# The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Least Squares Homotopy Perturbation Method

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Remark**

**1.**

**Proof.**

**Remark**

**2.**

## 3. Numerical Application

#### 3.1. The Case $Re=1$ and $Ha=0$

- First-term approximations:$$\begin{array}{c}{U}_{0}\left(y\right)=1-y\\ {V}_{0}\left(y\right)=3{y}^{2}-{y}^{3}\end{array}$$
- Second-term approximations:$$\begin{array}{c}{U}_{1}\left(y\right)={\displaystyle \frac{{y}^{5}}{5}}-{\displaystyle \frac{3{y}^{4}}{4}}+{y}^{3}-{\displaystyle \frac{29y}{20}}+1\\ {V}_{1}\left(y\right)={\displaystyle \frac{2{y}^{7}}{35}}-{\displaystyle \frac{{y}^{6}}{5}}+{\displaystyle \frac{3{y}^{5}}{10}}-{\displaystyle \frac{167{y}^{3}}{70}}+{\displaystyle \frac{113{y}^{2}}{35}}\end{array}$$
- Third-term approximations:$$\begin{array}{c}{U}_{2}\left(y\right)={\displaystyle \frac{2{y}^{9}}{315}}-{\displaystyle \frac{19{y}^{8}}{560}}+{\displaystyle \frac{{y}^{7}}{20}}-{\displaystyle \frac{{y}^{6}}{20}}+{\displaystyle \frac{23{y}^{5}}{70}}-{\displaystyle \frac{1643{y}^{4}}{1680}}+{\displaystyle \frac{113{y}^{3}}{105}}-{\displaystyle \frac{1763y}{1260}}+1\\ {V}_{2}\left(y\right)={\displaystyle \frac{4{y}^{11}}{5775}}-{\displaystyle \frac{2{y}^{10}}{525}}+{\displaystyle \frac{{y}^{9}}{210}}-{\displaystyle \frac{3{y}^{8}}{560}}+{\displaystyle \frac{97{y}^{7}}{1225}}-{\displaystyle \frac{533{y}^{6}}{2100}}+{\displaystyle \frac{121{y}^{5}}{350}}-{\displaystyle \frac{774469{y}^{3}}{323400}}+{\displaystyle \frac{2087479{y}^{2}}{646800}}\end{array}$$

- Second-term approximations:$\tilde{U}\left(y\right)=0.25139431098009168490{y}^{5}-0.89052033909540654567{y}^{4}$$+1.0440071928069991650{y}^{3}-1.4048811646916843042y+1$$\tilde{V}\left(y\right)=0.058317127462336779863{y}^{7}-0.21889392791262717538{y}^{6}$$+0.32522212582352969927{y}^{5}-2.3916763031317642956{y}^{3}+3.2270309777585249919{y}^{2}$
- Third-term approximations:$\tilde{U}\left(y\right)=-0.012783174216369037776{y}^{9}+0.079312017411340841618{y}^{8}$$-0.21264602372589301173{y}^{7}+0.26096763633169609359{y}^{6}$$+0.13236850157788926145{y}^{5}-0.90665763559833919190{y}^{4}+1.0653323749944480412{y}^{3}$$-1.4058936967747729964y+1$$\tilde{V}\left(y\right)=0.00077252197269867863129{y}^{11}-0.0033512835587865196224{y}^{10}$$+0.0028365610269174046158{y}^{9}-0.0031087019658224288029{y}^{8}$$+0.079915383004597381009{y}^{7}-0.25695066296630124619{y}^{6}$$+0.34706280997410686745{y}^{5}-2.3943088377575857329{y}^{3}+3.2271322102701755958{y}^{2}$

