Optimal Approximation of Fractional Order Brain Tumor Model Using Generalized Laguerre Polynomials

Iranian Journal of Science - Tập 47 - Trang 501-513 - 2023
Z. Avazzadeh1, H. Hassani2, M. J. Ebadi3, P. Agarwal2,4,5,6, M. Poursadeghfard7, E. Naraghirad8
1Department of Mathematical Sciences, University of South Africa, Florida, South Africa
2Anand International College of Engineering, Jaipur, India
3Department of Mathematics, Chabahar Maritime University, Chabahar, Iran
4International Center for Basic and Applied Sciences, Jaipur, India
5Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation
6Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, UAE
7Clinical Neurology Research Center, Shiraz University of Medical Sciences, Shiraz, Iran
8Department of Mathematics, Yasouj University, Yasouj, Iran

Tóm tắt

A brain tumor occurs when abnormal cells form within the brain. Glioblastoma (GB) is an aggressive and fast-growing type of brain tumor that invades brain tissue or spinal cord. GB evolves from astrocytic glial cells in the central nervous system. GB can occur at almost any age, but the occurrence increases with advancing age in older adults. Its symptoms may include nausea, vomiting, headaches, or even seizures. GB, formerly known as glioblastoma multiforme, currently has no cure with a high rate of resistance to therapy in clinical treatment. However, treatments can slow tumor progression or alleviate the signs and symptoms. In this paper, a fractional order brain tumor model was considered. The optimal solution of the model was obtained using an optimization method based on operational matrices. The solution to the problem under study was expanded in terms of generalized Laguerre polynomials (GLPs). The study problem was shifted to a system of nonlinear algebraic equations by the use of Lagrange multipliers combined with operational matrices based on GLPs. The analysis of convergence was discussed. In the end, some numerical examples were presented to justify theoretical statements along with the patterns of biological behavior.

Tài liệu tham khảo

Abo-Gabal H, Zaky MA, Doha EH (2022) Fractional Romanovski–Jacobi tau method for time-fractional partial differential equations with nonsmooth solutions. Appl Numer Math 182:214–234

Agrawal OP, Machado JAT, Sabatier J (2004) Nonlinear dynamics. Fractional derivatives and their applications, Kluwer Academic Publishers, Special Issue

Aizenshtadt VS, Krylov VI, Metel’skii AS (1966) Tables of Laguerre polynomials and functions. Mathematical tables series. Pergamon Press, Oxford

Alifieris C, Trafalis DT (2015) Glioblastoma multiform: Pathogenesis and treatment. Pharmacol Ther 152:63–82

Bavi O, Hosseininia M, Heydari MH, Bavi N (2022) SARS-CoV-2 rate of spread in and across tissue, groundwater and soil: a meshless algorithm for the fractional diffusion equation. Eng Anal Boundary Elem 138:108–117

Bhrawy AH, Alhamed YA, Baleanu D, Al-Zahrani AA (2014) New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract Calc Appl Anal 17:1137–1157

Bhrawy AH, Alhamed YA, Baleanu D, Al-Zahrani AA (2014) New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions,. Fract Calc Appl Anal 17:1137–1157

Blissitt PA (2014) Clinical practice guideline series update: care of the adult patient with a brain tumor. J Neurosci Nurs 46(6):367–8

Chen Y, Yu H, Meng X, Xie X, Hou M, Chevallier J (2021) Numerical solving of the generalized Black–Scholes differential equation using Laguerre neural network. Digital Signal Process 112:103003. https://doi.org/10.1016/j.dsp.2021.103003

Chi X, Jiang X (2021) Finite difference Laguerre-Legendre spectral method for the two-dimensional generalized Oldroyd-B fluid on a semi-infinite domain. Appl Math Comput 402:126138. https://doi.org/10.1016/j.amc.2021.126138

Cong ND, Tuan HT, Trinh H (2020) On asymptotic properties of solutions to fractional differential equations. J Math Anal Appl 484(2):123759

Cruywagen GC, Woodward DE, Tracqui P, Bartoo GT, Murray JD, Alvord EC Jr (1995) The modeling of diffusive tumours. J Biol Syst 3(4):937–45

Daşcıoǧlu A, Varol D (2021) Laguerre polynomial solutions of linear fractional integro-differential equations. Math Sci 15:47–54

Ellor SV, Pagano-Young TA, Avgeropoulos NG (2014) Glioblastoma: background, standard treatment paradigms, and supportive care considerations. J. Law Med. Ethics. 42(2):171–182

