3−15840cos14tsin14tt−35768t6−57595cost/4sint/4t216+53440cost/42sint/4t27+8288sint4t2−2723t53840−49t448472960cost/4381+1175cost42t2−363cost/43sint/44+5780sint/4t39879851sint/4cost/48−1288192tsint/427−126907tcost/424+21080cost/4t2345312tcost4−77760cost42−2923648cost/427−1298176sint/49116cost4t6+19cost/4t512−114sint4t5−313cost/4t46−184sint/4t43−2908cost/4t33−t86144−13t74608−t9294912+1459654481+23cost43t4−2720cost/43t29+1553cost/42t38−58cost44t3−5cost/42t564/3cost43t3−32cost42t4−1600cost/43t3+17920cost/42sint/49,y3,4t=cost4sint4t2+8192cost/42sint/427−800cost4sint4t+512sint4t2320cost4sint4−4160tsint4−1025024sint/427−t6960+13t5320+24cost44−256cost/43t9+36cost42t2−16cost4t3−23t4−60tcost42+176cost4t2−16t3−5016cost42+20032tcost/43−245t22−26880cost4+24725t9+31872.
To clarify the convergence ofyi,kt to yit,i=1,2,3, with respect to the number of iterations of the solution, we present numerical solutions of Example 5 graphically, where Figure 5 shows the exact solutions yit,i=1,2,3, and the approximate solution of LVIA for yi,kt,i=1,2,3 and k=1,3,5, respectively. We will test the accuracy of LVIA for Example 5 by using error analysis of yit, i=1,2,3, t∈0,1 for (30)-(31) with a step size of 0.2 as well as comparison among the absolute errors, relative errors, consecutive errors residual errors, and computing times (CPU) of 4th-order approximate LVIA solutions shown in Table 6.Figure 5
Plots of the LVIA solutionsyi,kt,i=1,2,3, k=1,3,5, respectively, and the exact solution yit, i=1,2,3 of Example 5 on 0,1, where yit is represented by a straight line and yi,kt is represented by a dashed line.Table 6
Error analysis and CPU times.
tAbsolute errorExt24relative errorRel24consecutive errorCon24residual errorRes34tCPUError analysis and CPU times ofy1t for Example 5 on 0,10.25.77×10−55.89×10−57.87×10−51.24×10−40.0580.44.16×10−54.52×10−52.27×10−31.37×10−30.0570.66.72×10−43.69×10−31.08×10−26.66×10−30.0510.82.57×10−33.69×10−33.41×10−22.23×10−20.0491.07.43×10−31.38×10−28.30×10−25.71×10−20.047Error analysis and CPU times ofy2t for Example 5 on 0,10.21.27×10−54.86×10−47.34×10−67.27×10−60.0540.46.88×10−53.91×10−31.56×10−46.16×10−50.0610.63.17×10−51.39×10−21.30×10−34.43×10−40.0530.81.18×10−43.64×10−25.49×10−31.95×10−30.0501.09.11×10−48.53×10−21.65×10−26.06×10−30.048Error analysis and CPU times ofy3t for Example 5 on 0,10.29.33×10−64.70×10−57.87×10−52.74×10−40.0770.41.88×10−44.84×10−42.27×10−32.38×10−30.0570.61.49×10−32.64×10−31.08×10−21.19×10−20.0570.86.57×10−39.15×10−33.14×10−23.97×10−30.0511.02.08×10−22.47×10−28.30×10−29.92×10−20.049Example 6.
Consider 3-dimensional nonlinear SMPDDEs [23]:(38)y1′t=y22t,y2′t=12y1t2,t≥0,y3′t=e5/2ty2t+9e2ty3tt3,subject to the initial conditions:(39)y10=1,y20=1,y30=0.
The exact solutions arey1t=et, y2t=et/2, and y3t=te3t. By comparing between the absolute errors of LVIA for four iterations approximations and Adomian decomposition method (ADM) [23] which are shown in Table 7, it is obvious that, from this table, the approximate solutions are identical to the exact solutions for all possible values of t in 0,1. Figure 6 shows that the residual error is the difference between both sides of equation (38), indicating that the two sides are identical with the increase of k, which shows the efficiency of the proposed method.Table 7
Comparison of (absolute error)Extik for Example 6.
ty1,4ty2,4ty3,4tADM [23]LVIAADM [23]LVIAADM [23]LVIA0.00000000.11.00×10−92.31×10−1101.39×10−123.00×10−92.57×10−110.29.10×10−81.59×10−91.00×10−99.13×10−114.30×10−82.08×10−90.31.06×10−61.90×10−81.60×10−81.07×10−93.00×10−93.00×10−80.46.03×10−61.14×10−79.10×10−86.14×10−95.05×10−72.14×10−70.52.34×10−54.60×10−73.52×10−72.40×10−82.87×10−61.04×10−60.67.084×10−51.47×10−61.06×10−67.36×10−81.01×10−53.94×10−60.71.81×10−43.92×10−62.69×10−61.90×10−73.16×10−51.26×10−50.84.10×10−49.28×10−66.03×10−64.35×10−78.86×10−51.36×10−50.98.45×10−42.00×10−51.23×10−59.03×10−72.28×10−49.26×10−51.01.62×10−34.00×10−52.33×10−51.74×10−65.40×10−42.22×10−4Figure 6
The residual errorResikt,i=1,2,3 of LVIA solutions of (38) at k=4,6,8,10, respectively, for Example 6.


## 4. Conclusion

The main objective of this paper is to adapt Laplace variational iteration algorithm to investigate SMPDDEs. This work aims to show the power of LVIA by reducing the numerical calculation which does not need any perturbations, discretization, and other restrictive hypotheses which may change the framework for the problem being solved. LVIA gives fast convergent successive approximations. We approve that the accuracy for LVIA gives it numerous wider applicability. Additionally, the results obtained by LVIA have the ability to be successful and furthermore helpful in linear and nonlinear cases since they do not require much computational work. It is obvious from the tables and figures that the absolute errors relief ask rises. In any case, the numerical results show that LVIA is a reliable technique for SMPDDEs. LVIA is a relatively new approach to provide a symbolic approximation, in a rapidly convergent series by using the initial conditions only and without tedious calculations which show the solution in a closed form, for linear and nonlinear problems and it changes SMPDDEs to the recurrence sequence of functions. Though the LVIA leads to convergent solutions for a wider region as in Examples 1 and 2, but in certain situations, the series converges very quickly in a very small zone or neighborhood of the boundary points, and it has a very slow convergence rate in the wider and/or outer zone.



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