Abstract
This paper studies a fractional Bloch equation pertaining to Hilfer fractional operator. Bloch equation is broadly applied in physics, chemistry, nuclear magnetic resonance (NMR), magnetic resonance imaging (MRI) and many more. The sumudu transform technique is applied to obtain the analytic solutions for nuclear magnetizationM= (Mx,My,Mz). The general solution of nuclear magnetizationMis shown in the terms of Mittag-Leffler (ML) type function. The influence of order and type of Hilfer fractional operator on nuclear magnetizationMis demonstrated in graphical form. The study of Bloch equation with composite fractional derivative reveals the new features of Bloch equation. The discussed fractional Bloch model provides crucial and applicable results to introduce novel information in scientific and technological fields.
Highlights
Nuclear magnetic resonance (NMR) is a physical occurrence broadly employed in physics, chemistry, medical science, and engineering for studying the complex materials
NMR is controlled with the aid of the Bloch equation, which connects a macroscopic system of magnetization to the employed radiofrequency, static magnetic fields and gradient
Novel features of FC are affected by different useful applications such as mathematical biology, fluid flow problems, turbulence, electrochemistry, controlled thermonuclear fusion, astrophysics, image processing, plasma physics, Keywords and phrases: Fractional order Bloch model, nuclear magnetic resonance, magnetization, Hilfer derivative, Sumudu transform, Mittag–Leffler function
Summary
The non-integer order model describes the physical system having more accuracy, with high-order dynamics and with complex nonlinear phenomena. It occurs because of two reasons such that (i) choice to choose the order of fractional order derivatives while it is not employable to derivative of integer order (ii) as derivative of integer order is local in nature, so it does not narrate complete history and physical nature of system while derivative of arbitrary order has a non-local characteristic, it yields the whole history and physical aspects of the real world problem. By analyzing the all above considered physical importance of Bloch equation, powerful memory nature of composite fractional derivative, the authors are encouraged to study the proposed research work.
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