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Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique

Year 2023, , 1658 - 1676, 03.11.2023

Azim Amiataei Mohammadhossein Derakhshan

https://doi.org/10.15672/hujms.1095502
Cited By: 1

Abstract

In the present study, a mathematical model of non-steady partial differential equation from the process of oxygen mass transport in the human pulmonary circulation is proposed. Mathematical modeling of this kind of problems lead to a non-steady partial differential equation and for its numerical simulation, we have used finite differences. The aim of the process is the exact numerical analysis of the study, wherein consistency, stability and convergence is proposed. The necessity of doing the process is that, we would like to increase the order of numerical solution to a higher order scheme. An increment in the order of numerical solution makes the numerical simulation more accurate, also makes the numerical simulation being more complicated. In addition, the process of numerical analysis of the study in this order of solution needs more research work.

Keywords

Non-steady partial differential equation Higher order finite difference scheme Axial diffusion Convergence Consistency Stability

References

  • [1] H. Ahmad, T.A. Khan, P.S. Stanimirović, Y.M. Chu and I. Ahmad, Modified variational iteration algorithm-II: convergence and applications to diffusion models, Complexity 2020, 1-14, 2020.
  • [2] H. Ahmad, T.A. Khan and C. Cesarano, Numerical solutions of coupled Burgers’ equations, Axioms 8 (4), 119, 2019.
  • [3] M.A. Akbar, L. Akinyemi, S.W. Yao, A. Jhangeer, H. Rezazadeh, M.M.A. Khater, H. Ahmad and M. Inc, Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method, Results Phys. 25, 104228, 2021.
  • [4] L. Akinyemi, H. Rezazadeh, S.W. Yao, M.A. Akbar, M.M.M. Khater, A. Jhangeer, M. Inc and H. Ahmad, Nonlinear dispersion in parabolic law medium and its optical solitons, Results Phys. 26, 104411, 2021.
  • [5] A. Aminataei, The study of diffusion equation in the pulmonary circulation, J. of Maths. & Soc. 1(2), 17-32, 2014.
  • [6] A. Aminataei, Numerical simulation of the process of oxygen mass transport in the human pulmonary capillaries incorporating the contribution of axial diffusion, J. Sci. Kharazmi Univ. 13 (3), 779-796, 2013.
  • [7] A. Aminataei, Simulation of the breathing gases in the airways, J. Advan. Math. Model. 1 (2), 51-66, 2011.
  • [8] A. Aminataei, A numerical simulation of the unsteady convective-diffusion equation, The. J. of Damghan Univ. of Basic Scis. 1 (2), 73-87, 2008.
  • [9] A. Aminataei, A mathematical model for oxygen dissociation curve in the blood, Euro. J. Scien. Res. 6 5, 2005.
  • [10] A. Aminataei, Blood oxygenation in the pulmonary circulation: a review, Euro. J. Scien. Res. 10 (55), 2005.
  • [11] A. Aminataei, Comparision of explicit and implicit approaches to numerical solution of uni-dimensional equation of diffusion, J. of Sci., Al-Zahra Univ. 1 (15), 2002.
  • [12] A. Aminataei, A numerical two layer model for blood oxygenation in lungs, Amir Kabir 12 (45), 63-85, 2001.
  • [13] A. Aminataei and S. Hassani, An efficient numerical method for the solution of initial and boundary values problems, J. of Sci. Al-Zahra Univ. 22 (2), 2009.
  • [14] A. Aminataei, M. Sharan and M.P. Singh, Two-layer model for the process of blood oxygenation in the pulmonary capillaries–parabolic profiles in the core as well as in the plasma layer, Appl. Math. Model. 12 (6), 601-609, 1988.
  • [15] A. Aminataei, M. Sharan and M.P. Singh, A numerical model for the process of gas exchange in the pulmonary capillaries, Indian J. Pure Appl. Math. 18, 1040-1060, 1987.
  • [16] A. Aminataei, M. Sharan and M.P. Singh, A numerical solution for the nonlinear convective facilitated-diffusion reaction problem for the process of blood oxygenation in the lungs, J.Nat.Acad.Math. 3, 182-187, 1985.
  • [17] C. Bridges, S. Karra and K.R. Rajagopal, On modeling the response of the synovial fluid: Unsteady flow of a shear-thinning, chemically-reacting fluid mixture, Comput. Math. Appl. 60 (8), 2333-2349, 2010.
