Fractional Bloch equation is a generalized form of the integer order Bloch equation. It governs the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance (NMR). Scale-3 (S-3) Haar wavelet operational matrix along with quasi-linearization is applied first time to detect the spin flow of fractional Bloch equations. A comparative analysis of performance of classical scale-2 (S-2) and novel scale-3 Haar wavelets (S-3 HW) has been carried out. The analysis shows that scale-3 Haar wavelets give better solutions on coarser grid point in less computation time. Error analysis shows that as we increase the level of the S-3 Haar wavelets, error goes to zero. Numerical experiments have been conducted on five test problems to illustrate the merits of the proposed novel scheme. Maximum absolute errors, comparison of exact solutions, and S-2 Haar wavelet and S-3 Haar wavelet solutions, are reported. The physical behaviors of computed solutions are also depicted graphically.
References
1.
Abergel
, D.
, 2002
, “Chaotic Solutions of the Feedback Driven Bloch Equations
,” Phys. Lett. A
, 302
(1
), pp. 17
–22
.2.
Ghosh
, D.
, Chowdhury
, A. R.
, and Saha
, P.
, 2008
, “Bifurcation Continuation, Chaos and Chaos Control in Nonlinear Bloch System
,” Commun. Nonlinear Sci. Numer. Simul.
, 13
(8
), pp. 1461
–1471
.3.
Abergel
, D.
, Joseph
, A. L.
, and Lallemand
, J. Y.
, 2002
, “Self-Sustained Maser Oscillations of a Large Magnetization Driven by a Radiation Damping-Based Electronic Feedback
,” J. Chem. Phys., Am. Inst. Phys.
, 116
, pp. 7073
–7080
.https://aip.scitation.org/doi/10.1063/1.14625834.
Yu
, Q.
, Liu
, F.
, Turner
, I.
, and Burrage
, K.
, 2014
, “Numerical Simulation of the Fractional Bloch Equations
,” J. Comput. Appl. Math.
, 255
, pp. 635
–651
.5.
Magin
, R.
, Feng
, X.
, and Baleanu
, D.
, 2009
, “Solving the Fractional Order Bloch Equation
,” Concepts Magn. Reson. Part A
, 34A
(1
), pp. 16
–23
.6.
Petras
, I.
, 2011
, “Modeling and Numerical Analysis of Fractional-Order Bloch Equations
,” Comput. Math. Appl.
, 61
, pp. 341
–356
.7.
Abuteen
, E.
, Momani
, S.
, and Alawneh
, A.
, 2014
, “Solving the Fractional Nonlinear Bloch System Using the Multi-Step Generalized Differential Transform Method
,” Comput. Math. Appl.
, 68
(12
), pp. 2124
–2132
.8.
Balac
, S.
, and Chupin
, L.
, 2008
, “Fast Approximate Solution of Bloch Equation for Simulation of RF Artifacts in Magnetic Resonance Imaging
,” Math. Comput. Modell.
, 48
(11–12
), pp. 1901
–1913
.9.
Park
, J. H.
, 2006
, “Chaos Synchronization of Nonlinear Bloch Equations
,” Chaos, Solitons Fractals
, 27
, pp. 357
–361
.10.
Kakmeni
, F. M. M.
, Nguenang
, J. P.
, and Kofane
, T. C.
, 2006
, “Chaos Synchronization in bi-Axialmagnets Modeled by Bloch Equation, Chaos
,” Solitons Fractals
, 30
(3
), pp. 690
–699
.11.
Qin
, S.
, Liu
, F.
, Turner
, I.
, Vegh
, V.
, Yu
, Q.
, and Yang
, Q.
, 2017
, “Multi-Term Time-Fractional Bloch Equations and Application in Magnetic Resonance Imaging
,” J. Comput. Appl. Math.
, 319
, pp. 308
–319
.12.
Bhalekar
, S.
, Gejji
, V. D.
, Baleanu
, D.
, and Magin
, R.
, 2012
, “Transient Chaos in Fractional Bloch Equations
,” Comput. Math. Appl.
, 64
(10
), pp. 3367
–3376
.13.
Magin
, R. L.
, Abdullah
, O.
, Baleanu
, D.
, and Zhou
, X. J.
, 2008
, “Anomalous Diffusion Expressed Through Fractional Order Differential Operators in the Bloch–Torrey Equation
,” J. Magn. Reson.
, 2
, pp. 255
–270
.https://www.sciencedirect.com/science/article/pii/S1090780707003473?via%3Dihub14.
Gómez-Aguilar
, J. F.
, 2019
, “Chaos in a Nonlinear Bloch System With Atangana-Baleanu Fractional Derivatives
,” Numer. Methods Partial Differential Equations
, 34
(5
), pp. 1716
–1738
.15.
Bhalekar
, S.
, Gejji
, V. D.
, Baleanu
, D.
, and Magin
, R. L.
, 2011
, “Fractional Bloch Equation With Delay
,” Comput. Math. Appl.
, 5
, pp. 1355
–1365
.https://www.sciencedirect.com/science/article/pii/S089812211100006X16.