#### 3.2. The Case $Re=1$ and $Ha=1$

- First-term approximations:$$\begin{array}{c}{U}_{0}\left(y\right)=1-y\\ {V}_{0}\left(y\right)=3{y}^{2}-2{y}^{3}\end{array}$$
- Second-term approximations:$$\begin{array}{c}{U}_{1}\left(y\right)={\displaystyle \frac{{y}^{5}}{5}}-{\displaystyle \frac{3{y}^{4}}{4}}+{\displaystyle \frac{5{y}^{3}}{6}}+{\displaystyle \frac{{y}^{2}}{2}}-{\displaystyle \frac{107y}{60}}+1\\ {V}_{1}\left(y\right)={\displaystyle \frac{2{y}^{7}}{35}}-{\displaystyle \frac{{y}^{6}}{5}}+{\displaystyle \frac{{y}^{5}}{5}}+{\displaystyle \frac{{y}^{4}}{4}}-{\displaystyle \frac{181{y}^{3}}{70}}+{\displaystyle \frac{459{y}^{2}}{140}}\end{array}$$
- Third-term approximations:$$\begin{array}{c}{U}_{2}\left(y\right)={\displaystyle \frac{2{y}^{9}}{315}}-{\displaystyle \frac{19{y}^{8}}{560}}+{\displaystyle \frac{9{y}^{7}}{140}}-{\displaystyle \frac{2{y}^{6}}{15}}+{\displaystyle \frac{2129{y}^{5}}{4200}}-{\displaystyle \frac{451{y}^{4}}{420}}+{\displaystyle \frac{401{y}^{3}}{504}}+{\displaystyle \frac{{y}^{2}}{2}}-{\displaystyle \frac{41129y}{25200}}+1\\ {V}_{2}\left(y\right)={\displaystyle \frac{4{y}^{11}}{5775}}-{\displaystyle \frac{2{y}^{10}}{525}}+{\displaystyle \frac{{y}^{9}}{140}}-{\displaystyle \frac{9{y}^{8}}{560}}+{\displaystyle \frac{1507{y}^{7}}{14700}}-{\displaystyle \frac{1129{y}^{6}}{4200}}+{\displaystyle \frac{317{y}^{5}}{1400}}+{\displaystyle \frac{153{y}^{4}}{560}}\\ -{\displaystyle \frac{838379{y}^{3}}{323400}}+{\displaystyle \frac{352623{y}^{2}}{107800}}\end{array}$$

- Second-term approximations:$\tilde{U}\left(y\right)=0.24252310472551437387{y}^{5}-0.79607170987751279666{y}^{4}$$+0.70039529434560076978{y}^{3}+0.51680281524640722276{y}^{2}$$-1.6636495044400095698y+1$$\tilde{V}\left(y\right)=0.058199385875154810585{y}^{7}-0.20176414373359385855{y}^{6}$$+0.17947791112252870094{y}^{5}+0.28462933749944678084{y}^{4}-2.5916327628078782832{y}^{3}$$+3.2710902720443418494{y}^{2}$
- Third-term approximations:$\tilde{U}\left(y\right)=-0.039570918805227895983{y}^{9}+0.18039801278862870509{y}^{8}$$-0.33738809541609613996{y}^{7}+0.25556139173308396128{y}^{6}+0.28898499486097221037{y}^{5}$$-0.98687905924276726841{y}^{4}+0.80192640217575596712{y}^{3}+0.50063075107844706510{y}^{2}$$-1.6636634791727966046y+1$$\tilde{V}\left(y\right)=0.00070249628967946626576{y}^{11}-0.0034005002946346048030{y}^{10}$$+0.0063398061612668565765{y}^{9}-0.017261143110682215908{y}^{8}$$+0.10727677845680360313{y}^{7}-0.27266199870132195652{y}^{6}+0.22694957760601929024{y}^{5}$$+0.27260900187074289280{y}^{4}-2.5917328827530869045{y}^{3}+3.2711788644752135727{y}^{2}$

