# On a New Model Based on Third-Order Nonlinear Multisingular Functional Differential Equations

Authors: Zulqurnain Sabir; Hatıra Günerhan; Juan L. G. Guirao
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this study, a novel mathematical model based on third-order nonlinear multisingular functional differential equations (MS-FDEs) is presented. The designed model is solved by using a well-known differential transformation (DT) scheme that is a very credible tool for solving the nonlinear third-order nonlinear MS-FDEs. In order to check the exactness, efficacy, and convergence of the scheme, some numerical examples are presented based on nonlinear third-order MS-FDEs and numerically solved by using DT scheme. The scheme of differential transformation allows us to find a complete solution and a closed approximate solution of the differential equation. The distinctive advantage of the computational technique is to deal with the complex and monotonous physical problems that are obtained in various branches of engineering and natural sciences. Moreover, a comparison of the obtained numerical outcomes from the exact solutions shows the correctness, accurateness, and exactness of the designed model as well as the presented scheme.


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## Body

## 1. Introduction

The singular study along with functional differential equations (FDEs) is considered very significant for the researcher’s community, and the implementations of the FDEs have been noticed in the sixth decade of the nineteen century. The FDEs have a huge variety of applications in many fields; to mention few of them are, models of population growth [1], electrodynamics [2], infection models of HIV-1 [3], models of tumor growth [4], models based on chemical kinetics [5], B-virus infection hepatitis models [6], models of the gene regulations [7], and models of viral infections [8], and many more [9–14]. The singular study based on the differential models is very interesting, complicated, experimental, and challenging for the researchers due to the singularity appearance at origin. There are many singular models in the literature; one of the famous models is Lane–Emden that represents singularity at x = 0 always. The model of the Lane–Emden is famous as its historic point of view and has been applied broadly due to its huge important and significant applications in the fields of science and technology. Some of the important applications of the Lane–Emden model are that it is used in various phenomena of mathematical physics structure and in the study of astrophysics, such as models of the stellar structure [15], study of thermal explosions model [16], study of the model of isothermal gas spheres [17], oscillating magnetic fields [18], and thermionic currents [19].In recent decades, the research community is interested to solve the singular nonlinear FDEs numerically due to the singularity and functionality in differential equations. For example, to present the solutions of these nonlinear FDEs, Kadalbajoo and Sharma [20, 21] applied a numerical scheme. In order to solve differential-difference based model, Mirzaee and Hoseini [22] implemented a numerical collocation scheme. Xu and Jin [23] explained the singularly functional perturbed differential model by applying the fractional steps and boundary functions. Geng et al [24] applied a numerical approach for the delay differential equations on the basis of singularly perturbed model.The purpose of the recent study is to present the model based on the nonlinear multisingular (MS) functional differential equations, i.e., MS-FDEs of order three. The modeled form of the third-order nonlinear MS-FDE along with its initial conditions (ICs) is written as(1)y‴x+τ1+αxy″x+τ2+βx2y′x+τ3+xyx+τ4=fx,y0=a,y′0=b,y″0=c.The parametersτi (i=1,2,3,4) and α,β,a,b,andc are the real constant values.The idea of the above model is achieved by extending the work of Sabir et al. [25] that is used to explain the nonlinear singular FDEs of second order. For the verification and correctness of the designed MS-FDEs model, three different examples have been modeled and numerically solved by using the well-known differential transformation (DT) scheme, and the obtained numerical outcomes of DT scheme are compared with the exact solutions. The DT scheme has been applied to solve many stiff, nonstiff, singular, nonsingular, linear, and nonlinear types of problems. Zhou [26], for the first time, presented the idea of the DT scheme at the end of the 19th century to solve the linear/nonlinear initial value problems based on the analysis of electrical circuit. The DT scheme is basically a numerical approach, which works on the basis of the expansion of TS, which constitutes a polynomial form of the analytic results. The quality of the numerical DT scheme is to require less work and does not require linearization as well as assumptions. This numerical scheme is designed on the basis of an analytical solution by using the polynomial expressions, such as the Taylor series (TS) expansion. But its procedure is more easier than the conventional higher-order TS scheme, which achieves symbolic computation of the necessary derivatives using the data-based functions. Three explanatory and illustrative examples based on model (1) are provided to show the efficacy of the obtained results from the DT scheme. These numerical outcomes are compared with the exact solutions that indicate the proficiency of the designed model as well as the proposed scheme.Some major key factors of the present study are summarized as follows:The mathematical modeled form of the third-order nonlinear MS-FDEs is presented successfully by extending the work of Sabir et al. [25]The designed nonlinear MS-FDEs based on the designed model are addressed numerically by using the famous DT scheme.Manipulation of the present scheme is to apply the brilliance-obtained outcomes for nonlinear MS-FDEs with better precision and outstanding consistency.The reliability and correctness of the designed model are authentic through the comparison of the numerical results obtained by the DT scheme and the exact results. The overlapping of these results indicates perfection, excellence, and faultlessness for the model.The third-order functional differential model given in equation (1) is not easy to solve because of nonlinearity, multisingularity, functionality, and harder in nature. DT scheme is the best choice and good selection to handle these types of complicated and complex models.The rest of the paper is described as follows: the designed detailed methodology on the basis of DT scheme is provided in Section2. Results and discussion are provided in Section 3. Conclusion along with future research direction is provided in the last section.


