This paper proposes Chebyshev-dependent inhomogeneous second order differential equation for the m-Boubaker polynomials (or Boubaker-Turki polynomials). This differential equation is also presented as a guide to applied physics studies. A concrete example is given through an attempt to solve the Bloch NMR flow equation inside blood vessels

Awojoyogbe, O. Bamidele

Bannour, Amine

Ben Mahmoud, Karem B.

Dada, Micahel

Hasler, Maximilian

English

Digital Commons @ PVAMU (Prairie View A&M Univ)

Applications and Applied Mathematics: An International Journal (AAM) Volume 3 Issue 2 Article 12 12-2008 Establishment of a Chebyshev-dependent Inhomogeneous Second Order Differential Equation for the Applied Physics-related Boubaker-Turki Polynomials Micahel Dada Federal University of Technology O. Bamidele Awojoyogbe Federal University of Technology Maximilian Hasler Université Antilles-Guyane Karem B. Ben Mahmoud Amine Bannour Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Other Physics Commons, and the Special Functions Commons Recommended Citation Dada, Micahel; Awojoyogbe, O. Bamidele; Hasler, Maximilian; Ben Mahmoud, Karem B.; and Bannour, Amine (2008). Establishment of a Chebyshev-dependent Inhomogeneous Second Order Differential Equation for the Applied Physics-related Boubaker-Turki Polynomials, Applications and Applied Mathematics: An International Journal (AAM), Vol. 3, Iss. 2, Article 12. Available at: https://digitalcommons.pvamu.edu/aam/vol3/iss2/12 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. 329      Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466  Vol. 3, Issue 6 (December 2008)  pp. 329 – 336 (Previously Vol. 3, No. 2) Applications and Applied Mathematics:  An International Journal (AAM)   Establishment of a Chebyshev-dependent Inhomogeneous  Second Order Differential Equation for the Applied Physics-related  Boubaker-Turki Polynomials   Micahel Dada O. Bamidele Awojoyogbe Department of Physics Federal University of Technology Minna, Niger-State, Nigeria  Maximilian Hasler Laboratoire A.O.C. Université Antilles-Guyane, D.S.I., B.P. 7209 97275 Schoelcher cedex. MARTINIQUE, FRANCE  Karem B. Ben Mahmoud Almas, Esstt, 1135 Naassan, Tunisia  Amine Bannour* FMD Cluj-Napoca, ROUMANIA *Corresponding author                   Abstract    This paper proposes Chebyshev-dependent inhomogeneous second order differential equation for the m-Boubaker polynomials (or Boubaker-Turki polynomials). This differential equation is also presented as a guide to applied physics studies. A concrete example is given through an attempt to solve the Bloch NMR flow equation inside blood vessels.     Keywords:  Polynomial expansion, Differential equation, Applied physics  AMS 2000 Subject Classification Numbers: 33E20, 41A30, 41A55   1Dada et al.: Chebyshev-dependent Inhomogeneous Second Order Differential EquationPublished by Digital Commons @PVAMU, 2008330                                                                                                                                           Dada, et al. 1.  Introduction  The Boubaker polynomials are the components of a special function which was established while studying an applied physics model by Chaouachi et al. (2007). This special function and its usefulness have been discussed in several studies by Boubaker (2007, 2008). The modified Boubaker polynomials (or Boubaker-Turki polynomials) is an enhanced form of these polynomials. Oppositely to the earlier defined polynomials, the Boubaker-Turki polynomials have a characteristic differential equation established by Labiadh et al. (2008) as well as an original ordinary generating function demonstrated by Awojoyogbe et al. (2009).  In this paper we tried to establish, for the first time, a Chebyshev-dependent differential equation for the Boubaker-Turki polynomials as a guide to particular applied physics studies.    2.  Historic preview    2.1.  The Boubaker Polynomials    According to their first definition in an attempt to solve the heat equation, the Boubaker polynomials have the explicit expression:   .)1.()()4(   )( 20pnpξ(n)pppnn XCpnpnXB   ,                    (1)  where the symbol   designates the floor function. Their coefficients could be defined through a recursive formula (2) that yields the first few polynomials (3):  2,0,0 ,1, 1 ,22 (( 1) 1) 1,4 22 1 1( ) ;  ;41;            ( 4);( 2 )( 2 1) ( 4 4);( 1)( 1) ( 4 )( 1) 2,               if n even,( 1) ( 2),     nnξ(n)n jn n jjn nn j n jnn nnn (( ) )B X b X ξ(n)b b nn j n j n jb bj n j n jbn                        if n odd.      (2)  2Applications and Applied Mathematics: An International Journal (AAM), Vol. 3 [2008], Iss. 2, Art. 12https://digitalcommons.pvamu.edu/aam/vol3/iss2/12AAM : Intern. J., Vol. 3, Issue 6  (December 2008)                                                                           331 012233445 356 4 267 5 378 6 289 7 5 391 2(X) 1;(X) ;(X) 2;(X) ;(X) 2;(X) 3 ;(X) 2 3 2;(X) 3 2 5 ;(X) 4 8 2;( ) 5 3 10 7 ;( ) . ( ) ( ),       for 2. m m mBB XB XB X XB XB X X XB X X XB X X X XB X X XB X X X X X XB X X B X B X m                                                  (3)                                                                                            2.2. The Modified Boubaker Polynomials    The Modified Boubaker polynomials have been proposed through a specialized study. They are defined by (4):   2 12 2 2 22 1( 4 ) 2 1 1( ) 2 2 ( 4) ( ) 2 ( 1)  , ( ) .! 4npξ(n)n n n n n p p n pnp j pn p n (( ) )B X X n X n j X ξ np                             (4)  They are solutions to a second order characteristic equation (5):  0)86²²²3(')23²(3")2²3)(1²(  ynnnXnynnXXynnXX .   (5)  The modified Boubaker polynomials have a recursive monomial definition expressed by equation (6):   22,02,0 ,1, 1 ,21,22 2( ) ; 2 ;            2 ( 4);( 2 )( 2 1) ( 4 4);( 1)( 1) ( 4 )( 1) 2,                      if  even,2( 1) 2( 2),   nn jn n jjn nn nn j n jnn nnB X b Xb b nn j n j n jb bj n j n jnbn                               if  odd.n                                                         (6)  3Dada et al.: Chebyshev-dependent Inhomogeneous Second Order Differential EquationPublished by Digital Commons @PVAMU, 2008332                                                                                                                                           Dada, et al. This definition allowed an establishment of a quasi-polynomial expression (7) of the m-Boubaker polynomials:     1838118381)( 222222 XXXXXXXXXXXBnnn.      (7)   3.  The Ordinary Generating Function of the Boubaker-Turki Polynomials   H. Labiadh et al. (2007) succeeded to establish an ordinary generating to the Boubaker-Turki polynomials that verifies (8):  nnn 0( , )  (X) .Bf X t B t            (8)  This ordinary generating function is expressed by (9):         12~ 2131),(  tXtttXfB .      (9)   4.  A Chebyshev-dependent Second Order Differential Equation for the Boubaker-Turki Polynomials  According to the recent results, the Boubaker-Turki polynomials can be expressed by (10):   )(2)(4)(~XTdXXdTnXXB nnn  ,        (10)  where Tn(X), for n > 2, are the Chebyshev  first order polynomials.  Equation (10) gives (11)  dXXdTndXXTdnXdXXBd nnn )(24)(4)(~22        (11)  and (12)  223322 )(28)(4)(~dXXTdndXXTdnXdXXBd nnn   ,      (12)  using the third order differential equation (13)   dXXdTndXXTdXdXXTdx nnn)(1)(3)()1( 233222  ,      (13)  we obtain the second order differential equation (14) 4Applications and Applied Mathematics: An International Journal (AAM), Vol. 3 [2008], Iss. 2, Art. 12https://digitalcommons.pvamu.edu/aam/vol3/iss2/12AAM : Intern. J., Vol. 3, Issue 6  (December 2008)                                                                           333  24 (1 ) " ( , ) ' ( , ) 2 ( , ) ( ),nX X y P X n y Q X n y Q X n T X                            (14) with  22 2( , ) 4 2 2 8( , ) 4 6 32.P X n X nX nQ X n X n n n            5. Application of the Boubaker-Turki Polynomials in Applied Physics Problems    5.1.  Magnetic resonance blood flow imaging   It is known, as confirmed by M.B. Scheidegger et al. (1992), P. Boesiger et al. (1992), Liu (1992), Stahlberg (1992) and Schmalbrock (1990), that there are three main ways of generating magnetic resonance (MR) signals: Free induction decay (FID), Spin Echo (SE), and Gradient Echo technique. Integrating these signal types with the gradient sequences necessary for spatial encoding produces the fundamental magnetic resonance imaging sequences. All other imaging Magnetic resonance sequences can be characterized using these three basic ways of MR signal generation.  In the following discussion, attention will be focused on quantitative blood flow imaging based on the phase contrast method.  