Garrappa R, Kaslik E (2020) On initial conditions for fractional delay differential equations. Commun Nonlinear Sci Numer Simul 90:105359

Hajimohammadi Z, Parand K (2021) Numerical learning approximation of time-fractional sub diffusion model on a semi-infinite domain. Chaos Solitons Fractals 142:110435. https://doi.org/10.1016/j.chaos.2020.110435

Hajiseyedazizi SN, Samei ME, Alzabut J, Chu Y-M (2021) On multistep methods for singular fractional q-integro-differential equations. Open Math. 19(1):1378–1405

Hammad HA, Agarwal P, Momani S, Alsharari F (2021) Solving a fractional-order differential equation using rational symmetric contraction mappings. Fractal Fract. 5(4):159. https://doi.org/10.3390/fractalfract5040159

Hassani H, Avazzadeh Z, Tenreiro Machado JA (2020) Numerical approach for solving variable-order space-time fractional telegraph equation using transcendental Bernstein series. Eng Comput 36:867–878

Hassani H, Avazzadeh Z, Tenreiro Machado JA, Agarwal P, Bakhtiar M (2022) Optimal solution of a fractional HIV/AIDS epidemic mathematical model. J Comput Biol 29(3):276–291

Hassani H, Tenreiro Machado JA, Avazzadeh Z (2019) An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique. Nonlinear Dyn 97:2041–2054

He Z-Y, Abbes A, Jahanshahi H, Alotaibi ND, Wang Y (2022) Fractional order discrete-time SIR epidemic model with vaccination: Chaos and complexity. Mathematics 10(2):165

Heydari MH, Atangana A (2022) A numerical method for nonlinear fractional reaction-advection-diffusion equation with piecewise fractional derivative. Math Sci https://doi.org/10.1007/s40096-021-00451-z

Heydari MH, Hooshmandasl MR, Maalek Ghaini FM (2014) An efficient computational method for solving fractional biharmonic equation. Comput Math Appl 68(3):269–287

Hosseininia M, Heydari MH (2019) Meshfree moving least squares method for nonlinear variable-order time fractional 2D telegraph equation involving Mittag-Leffler non-singular kernel. Chaos Solitons Fractals 127:389–399

Hussien HSh (2019) Efficient collocation operational matrix method for delay differential equations of fractional order. Iran J Sci Technol Trans A. Sci 43:1841–1850

Hussien HSh (2019) Efficient collocation operational matrix method for delay differential equations of fractional order. Iran J Sci Technol Trans A Sci 43:1841–1850

Jaros J, Kusano T (2014) On strongly monotone solutions of a class of cyclic systems of nonlinear differential equations. J Math Anal Appl 417:996–1017

Johnson DR, Fogh SE, Giannini C, Kaufmann TJ, Raghunathan A, Theodosopoulos PV, Clarke JL (2015) Case-based review: Newly diagnosed glioblastoma. Neurooncol Pract. 2(3):106–121

Karthikeyan K, Karthikeyan P, Baskonus HM, Venkatachalam K, Chu YM (2021) Almost sectorial operators on \(\psi -\)Hilfer derivative fractional impulsive integro - differential equations, Math. Methods Appl. Sci. 1-15

Kreyszig E (1978) Introductory functional analysis with applications. Wiley, Berlin

Kumar M, Upadhyay S, Rai KN (2018) A study of lung cancer using Modified Legendre wavelet Galerking method. J Therm Biol 78:356–366

Lorenzo CF, Hartley TT (2000) Initialized fractional calculus. Int J Appl Math 3(3):249–265

Nelson SJ, Cha S (2003) Imaging glioblastoma multiforme. Cancer J 9(2):134–145

Odibat Z (2019) On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations. Appl Numer Math 137:203–212

Perry J, Zinman L, Chambers A, Spithoff K, Lloyd N, Laperriere N (2006) The use of prophylactic anticonvulsants in patients with brain tumours-a systematic review. Curr Oncol 13(6):222–229

Phillips HS, Kharbanda S, Chen R, Forrest WF, Soriano RH, Wu TD, Misra A, Nigro JM, Colman H, Soroceanu L, Williams PM, Modrusan Z, Feuerstein BG, Aldape K (2006) Molecular subclasses of high-grade glioma predict prognosis, delineate a pattern of disease progression, and resemble stages in neurogenesis. Cancer Cell 9(3):157–173

Podlubny I (1999) Fractional differential equations. Academic Press, New York

Radmanesh M, Ebadi MJ (2020) A local mesh-less collocation method for solving a class of time-dependent fractional integral equations: 2D fractional evolution equation. Eng Anal Boundary Elem 113:372–381