## 2. Methodology (Differential Transform Scheme)

The mathematical definition of DT scheme usingyζ is given as(2)Yk=1k!dkyζdζkx=0.The original function in the above equation (2) is yζ, whereas the transformed function (TF) is denoted by Yk, which is also called the T-function. The inverse of DT scheme of Yk is provided as(3)yζ=∑k=0∞Ykζ−ζ0k≡D−1Yk.Using the results of equations (2) and (3), the obtained function becomes(4)yζ=∑k=0∞ζkk!dkyζdζkx=0.Equation (4) provides the concepts of DT scheme that are derived from the TS expansion, and this scheme has not been applied for symbolical assessment of the derivatives. Moreover, comparative derivative values are achieved by using the iterative procedure, which is defined by the transformed original function. In this study, the lowercase and uppercase letters are used to show the original function and the TF, respectively. Using the nature of the above two equations, one can easily prove the TFs have the basic mathematical values provided in Table 1.Table 1
The essential operations of DT scheme.
Unique function
TF
y
ζ
=
u
ζ
±
v
ζ
Y
k
=
U
k
±
V
k
y
ζ
=
c
u
ζ
Y
k
=
c
U
k
y
ζ
=
d
m
u
ζ
/
d
ζ
m
Y
k
=
k
+
1
+
k
+
2
⋯
k
+
m
U
k
+
m
y
ζ
=
u
c
ζ
Y
k
=
c
k
U
k
y
ζ
=
u
ζ
/
c
Y
k
=
U
k
/
c
k
y
ζ
=
d
m
/
d
ζ
m
u
c
ζ
Y
k
=
k
+
1
+
k
+
2
⋯
k
+
m
c
k
+
m
U
k
+
mIn real applications,yζ is obtained by a finite series and equation (3) can be described as(5)yζ=∑k=0mζkYk.Equation (5) shows the term ∑k=m+1∞ζkYk, which is very small and can be neglected, while m shows the convergence of natural frequency.For better explanation of the DT scheme, some important theorems are presented as follows:Theorem 1.
Iffζ=mζoζ, then FK=Mk⊗Ok=∑l=0kMlOk−l (here, ⊗ denotes the convolution):(6)fζ=∑k=0∞Mkζ−ζ0k×∑k=0∞Okζ−ζ0k=M0+M1ζ−ζ0+M2ζ−ζ02+⋯=M0O0+MU0O1+M1O0ζ−ζ0×O0+O1ζ−ζ0+O2ζ−ζ02+⋯+M0O2+M1O1+M2O0ζ−ζ02+⋯=∑k=0∞∑l=0kMlOk−lζ−ζ0k.By using equation (3), we get(7)FK=∑l=0kMlOk−l.Theorem 2.
Iffζ=ζϕ, then(8)Fh=δh−ϕ=1,h=ϕ,0,h≠ϕ.Proof.
By using equation (2), we have(9)Fh=1h!∂ζϕ∂ζht=0=1h!∂hζh∂ζh=h!h!=1,h=ϕ,1h!∂hζϕ∂ζh=0,h≠ϕ.Theorem 3.
By takingfζ=eζ+c, we have Fk=ec/k!.Proof.
Using equation (2), we get(10)Fk=1k!∂eζ+c∂ζkt=0=ec∂ek∂ζkt=0=eck!.Theorem 4.
Iffζ=qζ+m, then [27](11)Fk=∑i=kNmi−kikQiforN⟶∞.Proof.
By using differential inverse transform ofYk, we have(12)fζ=∑l=2∞Qkζ−ζ0+mk=Q0+Q1ζ−ζ0+m+Q2ζ−ζ0+m2+Q3ζ−ζ0+m3+⋯=Q0+Q1ζ−ζ0+Q1m+Q2ζ−ζ02+Q2m2+2Q2ζ−ζ0m+Q3m3+3Q3ζ−ζ0m2+3Q3ζ−ζ02m+Q3ζ−ζ03+⋯=Q0+Q1m+Q2m2+Q3m3+⋯+ζ−ζ0Q1+2Q2m+3Q3m2+⋯+x−x02Q2+3Q3m+⋯+ζ−ζ03Q3+⋯+⋯=∑l=0∞l!0!l−0!ml−0Ql+∑l=0∞l!1!l−1!ml−1Qlζ−ζ0+∑l=2∞l!2!l−2!ml−2Qlζ−ζ02+∑l=0∞l!3!l−3!ml−3Qlζ−ζ03+⋯+∑l=0∞l!k!l−k!ml−kQlζ−ζ0k=∑k=0∞∑l=k∞l!k!l−k!ml−kQlζ−ζ0k=∑k=0∞∑l=k∞lkml−kQlζ−ζ0k.By comparing equations (3) and (12), Yk becomes(13)Fk=∑l=kNlkml−kQlforN⟶∞.Theorem 5.
Ifyζ=dbuζ+m/dζb, then(14)Yh=h+b!h!∑l=h+bNml−h−blh+bUlforN⟶∞.Proof.
Supposeyζ=uζ+m, in equation (2), we have(15)Yh=1h!∂h∂ζhdbyζdζb=h+b!h!Yh+b.
Using theYk values from equation (13), it becomes(16)h+b!h!Yh+b=h+b!h!dh+b∑l=hNlhml−kUldζb=h+b!h!∑l=h+bNlh+bml−h+bUl,Yh=1h!∂h∂ζhdbyζdζb=1h!∂h∂ζhdbuζ+mdζb=h+b!h!∑l=h+bNlh+bml−h−bUl,N⟶∞.