  • [18] C.S. Desai, L.D. Johnson, Evaluation of some numerical schemes for consolidation, Int. J. Numer. Meth. Engng. 7 (3), 243-254, 1973.
  • [19] G. Evans, J. Blackledge and P. Yardleg, Numerical Methods for Partial Differential Equations; Springer-Verlag: London, UK, 2000.
  • [20] M. Hirabayashi, M. Ohta, D.A. Rüfenacht and B. Chopard, Characterization of flow reduction properties in an aneurysm due to a stent, Phys Rev E Stat Nonlin Soft Matter Phys. 68 (2), 021918, 2003.
  • [21] S.J. Hund and J.F. Antaki, An extended convection diffusion model for red blood cell- enhanced transport of thrombocytes and leukocytes, Phys. Med. Biol. 54 (20), 6415, 2009.
  • [22] A. Kaesler, M. Rosen, T. Schmitz-Rode, U. Steinseifer and J. Arens, Computational modeling of oxygen transfer in artificial lungs, Artif. Organs 42 (8),786-799, 2018.
  • [23] L. Lapidus and G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley & Sons: New York, NY, USA, 1982.
  • [24] P.D. Lax and R.D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (2), 267-293, 1956.
  • [25] E.F. Leonard and S.B. Jørgensen, The analysis of convection and diffusion in capillary beds, Annu. Rev. Biophys. Bioeng. 3 (0), 293-339, 1974.
  • [26] Y. Liu, Z. Liu, C.F. Wen, J.C. Yao and S. Zeng, Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems, Appl. Math. Optim. 84, 2037-2059, 2021.
  • [27] Y. Liu, S. Migórski, V.T. Nguyen and S. Zeng, Existence and convergence results for an elastic frictional contact problem with nonmonotone subdifferential boundary conditions, Acta Math. Sci. 41 (4), 1151-1168, 2021.
  • [28] Z. Liu, D. Motreanu and S. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim, 31 (2), 1158- 1183, 2021.
  • [29] M. Massoudi and J.F. Antaki, An anisotropic constitutive equation for the stress tensor of blood based on mixture theory, Math. Probl. Eng. 579172, 2008.
  • [30] R.V.N. Melnik and D.R. Jenkins, On computational control of flow in airblast atomisers for pulmonary drug delivery, Int. J. Pharm. 239 (1-2), 23-35, 2002.
  • [31] L. Mountrakis, E. Lorenz and A.G. Hoekstra, Where do the platelets go? A simulation study of fully resolved blood flow through aneurysmal vessels, Interface Focus 3 (2), 20120089, 2013.
  • [32] J. Noye, Numerical Simulation of Fluid Motion; North-Holland: Amsterdam, The Netherlands, 1978.
  • [33] M. Sharan, M.P. Singh and A. Aminataei, A numerical model for blood oxygenation in the pulmonary capillaries effect of pulmonary membrane resistance, BioSystems 20 (4), 355-364, 1987.
  • [34] T. Skorczewski, L.C. Erickson and A.L. Fogelson, Platelet motion near a vessel wall or thrombus surface in two-dimensional whole blood simulations, Biophys. J. 104 (8), 1764-1772, 2013.
  • [35] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press: Oxford, UK, 1993.
  • [36] L.H. Thomas, Elliptic Problems in Linear Difference Equations Over a Network Wat- son Scientific Computing Laboratory, New York, Columbia University, 1949.
  • [37] B. Weir, Unruptured intracranial aneurysms: a review, J. Neurosurg. 96 (1), 3-42, 2002.
  • [38] I.R. Whittle, N.W. Dorsch and M. Besser, Spontaneous thrombosis in giant intracranial aneurysms, J. Neurol. Neurosurg. Psychiatry. 45 (11), 1040-1047, 1982.
  • [39] W.T Wu, Y. Li, N. Aubry, M. Massoudi and J.F. Antaki, Numerical simulation of red blood cell-induced platelet transport in saccular aneurysms, Appl. Sci. 7 (5), 484, 2017.
  • [40] W.T. Wu and M. Massoudi, Heat transfer and dissipation effects in the flow of a drilling fluid, Fluids 1 (1), 4, 2016.
  • [41] S. Zeng, S. Migórski and A.A. Khan, Nonlinear quasi-hemivariational inequalities: existence and optimal control, SIAM J. Control Optim, 59 (2), 1246-1274, 2021.
  • [42] S. Zeng, S. Migórski and Z. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31 (4), 2829-2862, 2021.
  • [43] S. Zeng, S. Migórski and L. Zhenhai, Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation (in Chinese), Sci. China Math. 52, 331354, 2022.
  • [44] W. Zhan and C.H. Wang, Convection enhanced delivery of chemotherapeutic drugs into brain tumour, J. Control Release 271, 74-87, 2018.

