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Research Papers

A Numerical Algorithm to Capture Spin Patterns of Fractional Bloch Nuclear Magnetic Resonance Flow Models

[+] Author and Article Information
R. C. Mittal

Department of Mathematics,
Indian Institute of Technology Roorkee,
Roorkee, Uttarakhand 247667, India

Sapna Pandit

Department of Mathematics,
Indian Institute of Technology Roorkee,
Roorkee, Uttarakhand 247667, India
e-mail: sappu15maths@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 10, 2018; final manuscript received March 27, 2019; published online May 13, 2019. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 14(8), 081001 (May 13, 2019) (12 pages) Paper No: CND-18-1153; doi: 10.1115/1.4043565 History: Received April 10, 2018; Revised March 27, 2019

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.

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Figures

Grahic Jump Location
Fig. 1

Comparison of Haar solutions and exact solutions when α=0.7 and K1=1

Grahic Jump Location
Fig. 2

Plot of Mz(t) for α=0.5 to α=1 at T1′=1,M0=100,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 3

Plot of My(t) for α=1 at M0=100,T1′=1 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 4

Plot of My(t) for α=0.9 at M0=100,T1′=1 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 5

Plot of My(t) for α=0.8 at M0=100,T1′=1 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 6

Spin behavior ((a)–(c)) of Mx(t) versus My(t) for α=0.8,0.9 at M0=100,T1′=1 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 7

Spin behavior of Mzx(t),My(t), and Mz(t) for α=0.9 at M0=100,T1′=1  (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 8

Spin flow using s-3 Haar wavelet at α=1 at M0=100,T1′=1 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 9

Comparison of Haar solutions and exact solution AFBE at α=0.7 and K2=1

Grahic Jump Location
Fig. 10

Pattern behavior in two-dimensional plots of Mx(t) versus My(t) at α=0.99 and α=0.8 at M0=100,T1′=100 (ms)α,T2′=20 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 11

Spin flow using S-3 Haar wavelets on AFBE at α=0.8 and M0=100,T1′=100(ms)α,T2′=100(ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 12

Spin behavior in two-dimensional plots of Mx(t) versus My(t) when α=0.8 (a) and α=0.9(b) at M0=100,T1′=100 (ms)α,T2′=100 (ms)α, and ω0′=160*2π Hz

Grahic Jump Location
Fig. 13

Patterns of My(t) when α=0.9 (a) and Mz(t) when α=1 (b) with the initial condition M̃x(0)=0,M̃y(0)=0.001, and M̃z(0)=0 and with set of parameters being λ=35,γ=−1.26,ψ=0.173,τ1=5, and τ2=2.5

Grahic Jump Location
Fig. 14

Spin flow of Mx(t) along with My(t) when M̃x(0)=0,M̃y(0)=0.001, and M̃z(0)=0: α=0.97 (a) and α=1 (b) with set of parameters being λ=35,γ=−1.26,ψ=0.173,τ1=5, and τ2=2.5

Grahic Jump Location
Fig. 15

Spin flow of time varying magnetization components (Mx(0),My(0),Mz(0)) when M̃x(0)=0,M̃y(0)=0.001,M̃z(0)=0, α=0.99, and ν=1 with set of parameters being λ=35,γ=−1.26,ψ=0.173, and τ2=2.5

Grahic Jump Location
Fig. 16

Spin flow of time varying magnetization components (Mx(t),My(t),Mz(t)) when M̃x(0)=0.5,M̃y(0)=−0.5,M̃z=0.5, α=0.99, and ν=2 with set of parameters being λ=35,γ=−1.26,ψ=0.173, and τ1=2.5

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