## 4. Discussion of the Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

- Bota, C.; Caruntu, B. Approximate analytical solutions of nonlinear differential equations using the Least Squares Homotopy Perturbation Method. J. Math. Anal. Appl.
**2017**, 448, 401–408. [Google Scholar] [CrossRef] - Bota, C.; Caruntu, B.; Lazureanu, C. The Least Squares Homotopy Perturbation Method for boundary value problems. Appl. Comput. Math.
**2017**, 16, 39–47. [Google Scholar] - Qayyum, M.; Oscar, I. Least Square Homotopy Perturbation Method for Ordinary Differential Equations. J. Math.
**2021**, 2021, 7059194. [Google Scholar] [CrossRef] - Thabet, H.; Kendre, S. Modified least squares homotopy perturbation method for solving fractional partial differential equations. Malaya J. Mat.
**2018**, 6, 420–427. [Google Scholar] [CrossRef] [Green Version] - Kumar, R.; Koundal, R.; Shehzad, S.A. Generalized least square homotopy perturbation solution of fractional telegraph equations. Comput. Appl. Math.
**2019**, 38, 184. [Google Scholar] [CrossRef] - Zhang, J.; Wei, Z.; Li, L.; Zhou, C. Least-Squares Residual Power Series Method for the Time-Fractional Differential Equations. Complexity
**2019**, 2019, 6159024. [Google Scholar] [CrossRef] - Das, P.; Rana, S. Theoretical prospects of fractional order weakly singular Volterra Integro differential equations and their approximations with convergence analysis. Math. Methods Appl. Sci.
**2021**, 44, 9419–9440. [Google Scholar] [CrossRef] - Kumar, R.; Koundal, R.; Shehzad, S.A. Modified homotopy perturbation approach for the system of fractional partial differential equations: A utility of fractional Wronskian. Math. Methods Appl. Sci.
**2022**, 45, 809–826. [Google Scholar] [CrossRef] - He, J.H. Homotopy perturbation technique. Comput. Methods Appl. Mech. Eng.
**1999**, 178, 257–262. [Google Scholar] [CrossRef] - He, J.H. Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals
**2005**, 26, 695–700. [Google Scholar] [CrossRef] - He, J.H. Variational iteration method, a kind of nonlinear analytical technique. Some examples. Int. J. Non-Linear Mech.
**1999**, 34, 699–708. [Google Scholar] [CrossRef] - Parsa, A.B.; Rashidi, M.M.; Beg, O.A.; Sardi, S.M. Semi-computational simulation of magneto-hemodynamic flow in a semi-porous channel using optimal homotopy and differential transform methods. Comput. Biol. Med.
**2013**, 43, 1142–1153. [Google Scholar] [CrossRef] [PubMed] - Adomian, G.A. Review of the decomposition method in applied mathematics. J. Math. Anal. Appl.
**1998**, 135, 501–544. [Google Scholar] [CrossRef] [Green Version] - Marinca, V.; Herisanu, N.; Bota, C. An optimal homotopy asymptotic method applied to the steady flow of a fourth-grade fluid past a porous plate. Appl. Math. Lett.
**2009**, 22, 245–251. [Google Scholar] [CrossRef] [Green Version] - Srivastava, V.P. A Theoretical Model for Blood Flow in Small Vessels. Int. J. Appl. Appl. Math.
**2007**, 2, 51–65. [Google Scholar] - Wang, C.Y.; Bassingthwaighte, J.B. Blood Flow in Small Curved Tubes. J. Biomech. Eng.
**2003**, 125, 910–913. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bali, R.; Awasthi, U. Mathematical model of Blood Flow in Small Blood Vessel in the Presence of Magnetic Field. Appl. Math.
**2001**, 2, 264–269. [Google Scholar] [CrossRef] [Green Version] - Yamamoto, T.; Nagayama, Y.; Tamura, M. A blood-oxygenation-dependent increase in blood viscosity due to a static magnetic field. Phys. Med. Biol.
**2004**, 49, 3267. [Google Scholar] [CrossRef] [PubMed] - Alshare, A.; Tashtoush, B.; El-Khalil, H.H. Computational Modeling of Non-Newtonian Blood Flow Through Stenosed Arteries in the Presence of Magnetic Field. J. Biomech. Eng.