## 3. Results and Discussion

To present the numerical solutions based on the designed third-order nonlinear MS-FDE model, the nonlinear study is very important and many investigations have been provided in references [28–33]. Three different examples have been presented, and the solutions of the examples are performed by using the DT scheme.Example 1.
Consider the nonlinear third-order MS-FDE given as(17)y″ψ−1+1ψy″ψ+1+2ψ2y′ψ+2+ψyψ=eψ−1+1ψeψ+1+2ψ2eψ+2+ψeψ,y0=1,y′0=1,y″0=1.
Multiplied byψ2, the achieved form is given as(18)ψ2y‴ψ−1+ψy″ψ+1+2y′ψ+2+ψ3yψ=ψ2eψ−1+ψeψ+1+2eψ+2+ψ3eψ,
The DT scheme is applied to solve the model given in equation (17). By using the definitions of one-dimensional DT and the corresponding transformation of equation (17), the obtained system becomes(19)δk−2⊗k+3k+2k+1∑σ=k+3N−1σ−k−3σk+3Yσ+δk−1⊗k+2k+1∑σ=k+2N1σ−k−2σk+2Yσ+2k+1∑σ=k+1N2σ−k−1σk+1Yσ+δk−3⊗Yk=δk−2⊗1k!e−1+δk−1⊗1k!e+21k!e2+δk−3⊗1k!.
Theorem 1 is used in equation (19), we get(20)∑η=0k∑i=η+3Nδk−η−2η+3η+2η+1−1i−η−3ση+3Yσ+∑η=0k∑σ=v+2Nδk−η−1η+2η+1ση+2Yσ+2k+1∑i=k+1N2σ−k−1σk+1Yσ+∑η=0kδη−3Yk−η=∑η=0kδk−η−21η!e−1+∑η=0kδk−η−11η!e+21k!e2+∑η=0kδk−η−31η!.
Using the ICs given in equation (17), we have(21)Y0=1,Y1=1,Y2=12.
TakingN=4 and k=0and1, by using equations (20) and (21), the obtained linear algebraic equation system is written as(22)12Y3+32Y4+3=e2,30Y3+108Y4+3=e+2e2.
By solving the above coupled equations given in (22), we have(23)y3=−57−8e+11e284,y4=9+2e−e256.
By using the values ofYk for k=0​ and1 in yψ, i.e., the inverse-reduced DT, the results are written as(24)yψ=∑k=0∞Ykψk=1+ψ+12ψ2+−57−8e+11e284ψ3+9+2e−e256ψ4+Oψ5.
Repeat the process by using the equations (20) and (25) for N=6 and k=0,1,and2. The solution of the obtained linear algebraic equations system is given as(25)y3=−57−8e+11e284,y4=9+2e−e256,y5=128e2−128e−49517120,y6=−8e2+8e+515136.
By using the inverse-reduced DTyk, the solutions will be as follows:(26)yψ=∑k=0∞Ykψk=1+ψ+12ψ2+−57−8e+11e284ψ3+9+2e−e256ψ4+128e2−128e−49517120ψ5+−8e2+8e+515136ψ6+Oψ7.
Table2 shows the comparison of the present numerical results for N=4 and N=6 with the exact solutions. The yψ results are slightly varied by changing the N parameter values. It is clear in Table 2 that the proposed and exact solutions overlapped each other.Table 2
Comparison of the obtained results and exact solutions forN = 4 and N = 6.
ψ
DT (N=4)
DT (N=6)
Exact solution
0.01
1.01005
1.01005
1.01005
0.02
1.02020
1.02020
1.02020
0.03
1.03045
1.03045
1.03045
0.04
1.04080
1.04080
1.04081
0.05
1.05125
1.05125
1.05127
0.06
1.06180
1.06180
1.06183
0.07
1.07246
1.07246
1.07250
0.08
1.08322
1.09408
1.08328
0.09