Year 2023, , 1658 - 1676, 03.11.2023

Azim Amiataei Mohammadhossein Derakhshan

https://doi.org/10.15672/hujms.1095502
Cited By: 1

Abstract

References

  • [1] H. Ahmad, T.A. Khan, P.S. Stanimirović, Y.M. Chu and I. Ahmad, Modified variational iteration algorithm-II: convergence and applications to diffusion models, Complexity 2020, 1-14, 2020.
  • [2] H. Ahmad, T.A. Khan and C. Cesarano, Numerical solutions of coupled Burgers’ equations, Axioms 8 (4), 119, 2019.
  • [3] M.A. Akbar, L. Akinyemi, S.W. Yao, A. Jhangeer, H. Rezazadeh, M.M.A. Khater, H. Ahmad and M. Inc, Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method, Results Phys. 25, 104228, 2021.
  • [4] L. Akinyemi, H. Rezazadeh, S.W. Yao, M.A. Akbar, M.M.M. Khater, A. Jhangeer, M. Inc and H. Ahmad, Nonlinear dispersion in parabolic law medium and its optical solitons, Results Phys. 26, 104411, 2021.
  • [5] A. Aminataei, The study of diffusion equation in the pulmonary circulation, J. of Maths. & Soc. 1(2), 17-32, 2014.
  • [6] A. Aminataei, Numerical simulation of the process of oxygen mass transport in the human pulmonary capillaries incorporating the contribution of axial diffusion, J. Sci. Kharazmi Univ. 13 (3), 779-796, 2013.
  • [7] A. Aminataei, Simulation of the breathing gases in the airways, J. Advan. Math. Model. 1 (2), 51-66, 2011.
  • [8] A. Aminataei, A numerical simulation of the unsteady convective-diffusion equation, The. J. of Damghan Univ. of Basic Scis. 1 (2), 73-87, 2008.
  • [9] A. Aminataei, A mathematical model for oxygen dissociation curve in the blood, Euro. J. Scien. Res. 6 5, 2005.
  • [10] A. Aminataei, Blood oxygenation in the pulmonary circulation: a review, Euro. J. Scien. Res. 10 (55), 2005.
  • [11] A. Aminataei, Comparision of explicit and implicit approaches to numerical solution of uni-dimensional equation of diffusion, J. of Sci., Al-Zahra Univ. 1 (15), 2002.
  • [12] A. Aminataei, A numerical two layer model for blood oxygenation in lungs, Amir Kabir 12 (45), 63-85, 2001.
  • [13] A. Aminataei and S. Hassani, An efficient numerical method for the solution of initial and boundary values problems, J. of Sci. Al-Zahra Univ. 22 (2), 2009.
  • [14] A. Aminataei, M. Sharan and M.P. Singh, Two-layer model for the process of blood oxygenation in the pulmonary capillaries–parabolic profiles in the core as well as in the plasma layer, Appl. Math. Model. 12 (6), 601-609, 1988.
  • [15] A. Aminataei, M. Sharan and M.P. Singh, A numerical model for the process of gas exchange in the pulmonary capillaries, Indian J. Pure Appl. Math. 18, 1040-1060, 1987.
  • [16] A. Aminataei, M. Sharan and M.P. Singh, A numerical solution for the nonlinear convective facilitated-diffusion reaction problem for the process of blood oxygenation in the lungs, J.Nat.Acad.Math. 3, 182-187, 1985.
  • [17] C. Bridges, S. Karra and K.R. Rajagopal, On modeling the response of the synovial fluid: Unsteady flow of a shear-thinning, chemically-reacting fluid mixture, Comput. Math. Appl. 60 (8), 2333-2349, 2010.
  • [18] C.S. Desai, L.D. Johnson, Evaluation of some numerical schemes for consolidation, Int. J. Numer. Meth. Engng. 7 (3), 243-254, 1973.
  • [19] G. Evans, J. Blackledge and P. Yardleg, Numerical Methods for Partial Differential Equations; Springer-Verlag: London, UK, 2000.
  • [20] M. Hirabayashi, M. Ohta, D.A. Rüfenacht and B. Chopard, Characterization of flow reduction properties in an aneurysm due to a stent, Phys Rev E Stat Nonlin Soft Matter Phys. 68 (2), 021918, 2003.
  • [21] S.J. Hund and J.F. Antaki, An extended convection diffusion model for red blood cell- enhanced transport of thrombocytes and leukocytes, Phys. Med. Biol. 54 (20), 6415, 2009.