**2013**, 135, 1145–1153. [Google Scholar] [CrossRef] - Weng, H.C. Hydrodynamic Modeling of Targeted Magnetic-Particle Delivery in a Blood Vessel. J. Biomech. Eng.
**2013**, 135, 034504. [Google Scholar] [CrossRef] [PubMed] - Mekheimer, K.S. Peristaltic flow of blood under effect of a magnetic field in a non-uniform channels. Appl. Math. Comput.
**2004**, 153, 763–777. [Google Scholar] [CrossRef] - Tenforde, T.S. Magnetically induced electric fields and currents in the circulatory system. Prog. Biophys. Mol. Biol.
**2005**, 87, 279–288. [Google Scholar] [CrossRef] [PubMed] - Bhatti, M.M.; Zeeshan, A.; Bashir, F.; Sait, S.M.; Ellahi, R. Sinusoidal motion of small particles through a Darcy-Brinkman-Forchheimer microchannel filled with non-Newtonian fluid under electro-osmotic forces. J. Taibah Univ. Sci.
**2021**, 15, 514–529. [Google Scholar] [CrossRef] - Rostami, S.; Ellahi, R.; Oztop, H.F.; Goldanlou, A.S. A study on the effect of magnetic field and the sinusoidal boundary condition on free convective heat transfer of non-Newtonian power-law fluid in a square enclosure with two constant-temperature obstacles using lattice Boltzmann method. J. Therm. Anal. Calorim.
**2021**, 144, 2557–2573. [Google Scholar] [CrossRef] - Khan, A.S.; Xu, H.Y.; Khan, W. Magnetohydrodynamic Hybrid Nanofluid Flow Past an Exponentially Stretching Sheet with Slip Conditions. Mathematics
**2021**, 9, 3291. [Google Scholar] [CrossRef] - Rehman, A.; Salleh, Z. Influence of Marangoni Convection on Magnetohydrodynamic Viscous Dissipation and Heat Transfer on Hybrid Nanofluids in a Rotating System among Two Surfaces. Mathematics
**2021**, 9, 2242. [Google Scholar] [CrossRef] - Ali, B.; Naqvi, R.A.; Haider, A.; Hussain, D.; Hussain, S. Finite Element Study of MHD Impacts on the Rotating Flow of Casson Nanofluid with the Double Diffusion Cattaneo—Christov Heat Flux Model. Mathematics
**2020**, 8, 1555. [Google Scholar] [CrossRef] - Ryu, J.; Hu, X.; Shadden, S.C. A Coupled Lumped-Parameter and Distributed Network Model for Cerebral Pulse-Wave Hemodynamics. J. Biomech. Eng.
**2015**, 137, 101009. [Google Scholar] [CrossRef] [Green Version] - Srinivasacharya, D.; Rao, G.M. Mathematical model for blood flow through a bifurcated artery using couple stress fluid. Math. Biosci.
**2016**, 278, 37–47. [Google Scholar] [CrossRef] - Sinha, A. MHD flow and heat transfer of a third order fluid in a porous channel with stretching wall: Application to hemodynamics. Alex. Eng. J.
**2015**, 54, 1243–1252. [Google Scholar] [CrossRef] [Green Version] - Zaman, A.; Ali, N.; Sajid, M. Numerical simulation of pulsatile flow of blood in a porous-saturated overlapping stenosed artery. Math. Comput. Simul.
**2017**, 134, 1–16. [Google Scholar] [CrossRef] - Caruntu, B.; Bota, C.; Bundau, O. Analytical simulation of magneto-hemodynamic flow in a semi-porous channel using the Polynomial Least Squares Method. ITM Web Conf.
**2019**, 29, 1–13. [Google Scholar] [CrossRef] [Green Version] - Skalak, F.M.; Wang, C.Y. On the non-unique solutions of laminar flow through a porous tube or channel. SIAM J. Appl. Math.
**1978**, 34, 535–544. [Google Scholar] [CrossRef] - Quaile, J.P.; Levy, E.K. Laminar flow in a porous tube with suction. Int. J. Heat Mass Transf.
**1975**, 97, 223–243. [Google Scholar] [CrossRef] - Evans, E.A.; Skalak, R. Mechanics and Thermodynamics of Biomembranes. In Elsevier Biomedical; Hue, L., Van de Werve, G., Eds.; Elsevier Biomedical: Amsterdam, NY, USA, 1981; 464p. [Google Scholar]