1.09408
1.09408
1.09417
0.1
1.10504
1.10504
1.10517
0.2
1.22044
1.22044
1.22140
0.3
1.34683
1.34685
1.34985
0.4
1.48515
1.48522
1.49182
0.5
1.63663
1.63686
1.64872
0.6
1.80282
1.80341
1.82211
0.7
1.98556
1.98688
2.01375
0.8
2.18698
2.18965
2.22554
0.9
2.40955
2.41451
2.45960Example 2.
Consider the third-order MS-FDEs with its ICs:(27)y‴ψ−1+1ψy″ψ+1+2ψ2y′ψ+2+ψyψ=ψ5+45ψ+48+108ψ+64ψ2,y0=1,y′0=0,y″0=0.
Equation (27) becomes(28)ψ2y‴ψ−1+ψy″ψ+1+2y′ψ+2+ψ3yψ=ψ7+45ψ3+48ψ2+108ψ+64.
The definition of one-dimensional DT scheme is applied and taking the consistent transform of equation (27), the system is given as(29)δk−2⊗k+3k+2k+1∑σ=k+3N−1σ−k−3σk+3Yσ+δk−1⊗k+2k+1∑σ=k+2N1σ−k−2σk+2Yσ+2k+1∑σ=k+1N2σ−k−1σk+1Yσ+δk−3⊗Yk=δk−7+45δk−3+48δk−2+108δk−1+64δk.
Appling Theorem1, we have(30)∑η=0k∑σ=η+3Nδk−η−2η+3η+2η+1−1σ−η−3ση+3Yσ+∑η=0k∑σ=η+2Nδk−η−1η+2η+1σl+2Yσ+2k+1∑σ=k+1N2σ−k−1σk+1Yσ+∑η=0kδη−3Yk−η=δk−7+45δk−3+48δk−2+108δk−1+64δk.
Using the ICs given in equation (27), we have(31)Y0=1,Y1=0,Y2=0.
Taking the values ofN=4 and k=0and1 in equations (30) and (31), the obtained linear algebraic equations system is given as(32)12Y3+32Y4=32,30Y3+108Y4=108.
Solving the coupled equations given in system (32), we have(33)y3=0,y4=1.
Using theYk values for k=0and1 in the inverse-reduced DT yψ, the obtained results are given as(34)yψ=∑k=0∞Ykψk=1+ψ4.
Repeat the same procedure using equations (30) and (31), for N=5 and k=0,1,and2. The following linear algebraic system becomes(35)30Y3+108Y4+340Y5=108,12Y3+32Y4+80Y5=32,3Y3+8Y4+60Y5=8.
By solving the equation system, we have(36)y3=0,y4=1,y5=0.
By using the inverse-reduced DTyk, the solutions will be as follows:(37)yψ=∑k=0∞Ykψk=1+ψ4.This is the exact solution of the Example 2.Example 3.
Consider the third-order MS-FDEs(38)y‴ψ−1+1ψy″ψ+1+2x2y′ψ+2+ψyψ=ψ4+ψ+18+30ψ+24ψ2,y0=1,y′0=0,y″0=0.
Multiplying byψ2, equation (38) takes the form as(39)ψ2y‴ψ−1+ψy″ψ+12y′ψ+2+ψ3yψ=ψ6+ψ3+18ψ2+30ψ+24.
Using the definitions of the one-dimensional DT scheme, we get(40)δk−2⊗k+3k+2k+1∑σ=k+3N−1σ−k−3σk+3Yσ+δk−1⊗k+2k+1∑σ=k+2N1σ−k−2σk+2Yσ+2k+1∑σ=k+1N2σ−k−1σk+1Yσ+δk−3⊗Yk=δk−6+δk−3+18δk−2+30δk−1+24δk.
By using Theorem1, we get(41)∑η=0k∑σ=η+3Nδk−η−2η+3η+2η+1−1σ−l−3ση+3Yσ+∑η=0k∑σ=η+2Nδk−η−1η+2η+1ση+2Yσ+2k+1∑σ=k+1N2σ−k−1σk+1Yσ+∑η=0kδη−3Yk−η=δk−6+δk−3+18δk−2+30δk−1+24δk.
Using the ICs of equation (38), we have(42)Y0=1,Y1=0,Y2=0.
TakingN=4 and k=0and1 in equations (41) and (42), the linear algebraic equations system is achieved as(43)12Y3+32Y4=12,30Y3+108Y4=30.
Solving the above system, we get(44)y3=1,y4=0.
Using the values ofYk for k=0and1 into the inverse-reduced DT of yψ, the solution will be as follows:(45)yψ=∑k=0∞Ykψk=1+ψ3.