Phase contrast technique employs the phase shift in the MR signal that is induced by the flowing blood. Blood spins moving along an applied gradient acquire a phase shift which is proportional to the strength and duration of the gradient and the motion of the spins. Therefore, with phase contrast method, complete suppression of stationary tissue can be achieved. This means that small blood vessels can be clearly visualized, even with slowly flowing blood.   5.2.  Spatial magnetization equation   The Spatial magnetization component M is a solution of the equation (15) as mentioned by Awojoyogbe (2002, 2003, 2004):     22 2 2 1121 2 1 2 1( . )1 1 1( , ) ,oM B xtM Mv v B x t Mx T T x TT T                                                          (15)                           where  is the gyromagnetic ratio of fluid spins, B1(x,t) is the rotating field,  T1  T2 are the spin lattice and spin-spin relaxation times respectively and v is the fluid velocity.  If we assume that the rotating field, which is a controlled item, can be expressed as a proportional function to the right term in the expression (14), the m-Boubaker polynomials in equation (14) are consequently solutions of a general second order non homogeneous differential equation (15) derived from the Bloch NMR flow equation.  Identification of the left term of equation (15) allows assuming that we can write (16) 5Dada et al.: Chebyshev-dependent Inhomogeneous Second Order Differential EquationPublished by Digital Commons @PVAMU, 2008334                                                                                                                                           Dada, et al.  12 211 1 2111 1 212211( , ) ,       position of spins1 1,                      velocity of blood flow ,                                   accelooooMB B x tT TTMB vT T TMB vT                         3 4 5 6 7eration of blood flow , , , , .    the higher order terms of the phase of the MRI signal. B B B B B        (16)  Hence, the properties of the established generating function (9), the Boubaker-Turki polynomials and the Bloch NMR flow equations discussed above can be very significant to extract relevant flow parameters for quantitative analysis of blood flow especially under pathological conditions   5.3.  A Solution to the Heat Transfer Equation   Ghanouchi et al. (2007) proposed a solution of a heat transfer problem in the case of a modulated heat supply targeting a plate surface. In this study, the source term in the main heat transfer equation was decomposed in a spectral domain of Boubaker-Turki polynomials. The authors took benefit from the arithmetic and differential properties of these polynomials in order to discuss a particular problem that several precedent studies tried to explain: the establishment of non-Gaussian isothermal generative lines beneath a plate surface targeted by a Gaussian beam.   6.  Conclusion    The present work is a continuation of the previous publications on the Boubaker polynomials. The obtained differential equation was inhomogeneous with a Chebyshev-dependent second term. This equation can be a first order supply to investigations of mathematical models involving mathematical modeling of physical and biophysical systems like the studies published by Riahi (2007), Slama et al. (2008), Srivastava (2007) and Marik (2006). Further investigations are focused on properties that lead to a characteristic homogenous equation. It was interesting to note that equations (9), (14) and (15) can play fundamental roles in the search for the best possible data obtainable on highly complex blood flow conditions. This will be the focus of our next investigation. It may be particularly interesting to note that the NMR flow parameters such as the flow velocity, T1  T2 NMR relaxation parameters for a sample can be uniquely determined through the first few m-Boubaker polynomials.     6Applications and Applied Mathematics: An International Journal (AAM), Vol. 3 [2008], Iss. 2, Art. 12https://digitalcommons.pvamu.edu/aam/vol3/iss2/12AAM : Intern. J., Vol. 3, Issue 6  (December 2008)                                                                           335 REFERENCES  Awojoyogbe O.B. (2002). A Mathematical Model of Bloch NMR Equations for Quantitative Analysis of Blood Flow in Blood Vessels with Changing Cross-section I. Physica A, Vol. 303, pp. 163-175.   Awojoyogbe O.B. (2003). A Mathematical Model of Bloch NMR Equations for Quantitative Analysis of Blood Flow in Blood Vessels with Changing Cross-section  II. Physica A,Vol. 323c, pp. 534-550.  