**Figure 7.**The combined influence of $Re$ and $Ha$ on U. The red surface corresponds to $y=0.2$, the blue surface to $y=0.4$, the yellow surface to $y=0.6$ and the green one to $y=0.8$.

**Figure 8.**The combined influence of $Re$ and $Ha$ on V. The red surface corresponds to $y=0.2$, the blue surface to $y=0.4$, the yellow surface to $y=0.6$, and the green one to $y=0.8$.

**Table 1.**Comparison of the absolute errors of the approximate solutions U in case $Re=1$ and $Ha=0$.

y | ${\mathit{U}}_{\mathit{HAM}}$ | ${\mathit{U}}_{\mathit{DTM}}$ | ${\mathit{U}}_{\mathit{PLSM}}$ | ${\mathit{U}}_{\mathit{HPM}2\mathit{term}.}$ | ${\mathit{U}}_{\mathit{HPM}3\mathit{t}.}$ | ${\mathit{U}}_{\mathit{LSHPM}2\mathit{t}.}$ | ${\mathit{U}}_{\mathit{LSHPM}3\mathit{t}.}$ |
---|---|---|---|---|---|---|---|

0.1 | $9.04\times {10}^{-3}$ | $1.08\times {10}^{-2}$ | $8.06\times {10}^{-4}$ | $4.45\times {10}^{-3}$ | $6.77\times {10}^{-4}$ | $8.61\times {10}^{-5}$ | $3.68\times {10}^{-6}$ |

0.2 | $1.77\times {10}^{-2}$ | $1.77\times {10}^{-2}$ | $1.90\times {10}^{-3}$ | $9.08\times {10}^{-3}$ | $1.35\times {10}^{-3}$ | $8.00\times {10}^{-5}$ | $1.56\times {10}^{-6}$ |

0.3 | $2.71\times {10}^{-2}$ | $1.98\times {10}^{-2}$ | $2.14\times {10}^{-3}$ | $1.37\times {10}^{-2}$ | $2.01\times {10}^{-3}$ | $4.18\times {10}^{-6}$ | $3.41\times {10}^{-6}$ |

0.4 | $3.45\times {10}^{-2}$ | $1.79\times {10}^{-2}$ | $1.34\times {10}^{-3}$ | $1.78\times {10}^{-2}$ | $2.64\times {10}^{-3}$ | $9.71\times {10}^{-5}$ | $9.99\times {10}^{-8}$ |

0.5 | $3.73\times {10}^{-2}$ | $1.39\times {10}^{-2}$ | $8.14\times {10}^{-5}$ | $2.10\times {10}^{-2}$ | $3.16\times {10}^{-3}$ | $1.29\times {10}^{-4}$ | $2.42\times {10}^{-6}$ |

0.6 | $3.46\times {10}^{-2}$ | $9.33\times {10}^{-3}$ | $1.43\times {10}^{-3}$ | $2.24\times {10}^{-2}$ | $3.50\times {10}^{-3}$ | $7.69\times {10}^{-5}$ | $1.22\times {10}^{-6}$ |

0.7 | $2.77\times {10}^{-2}$ | $5.39\times {10}^{-3}$ | $2.09\times {10}^{-3}$ | $2.15\times {10}^{-2}$ | $3.54\times {10}^{-3}$ | $2.59\times {10}^{-5}$ | $9.10\times {10}^{-7}$ |