Repeat the same process forN=5 and k=0,1,and2 by using the above equations. The obtained linear algebraic equation system is given as(46)30Y3+108Y4+340Y5=30,12Y3+32Y4+80Y5=12,3Y3+8Y4+60Y5=3.
The solution of the obtained system of equations is(47)y3=1,y4=0,y5=0.
By using the inverse-reduced DT ofyk, the solutions becomes(48)yψ=∑k=0∞Ykψk=1+ψ3.which is the exact solution. yψ is calculated for different values of N and shown in Figure 1(c).
For more clear understanding, Figure1 is plotted that has been drawn between 0.01 and 0.9. The values of N are taken as 4 and 6. One can see that the exact and present solutions for N=4 and N=6 are overlapped to each other in the range of 0.01 to 0.09. However, by increasing a slight value in the step size, the results are slightly different but accurate. So it can be concluded that small step size gives more accurate values as compared to large step size.Figure 1
Plots ofyψ for Examples 1–3 in the range of 0.01<ψ<0.9. Comparison of numerical results of yψ for (a) Example 1, (b) Example 2, and (c) Example 3.
(a)
(b)
(c)


## 4. Conclusion

The task to model the third-order MS-FDEs is very difficult to handle as well as construct the differential equations of the designed model. The numerical differential transformation scheme is applied successfully to check the correctness and the accurateness of the designed model. The traditional/conventional techniques fail to solve such multisingular, nonlinear, functionality, and harder nature models. The numerical differential transformation scheme is a good choice to solve such types of complicated, nonlinear, and multisingular models. Consequently, the adopted scheme is effective as well as suitable too. The present study shows that the DTM is an effective and suitable technique to solve such types of equations that we have investigated here. The comparison of the exact and solutions obtained from the differential transformation scheme has also been presented in tabular form as well as graphically. The overlapping of the results shows the perfection of the designed model and establishes the worth of the designed scheme. However, it is observed that when the step size is small, the results are more accurate, but making a slight increase in the step size, the results are overlapped and the error is reduced.In future, a system of third-order and fourth-order multisingular functional models will be modeled and it will be verified by using the differential transformation scheme as well as famous artificial neural networks [34–39].



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