New aspects of fractional Bloch model associated with composite fractional derivative

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.


Introduction
Nuclear magnetic resonance (NMR) is a physical occurrence broadly employed in physics, chemistry, medical science, and engineering for studying the complex materials.
The Bloch equations are a collection of the macroscopic equations which are applied to compute nuclear magnetization ) , , ( turbulence, electrochemistry, controlled thermonuclear fusion, astrophysics, image processing, plasma physics, stochastic dynamical processes, control theory and several others. In view of the above considered details, it is worthy that FC has emerged as a crucial new mathematical key for the solution of distinct issues in the field of science and engineering. The principal benefit of calculus of arbitrary order is to model real world problems with complete memory influence. Many researchers have studied principal outcomes in the structure of monogram, books and quality research articles connected to applications of FC in distinct important directions, for further information; see . Generally physical problems related to fractional operators capable to represent the effective physical interpretation of the model in comparison to integer order derivatives and integrals. 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, therefore, it yields the whole history and physical aspects of the real world problem.
To mange above analyzed facts, we examine the physical model having arbitrary order derivative. Physical models of arbitrary orders have been nicely controlled. It has been shown that the physical models demonstrate the novel characteristic with full history of the problem, for complete study and implementation of the outcomes, see . Several Lastly, in Section fifth the conclusion moreover future physical results are pointed out.

Mathematical Overture
Some principal mathematical postulate which are applied in the present paper are presented as

Riemann-Liouville integral operator
For a function ) ( h , the Riemann-Liouville (RL) type integral operator [11] of order 0   is given as

RL fractional derivative
then RL derivative of non-integer order [11] is represented in the following manner

Caputo fractional operator
then Caputo type operator of arbitrary order [10] is given as

Hilfer derivative of fractional order
then Hilfer derivative of non-integer order [8] of a function ) ( h is given in the following way If set 0   and , 1   in these cases, Hilfer derivative of fractional order converts to the RL derivative of arbitrary order described as Eq. (4) and in Caputo derivative of fractional order given as Eq. (5), respectively.

ST of Hilfer derivative of fractional order
The ST of Hilfer derivative of fractional order [36,37] is expressed as To analyze the more characteristic of Hilfer derivative, see [9].

Construction of fractional Bloch Equation
The replacement of time derivative by fractional derivative in Bloch equation recommends a number of attractive and useful possibilities concerning magnetization relaxation and spin dynamics. In the present paper, we consider the standard Bloch equations [1,4].
represents the system magnetization of   (10) to control the consistent set of units for magnetization. Model (9) can also be represented in the following way The initial conditions (ICs) as

Analytical results of fractional Bloch Equation associated with Sumudu transform method
Here to analyze the new physical aspects and to evaluate the consequences of fractional parameters on the solution of discussed model (10), we solve system (10) by employing a poweful in addition efficient ST method. Primarily we use ST on model (10), then we get Now by utilizing Eq. (11) in to Eq. (12) and on rationalization, we get Further, by using the inverse ST technique on both sides Eq. (13) and applying outcomes of Chaurasia and Singh [38] moreover on rationalizing the deriving equations, we obtain The solutions for ) ( can be obtained by solving the corresponding differential equations of arbitrary order. It is supposed that Now we can merge two equations for x component and y component of the magnetization, discussed above to provide By using Eq. (15), we have Taking ST on both sides of Eq. (18), we get Now taking inverse ST on both the sides of Eq. (19) and utilizing Eq. (15), we have On comparing the real and imaginary parts, we get ), and ).

Particular cases of fundamental outcomes
Hilfer derivative of fractional order has two very special cases as follows  [8,39].

Numerical simulation and discussion for fractional Bloch model
Here, we discuss about Bloch equation pertaining to Hilfer operator. The analytical expressions for the magnetization components ) , , (