0.8 | $1.87\times {10}^{-2}$ | $2.63\times {10}^{-3}$ | $1.74\times {10}^{-3}$ | $1.78\times {10}^{-2}$ | $3.12\times {10}^{-3}$ | $9.63\times {10}^{-5}$ | $7.61\times {10}^{-7}$ |

0.9 | $9.27\times {10}^{-3}$ | $9.33\times {10}^{-4}$ | $6.99\times {10}^{-4}$ | $1.07\times {10}^{-2}$ | $2.02\times {10}^{-3}$ | $6.71\times {10}^{-5}$ | $4.05\times {10}^{-7}$ |

**Table 2.**Comparison of the absolute errors of the approximate solutions V in case $Re=1$ and $Ha=0$.

y | ${\mathit{V}}_{\mathit{HAM}}$ | ${\mathit{V}}_{\mathit{DTM}}$ | ${\mathit{V}}_{\mathit{PLSM}}$ | ${\mathit{V}}_{\mathit{HPM}2\mathit{term}.}$ | ${\mathit{V}}_{\mathit{HPM}3\mathit{t}.}$ | ${\mathit{V}}_{\mathit{LSHPM}2\mathit{t}.}$ | ${\mathit{V}}_{\mathit{LSHPM}3\mathit{t}.}$ |
---|---|---|---|---|---|---|---|

0.1 | $2.21\times {10}^{-4}$ | $1.19\times {10}^{-6}$ | $1.48\times {10}^{-7}$ | $2.25\times {10}^{-5}$ | $2.13\times {10}^{-6}$ | $1.41\times {10}^{-6}$ | $2.20\times {10}^{-8}$ |

0.2 | $5.75\times {10}^{-4}$ | $4.10\times {10}^{-6}$ | $7.17\times {10}^{-7}$ | $1.14\times {10}^{-4}$ | $6.55\times {10}^{-6}$ | $1.21\times {10}^{-5}$ | $9.78\times {10}^{-9}$ |

0.3 | $6.47\times {10}^{-2}$ | $7.91\times {10}^{-6}$ | $1.34\times {10}^{-6}$ | $2.83\times {10}^{-4}$ | $9.89\times {10}^{-6}$ | $3.20\times {10}^{-5}$ | $4.69\times {10}^{-9}$ |

0.4 | $3.03\times {10}^{-4}$ | $1.17\times {10}^{-5}$ | $1.26\times {10}^{-6}$ | $4.95\times {10}^{-4}$ | $9.25\times {10}^{-6}$ | $5.07\times {10}^{-5}$ | $1.40\times {10}^{-8}$ |

0.5 | $2.63\times {10}^{-4}$ | $1.49\times {10}^{-5}$ | $4.45\times {10}^{-7}$ | $6.84\times {10}^{-4}$ | $3.56\times {10}^{-6}$ | $5.66\times {10}^{-5}$ | $1.53\times {10}^{-8}$ |

0.6 | $7.02\times {10}^{-4}$ | $1.65\times {10}^{-5}$ | $3.80\times {10}^{-7}$ | $7.75\times {10}^{-4}$ | $5.54\times {10}^{-6}$ | $4.58\times {10}^{-5}$ | $1.84\times {10}^{-8}$ |

0.7 | $7.62\times {10}^{-4}$ | $1.59\times {10}^{-5}$ | $5.63\times {10}^{-7}$ | $7.12\times {10}^{-4}$ | $1.37\times {10}^{-5}$ | $2.53\times {10}^{-5}$ | $1.67\times {10}^{-8}$ |

0.8 | $4.73\times {10}^{-4}$ | $1.23\times {10}^{-5}$ | $2.24\times {10}^{-7}$ | $4.87\times {10}^{-4}$ | $1.54\times {10}^{-5}$ | $7.48\times {10}^{-6}$ | $1.46\times {10}^{-8}$ |

0.9 | $1.25\times {10}^{-4}$ | $5.88\times {10}^{-6}$ | $2.11\times {10}^{-8}$ | $1.80\times {10}^{-4}$ | $8.05\times {10}^{-6}$ | $2.77\times {10}^{-7}$ | $1.41\times {10}^{-8}$ |