Awojoyogbe O.B. (2004). Analytical Solution of the Time Dependent Bloch NMR   Equations: A Translational Mechanical Approach. Physica A, Vol. 339, pp. 437-460.   Awojoyogbe O.B. and Boubaker K. (2009). A solution to Bloch NMR flow equations for the analysis of homodynamic functions of blood flow system using m-Boubaker polynomials, International Journal of Current Applied Physics Vol. 9, pp. 278-288.  Boesiger P., Scheidegger M.B., Maier S.E., Liu K. and Maier D. (1992). Visualization and quantification of the human blood flow by magnetic resonance imaging. J. Biomed. Engn., 25(1) 55-67 .  Boubaker K. March (2007). Les Polynômes de Boubaker, Une classe  polynomiale qui émane d’un essai de résolution de l’équation de la chaleur, Deuxièmes Journées Méditerranéennes de Mathématiques Appliquées  JMMA02, Monastir, Tunisie.  Boubaker K. (2007). On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation, Trends in Applied Science Research, by Academic Journals, ‘aj’ New York; USA, Vol. 2, (6) pp. 540-544.   Boubaker K.  (2008).  The Boubaker polynomials, a new function class for solving bi-varied second order differential equations, F. E. J. of Applied Mathematics, Vol. 31, Issue 3 pp. 273-436.  Chaouachi A., Boubaker K., Amlouk M. and Bouzouita H. (2007). Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition, Eur. Phys. J. Appl. Phys. Vol. 37, pp. 105-109.  Labiadh H., Dada M., Awojoyogbe O.B., Ben Mahmoud K. B. and Bannour A. (2008). Establishment of an ordinary generating function and a Christoffel-Darboux type first-order differential equation for the heat equation related Boubaker-Turki polynomials, Journal of Differential Equations and C.P. vol. (1), pp. 51-66.  Labiadh H. (2007). A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial expression to the Boubaker polynomials, Journal of Differential equations and control processes, 2 reg. № P2375, ISSN 1817-2172, pp. 118-133  7Dada et al.: Chebyshev-dependent Inhomogeneous Second Order Differential EquationPublished by Digital Commons @PVAMU, 2008336                                                                                                                                           Dada, et al. Liu K. (1992). Magnetic Resonance Imaging for Imaging for the Acquisition of Vectorial Flow Velocity Pattern and Accurate Vessel Geometry, Ph.D. Thesis, Inst. of Biomedical Eng. and Medical Informatics, University and ETH Zurich.  Marik R. (2006). Interval – type Oscillation Criteria for Half – Linear PDE with Damping, Applications and Applied Mathematics: an Int. Journal Vol.  1, No. 1, pp. 1-10.  Riahi D. N. (2007). On Mathematical Modeling, Nonlinear Properties and Stability of Secondary Flow in a Dendrite Layer, Applications and Applied Mathematics: an Int. Journal Vol. 2, No. 2, pp. 122 – 135.  Scheidegger M.B., Maier S.E. and Boesiger P. (1992). FID-Acquired Echoes (FAcE), A Short Echo Time Imaging Method for Flow Artifact Suppression, Magn. Reson.           Imaging, pp. 517-534  Schmalbrock P., Yuan C., Chakers D.W., Koli J., Pelc N.j. (1990). Volume MR Angiography: Methods to Achieve Very Short Echo Times. Radiology, 175, 861-865.   Slama S., Bessrour J., Karem B. and Bouhafs M., (2008). Investigation of A3 point maximal front spatial evolution during resistance spot welding using 4q-Boubaker polynomial sequence, Proceedings of COTUME 2008, pp.79-80.  Stahlberg F. (1992). Quantification of Complex Flow Using MR Phase Imaging - a Study of Parameters Influencing the Phase/Velocity Relation, Magn. Reson.  Imaging, 10, 13-23.  Srivastava V. P. (2007). A Theoretical Model for Blood Flow in small vessels, Applications and Applied Mathematics: an Int. Journal Vol. 2, No. 1, pp. 51-65.    Werli F.W., Chao P.W., Youssen D. M. (1989): Parameter Dependence of           Susceptibility-Induced Signal Losses in Gradient-echo, Imaging. Mag. Reson. Imaging 7, 139. 8Applications and Applied Mathematics: An International Journal (AAM), Vol. 3 [2008], Iss. 2, Art. 12https://digitalcommons.pvamu.edu/aam/vol3/iss2/12

Establishment of a Chebyshev-dependent Inhomogeneous Second Order Differential Equation for the Applied Physics-related Boubaker-Turki Polynomials

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Establishment of a Chebyshev-dependent Inhomogeneous Second Order Differential Equation for the Applied Physics-related Boubaker-Turki Polynomials

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