  • [22] A. Kaesler, M. Rosen, T. Schmitz-Rode, U. Steinseifer and J. Arens, Computational modeling of oxygen transfer in artificial lungs, Artif. Organs 42 (8),786-799, 2018.
  • [23] L. Lapidus and G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Engineering, John Wiley & Sons: New York, NY, USA, 1982.
  • [24] P.D. Lax and R.D. Richtmyer, Survey of the stability of linear finite difference equations, Comm. Pure Appl. Math. 9 (2), 267-293, 1956.
  • [25] E.F. Leonard and S.B. Jørgensen, The analysis of convection and diffusion in capillary beds, Annu. Rev. Biophys. Bioeng. 3 (0), 293-339, 1974.
  • [26] Y. Liu, Z. Liu, C.F. Wen, J.C. Yao and S. Zeng, Existence of solutions for a class of noncoercive variational-hemivariational inequalities arising in contact problems, Appl. Math. Optim. 84, 2037-2059, 2021.
  • [27] Y. Liu, S. Migórski, V.T. Nguyen and S. Zeng, Existence and convergence results for an elastic frictional contact problem with nonmonotone subdifferential boundary conditions, Acta Math. Sci. 41 (4), 1151-1168, 2021.
  • [28] Z. Liu, D. Motreanu and S. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim, 31 (2), 1158- 1183, 2021.
  • [29] M. Massoudi and J.F. Antaki, An anisotropic constitutive equation for the stress tensor of blood based on mixture theory, Math. Probl. Eng. 579172, 2008.
  • [30] R.V.N. Melnik and D.R. Jenkins, On computational control of flow in airblast atomisers for pulmonary drug delivery, Int. J. Pharm. 239 (1-2), 23-35, 2002.
  • [31] L. Mountrakis, E. Lorenz and A.G. Hoekstra, Where do the platelets go? A simulation study of fully resolved blood flow through aneurysmal vessels, Interface Focus 3 (2), 20120089, 2013.
  • [32] J. Noye, Numerical Simulation of Fluid Motion; North-Holland: Amsterdam, The Netherlands, 1978.
  • [33] M. Sharan, M.P. Singh and A. Aminataei, A numerical model for blood oxygenation in the pulmonary capillaries effect of pulmonary membrane resistance, BioSystems 20 (4), 355-364, 1987.
  • [34] T. Skorczewski, L.C. Erickson and A.L. Fogelson, Platelet motion near a vessel wall or thrombus surface in two-dimensional whole blood simulations, Biophys. J. 104 (8), 1764-1772, 2013.
  • [35] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods, Oxford University Press: Oxford, UK, 1993.
  • [36] L.H. Thomas, Elliptic Problems in Linear Difference Equations Over a Network Wat- son Scientific Computing Laboratory, New York, Columbia University, 1949.
  • [37] B. Weir, Unruptured intracranial aneurysms: a review, J. Neurosurg. 96 (1), 3-42, 2002.
  • [38] I.R. Whittle, N.W. Dorsch and M. Besser, Spontaneous thrombosis in giant intracranial aneurysms, J. Neurol. Neurosurg. Psychiatry. 45 (11), 1040-1047, 1982.
  • [39] W.T Wu, Y. Li, N. Aubry, M. Massoudi and J.F. Antaki, Numerical simulation of red blood cell-induced platelet transport in saccular aneurysms, Appl. Sci. 7 (5), 484, 2017.
  • [40] W.T. Wu and M. Massoudi, Heat transfer and dissipation effects in the flow of a drilling fluid, Fluids 1 (1), 4, 2016.
  • [41] S. Zeng, S. Migórski and A.A. Khan, Nonlinear quasi-hemivariational inequalities: existence and optimal control, SIAM J. Control Optim, 59 (2), 1246-1274, 2021.
  • [42] S. Zeng, S. Migórski and Z. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31 (4), 2829-2862, 2021.
  • [43] S. Zeng, S. Migórski and L. Zhenhai, Nonstationary incompressible Navier-Stokes system governed by a quasilinear reaction-diffusion equation (in Chinese), Sci. China Math. 52, 331354, 2022.
  • [44] W. Zhan and C.H. Wang, Convection enhanced delivery of chemotherapeutic drugs into brain tumour, J. Control Release 271, 74-87, 2018.
There are 44 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Azim Amiataei This is me 0000-0001-5247-4492