**Table 3.**Comparison of the absolute errors of the approximate solutions for U for the case $Re=1$ and $Ha=1$.

y | ${\mathit{U}}_{\mathit{HAM}}$ | ${\mathit{U}}_{\mathit{DTM}}$ | ${\mathit{U}}_{\mathit{PLSM}}$ | ${\mathit{U}}_{\mathit{HPM}2\mathit{term}.}$ | ${\mathit{U}}_{\mathit{HPM}3\mathit{t}.}$ | ${\mathit{U}}_{\mathit{LSHPM}2\mathit{t}.}$ | ${\mathit{U}}_{\mathit{LSHPM}3\mathit{t}.}$ |
---|---|---|---|---|---|---|---|

0.1 | $1.19\times {10}^{-2}$ | $3.18\times {10}^{-2}$ | $8.22\times {10}^{-5}$ | $1.19\times {10}^{-2}$ | $3.13\times {10}^{-3}$ | $8.03\times {10}^{-5}$ | $4.19\times {10}^{-7}$ |

0.2 | $1.59\times {10}^{-2}$ | $5.15\times {10}^{-2}$ | $1.17\times {10}^{-4}$ | $2.33\times {10}^{-2}$ | $6.14\times {10}^{-3}$ | $1.14\times {10}^{-4}$ | $1.14\times {10}^{-6}$ |

0.3 | $1.25\times {10}^{-2}$ | $5.95\times {10}^{-2}$ | $3.15\times {10}^{-5}$ | $3.35\times {10}^{-2}$ | $8.85\times {10}^{-3}$ | $2.72\times {10}^{-5}$ | $1.41\times {10}^{-7}$ |

0.4 | $5.92\times {10}^{-3}$ | $5.84\times {10}^{-2}$ | $9.22\times {10}^{-5}$ | $4.14\times {10}^{-2}$ | $1.10\times {10}^{-2}$ | $9.64\times {10}^{-5}$ | $1.32\times {10}^{-6}$ |

0.5 | $1.62\times {10}^{-4}$ | $5.14\times {10}^{-2}$ | $1.48\times {10}^{-4}$ | $4.60\times {10}^{-2}$ | $1.25\times {10}^{-2}$ | $1.52\times {10}^{-4}$ | $3.46\times {10}^{-7}$ |

0.6 | $2.79\times {10}^{-3}$ | $4.11\times {10}^{-2}$ | $9.29\times {10}^{-5}$ | $4.65\times {10}^{-2}$ | $1.29\times {10}^{-2}$ | $9.57\times {10}^{-5}$ | $1.10\times {10}^{-6}$ |

0.7 | $3.20\times {10}^{-3}$ | $2.99\times {10}^{-2}$ | $3.03\times {10}^{-5}$ | $4.24\times {10}^{-2}$ | $1.22\times {10}^{-2}$ | $2.86\times {10}^{-5}$ | $3.63\times {10}^{-7}$ |

0.8 | $2.34\times {10}^{-3}$ | $1.92\times {10}^{-2}$ | $1.16\times {10}^{-4}$ | $3.33\times {10}^{-2}$ | $9.21\times {10}^{-3}$ | $1.15\times {10}^{-4}$ | $7.95\times {10}^{-7}$ |

0.9 | $1.21\times {10}^{-3}$ | $9.05\times {10}^{-3}$ | $8.04\times {10}^{-5}$ | $1.92\times {10}^{-2}$ | $5.92\times {10}^{-3}$ | $8.01\times {10}^{-5}$ | $1.05\times {10}^{-7}$ |