Rashid S, Sultana S, Karaca Y, Khalid A, Chu Y-M (2022) Some further extensions considering discrete proportional fractional operators. Fractals 30(1):2240026

Roohi R, Heydari MH, Bavi O, Emdad H (2021) Chebyshev polynomials for generalized Couette flow of fractional Jeffrey nanofluid subjected to several thermochemical effects. Eng Comput 37:579–595

S. Rashid, E. I. Abouelmagd, A. Khalid, F. B. Farooq, Y. -M. Chu, Some recent developments on dynamical h-discrete fractional type inequalities in the frame of nonsingular and nonlocal kernels. Fractals,30 (2) 2240110 (2022)

Sabermahani S, Ordokhani Y, Lima PM (2020) A novel lagrange operational matrix and tau-collocation method for solving variable-order fractional differential equations. Iran J Sci Technol Trans A Sci 44:127–135

Shahni J, Singh R (2022) Numerical simulation of Emden-Fowler integral equation with Green’s function type kernel by Gegenbauer-wavelet, Taylor-wavelet and Laguerre-wavelet collocation methods. Math Comput Simul 194:430–444

Shiralashetti, S. C. Kumbinarasaiah S (2020) Laguerre wavelets exact parseval frame-based numerical method for the solution of system of differential equations, Int J Appl Comput Math https://doi.org/10.1007/s40819-020-00848-9

Singh R, Rehman AU, Masud M, Alhumyani HA, Mahajan S, Pandit AK, Agarwal P (2022) Fractional order modeling and analysis of dynamics of stem cell differentiation in complex network. AIMS Math 7(4):5175–5198

Singla K (2021) Existence of series solutions for certain nonlinear systems of time fractional partial differential equations. J Geom Phys 167:104301

Sun H, Chen W, Wei H, Chen Y (2011) A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems. Eur Phys J Spec Top 193:185–192

Thakkar JP, Dolecek TA, Horbinski C, Ostrom QT, Lightner DD, Barnholtz-Sloan JS, Villano JL (2014) Epidemiologic and molecular prognostic review of glioblastoma. Cancer Epidemiol Biomark Prev 23(10):1985–1996

Tracqui P, Cruywagen GC, Woodward DE, Bartoo GT, Murray JD, Alvord EC Jr (1995) A mathematical model of glioma growth: the effect of chemotherapy on spatio-temporal growth. Cell Prolif 28(1):17–31

Urbańska K, Sokołowska J, Szmidt M, Sysa P (2014) Glioblastoma multiforme - an overview. Contemp Oncol (Pozn). 18(5):307–312

Vargas AM (2022) Finite difference method for solving fractional differential equations at irregular meshes. Math Comput Simul 193:204–216

Wang F-Z, Khan MN, Ahmad I, Ahmad H, Abu-Zinadah H, Chu Y-M (2022) Numerical solution of traveling waves in chemical kinetics: time-fractional fishers equations. Fractals 30(2):22400051

Woodward DE, Cook J, Tracqui P, Cruywagen GC, Murray JD, Alvord EC Jr (1996) A mathematical model of glioma growth: the effect of extent of surgical resection. Cell Prolif 29(6):269–288

Xu Y, Zhang Y, Zhao J (2019) Error analysis of the Legendre-Gauss collocation methods for the nonlinear distributed-order fractional differential equation. Appl Numer Math 142:122–138

Yu H, Wu B, Zhang D (2019) The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains. Numer Algorithms 82:1221–1250

Z. Abdollahi M, Mohseni Moghadam H, Saeedi MJ, Ebadi A (2021) computational approach for solving fractional Volterra integral equations based on two dimensional Haar wavelet method. Int J Comput Math. https://doi.org/10.1080/00207160.2021.1983549

Zaky MA (2019) Recovery of high order accuracy in Jacobi spectral collocation methods for fractional terminal value problems with non-smooth solutions. J Comput Appl Math 357:103–122

Zaky MA (2020) An accurate spectral collocation method for nonlinear systems of fractional differential equations and related integral equations with nonsmooth solutions. Appl Numer Math 154:205–222

Zhang D, Miao X (2017) New unconditionally stable scheme for telegraph equation based on weighted Laguerre polynomials. Numer Methods Partial Differ Equ 33(5):1603–1615

Thông tin
Thông tin xuất bản

Iranian Journal of Science

Tập 47

501-513

Thông tin tác giả