Mohammadhossein Derakhshan 0000-0001-6464-7338

Early Pub Date August 15, 2023
Publication Date November 3, 2023
Published in Issue Year 2023

Cite

APA Amiataei, A., & Derakhshan, M. (2023). Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique. Hacettepe Journal of Mathematics and Statistics, 52(6), 1658-1676. https://doi.org/10.15672/hujms.1095502
AMA Amiataei A, Derakhshan M. Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique. Hacettepe Journal of Mathematics and Statistics. November 2023;52(6):1658-1676. doi:10.15672/hujms.1095502
Chicago Amiataei, Azim, and Mohammadhossein Derakhshan. “Stability Analysis and Numerical Simulation of Non-Steady Partial Differential Model in the Human Pulmonary Capillaries Using Finite Differences Technique”. Hacettepe Journal of Mathematics and Statistics 52, no. 6 (November 2023): 1658-76. https://doi.org/10.15672/hujms.1095502.
EndNote Amiataei A, Derakhshan M (November 1, 2023) Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique. Hacettepe Journal of Mathematics and Statistics 52 6 1658–1676.
IEEE A. Amiataei and M. Derakhshan, “Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique”, Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, pp. 1658–1676, 2023, doi: 10.15672/hujms.1095502.
ISNAD Amiataei, Azim - Derakhshan, Mohammadhossein. “Stability Analysis and Numerical Simulation of Non-Steady Partial Differential Model in the Human Pulmonary Capillaries Using Finite Differences Technique”. Hacettepe Journal of Mathematics and Statistics 52/6 (November 2023), 1658-1676. https://doi.org/10.15672/hujms.1095502.
JAMA Amiataei A, Derakhshan M. Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique. Hacettepe Journal of Mathematics and Statistics. 2023;52:1658–1676.
MLA Amiataei, Azim and Mohammadhossein Derakhshan. “Stability Analysis and Numerical Simulation of Non-Steady Partial Differential Model in the Human Pulmonary Capillaries Using Finite Differences Technique”. Hacettepe Journal of Mathematics and Statistics, vol. 52, no. 6, 2023, pp. 1658-76, doi:10.15672/hujms.1095502.
Vancouver Amiataei A, Derakhshan M. Stability analysis and numerical simulation of non-steady partial differential model in the human pulmonary capillaries using finite differences technique. Hacettepe Journal of Mathematics and Statistics. 2023;52(6):1658-76.

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