Mittal
, R. C.
, and Pandit
, S.
, 2017
, “Quasilinearized Scale-3 Haar Wavelets-Based Algorithm for Numerical Simulation of Fractional Dynamical systems
,” Eng. Comput.
, 35
(5
), pp. 1907
–1931
.17.
Saeed
, U.
, and Rehman
, M.
, 2013
, “Haar Wavelet-Quasilinearization Technique for Fractional Nonlinear Differential Equations
,” Appl. Math. Comput.
, 220
, pp. 630
–648
.https://www.sciencedirect.com/science/article/pii/S009630031300768618.
Yi
, I. M.
, and Huang
, J.
, 2014
, “Wavelet Operational Matrix Method for Solving Fractional Differential Equations With Variable Coefficients
,” Appl. Math. Comput.
, 230
, pp. 383
–394
.https://www.sciencedirect.com/science/article/pii/S009630031301218619.
Chui
, C. K.
, and Lian
, J. A.
, 1995
, “Construction of Compactly Supported Symmetric and Anti Symmetric Orthonormal Wavelets With Scale-3
,” Appl. Comput. Harmonic Anal.
, 2
(1
), pp. 21
–51
.20.
Kumar
, M.
, and Pandit
, S.
, 2014
, “A Composite Numerical Scheme for the Numerical Simulation of Coupled Burgers' Equation
,” Comput. Phys. Commun.
, 185
(3
), pp. 809
–817
.21.
Pandit
, S.
, Kumar
, M.
, and Tiwari
, S.
, 2015
, “Numerical Simulation of Second-Order Hyperbolic Telegraph Type Equations With Variable Coefficients
,” Comput. Phys. Commun.
, 187
, pp. 83
–90
.22.
Chen
, Y.
, Yi
, M.
, and Yu
, C.
, 2012
, “Error Analysis for Numerical Solution of Fractional Differential Equation by Haar Wavelets Method
,” J. Comput. Sci.
, 3
(5
), pp. 367
–373
.23.
Kaur
, H.
, Mittal
, R. C.
, and Mishra
, V.
, 2011
, “Haar Wavelet Quasilinearization Approach for Solving Nonlinear Boundary Value Problems
,” Am. J. Comput. Math.
, 01
(3
), pp. 176
–182
.24.
Jiwari
, R.
, 2015
, “A Hybrid Numerical Scheme for the Numerical Solution of the Burgers' Equation
,” Comput. Phys. Commun.
, 188
, pp. 59
–67
.25.
Jiwari
, R.
, 2012
, “Haar Wavelet Quasilinearization Approach for Numerical Simulation of Burgers' Equation
,” Comput. Phys. Commun.
, 183
(11
), pp. 2413
–2423
.26.
Pandit
, S.
, and Kumar
, M.
, 2014
, “Haar Wavelet Approach for Numerical Solution of Two Parameters Singularly Perturbed Boundary Value Problems
,” Appl. Math. Inf. Sci.
, 8
, pp. 2965
–2974
.27.
Mittal
, R. C.
, and Pandit
, S.
, 2018
, “New Scale-3 Haar Wavelets Algorithm for Numerical Simulation of Second Order Ordinary Differential Equations
,” Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.
(epub).28.
Mittal
, R. C.
, and Pandit
, S.
, 2017
, “Sensitivity Analysis of Shock Wave Burgers' Equation Via a Novel Algorithm Based on Scale-3 Haar Wavelets
,” Int. J. Comput. Math.
, 95
(3
), pp. 601
–625
.https://www.tandfonline.com/doi/abs/10.1080/00207160.2017.1293820?tab=permissions&scroll=top29.
Mittal
, R. C.
, and Pandit
, S.
, 2017
, “Numerical Simulation of Unsteady Squeezing Nano-Fluid and Heat Flow Between Two Parallel Plates Using Wavelets
,” Int. J. Therm. Sci.
, 118
, pp. 410
–422
.30.
Podlubny
, I.
, 1999
, Fractional Differential Equations
, Academic Press
, NewYork
.31.
Kumar
, D.
, Singh
, J.
, and Baleanu
, D.
, 2018
, “A New Analysis of the Fornberg-Whitham Equation Pertaining to a Fractional Derivative With Mittag-Leffler–Type Kernel
,” Eur. Phys. J. Plus
, 133
, p. 70
.32.
Kumar
, D.
, Singh
, J.
, and Baleanu
, D.
, 2018
, “Modified Kawahara Equation Within a Fractional Derivative, Modified Kawahara Equation Within a Fractional Derivative With Non-Singular Kernel
,” Therm. Sci.
, 22
(2
), pp. 789
–796
.33.
Kumar
, D.
, Agarwal
, R. P.
, and Singh
, J.
, 2018
, “Modified Numerical Scheme and Convergence Analysis for Fractional Model of Lienard's Equation
,” J. Comput. Appl. Math.
, 339
, pp. 405
–413
.https://www.sciencedirect.com/science/article/pii/S0377042717301309Copyright © 2019 by ASME
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