**Table 4.**Comparison of the absolute errors of the approximate solutions for V for the case $Re=1$ and $Ha=1$.

y | ${\mathit{V}}_{\mathit{HAM}}$ | ${\mathit{V}}_{\mathit{DTM}}$ | ${\mathit{V}}_{\mathit{PLSM}}$ | ${\mathit{V}}_{\mathit{HPM}2\mathit{term}.}$ | ${\mathit{V}}_{\mathit{HPM}3\mathit{t}.}$ | ${\mathit{V}}_{\mathit{LSHPM}2\mathit{t}.}$ | ${\mathit{V}}_{\mathit{LSHPM}3\mathit{t}.}$ |
---|---|---|---|---|---|---|---|

0.1 | $4.90\times {10}^{-4}$ | $2.70\times {10}^{-4}$ | $3.06\times {10}^{-8}$ | $7.74\times {10}^{-5}$ | $1.53\times {10}^{-6}$ | $1.14\times {10}^{-8}$ | $1.91\times {10}^{-8}$ |

0.2 | $1.15\times {10}^{-3}$ | $5.83\times {10}^{-4}$ | $8.27\times {10}^{-8}$ | $3.03\times {10}^{-4}$ | $8.01\times {10}^{-6}$ | $5.24\times {10}^{-6}$ | $7.27\times {10}^{-9}$ |

0.3 | $1.19\times {10}^{-3}$ | $5.51\times {10}^{-4}$ | $1.37\times {10}^{-7}$ | $6.22\times {10}^{-4}$ | $2.06\times {10}^{-6}$ | $1.87\times {10}^{-5}$ | $4.18\times {10}^{-10}$ |

0.4 | $4.36\times {10}^{-4}$ | $1.45\times {10}^{-4}$ | $1.04\times {10}^{-7}$ | $9.38\times {10}^{-4}$ | $3.79\times {10}^{-6}$ | $3.38\times {10}^{-5}$ | $3.16\times {10}^{-9}$ |

0.5 | $7.25\times {10}^{-4}$ | $4.47\times {10}^{-4}$ | $1.51\times {10}^{-8}$ | $1.14\times {10}^{-3}$ | $5.51\times {10}^{-6}$ | $4.05\times {10}^{-5}$ | $1.35\times {10}^{-9}$ |

0.6 | $1.66\times {10}^{-3}$ | $9.52\times {10}^{-4}$ | $9.11\times {10}^{-8}$ | $1.16\times {10}^{-3}$ | $6.56\times {10}^{-6}$ | $3.41\times {10}^{-5}$ | $3.91\times {10}^{-10}$ |

0.7 | $1.90\times {10}^{-3}$ | $1.12\times {10}^{-3}$ | $6.89\times {10}^{-8}$ | $9.70\times {10}^{-4}$ | $6.30\times {10}^{-5}$ | $1.91\times {10}^{-5}$ | $1.24\times {10}^{-9}$ |

0.8 | $1.35\times {10}^{-3}$ | $8.72\times {10}^{-4}$ | $1.55\times {10}^{-8}$ | $6.05\times {10}^{-4}$ | $4.48\times {10}^{-5}$ | $5.55\times {10}^{-6}$ | $7.28\times {10}^{-10}$ |

0.9 | $4.73\times {10}^{-4}$ | $3.42\times {10}^{-4}$ | $6.71\times {10}^{-10}$ | $2.04\times {10}^{-4}$ | $1.71\times {10}^{-6}$ | $1.06\times {10}^{-7}$ | $3.35\times {10}^{-9}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Paşca, M.S.; Bundău, O.; Juratoni, A.; Căruntu, B.
The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model. *Mathematics* **2022**, *10*, 546.
https://doi.org/10.3390/math10040546

**AMA Style**

Paşca MS, Bundău O, Juratoni A, Căruntu B.
The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model. *Mathematics*. 2022; 10(4):546.
https://doi.org/10.3390/math10040546

**Chicago/Turabian Style**

Paşca, Mădălina Sofia, Olivia Bundău, Adina Juratoni, and Bogdan Căruntu.
2022. "The Least Squares Homotopy Perturbation Method for Systems of Differential Equations with Application to a Blood Flow Model" *Mathematics* 10, no. 4: 546.
https://doi.org/10.3390/math10040546