## 1 Introduction  
The Bloch equation is a system of differential equations. It is mainly valuable for studying expensive biological samples like RNA, DNA, proteins and nucleic acids. It has many real-life applications like process control, liquid media, petrochemical plants and process optimization in oil refineries. Surface magnetic resonance is based on the principle of NMR, and the measurements can be used to indirectly estimate the water content of the saturated and unsaturated zones. The standard system of Bloch equations is given as follows:  
\[ \begin{array}{l}\frac{dN_{x}(t)}{dt}=w_{0}N_{y}(t)-\frac{N_{x}(t)}{T_{2}}\\ \frac{dN_{y}(t)}{dt}=w_{0}N_{x}(t)-\frac{N_{y}(t)}{T_{2}}\\ \\ \frac{dN_{z}(t)}{dt}=\frac{N_{0}-N_{z}(t)}{T_{1}},\end{array}  \tag{1}\]  
with the initial conditions \(N_{x}(0)=a_{1}\), \(N_{y}(0)=a_{2}\) and \(N_{z}(0)=a_{3}\). Here \(N_{x}(t)\), \(N_{y}(t)\) and \(N_{z}(t)\) denote the system magnetisation in \(x\), \(y\) and \(z\) components, respectively; \(w_{0}\) is the resonant frequency given by the Larmor relationship \(w_{0}=\gamma M_{0}\), where \(M_{0}\) is the static magnetic field in \(z-\)component, \(N_{0}\) is the equilibrium magnetisation, \(T_{1}\) and \(T_{2}\) are the spin-lattice and spin-spin relaxation time, respectively, and \(a_{1},\ a_{2}\) and \(a_{3}\) are real constants. The analytical solution of Equation (1) is given by  
\[ \begin{array}{l}N_{x}(t)=e^{-\frac{t}{T_{2}}}\Big{(}N_{x}(0)\cos w_{0}t+N_{y }(0)\sin w_{0}t\Big{)}\\ N_{y}(t)=e^{-\frac{t}{T_{2}}}\Big{(}N_{y}(0)\cos w_{0}t-N_{x}(0)\sin w_{0}t \Big{)}\end{array}  \tag{2}\]  
\[N_{z}(t)=N_{z}(0)e^{-\frac{t}{T_{1}}}+N_{0}\Big{(}1-e^{-\frac{t}{T_{1}}}\Big{)}\]  
Real-life applications of fractional calculus are in subjects like biology [1], viscoelasticity [2; 3; 4], signal processing [5], control theory [6], fluid dynamics [7]. For more details, the reader should refer to [8]. There are many magnetic resonance systems which are modelled by the fractional-order Bloch equation, and it is well known that the fractional-order derivatives are non-local in nature. Therefore, we will replace the integer-order Bloch equation by the fractional-order Bloch equation with a view to further understand the resulting magnetic resonance systems. Therefore, we replace the integer-order time-derivative by the non-integer- order time-derivative:  
\[ \begin{array}{l}\frac{d^{n}N_{x}(t)}{dt^{n}}=w_{0}N_{y}(t)-\frac{N_{x}(t)} {T_{2}}\\ \frac{d^{n}N_{y}(t)}{dt^{n}}=w_{0}N_{x}(t)-\frac{N_{y}(t)}{T_{2}}\end{array}  \tag{3}\]  
\[\frac{d^{\prime}N_{z}(t)}{dt^{\gamma}}=\frac{N_{0}-N_{z}(t)}{T_{1}},\]  
where \(0<\alpha,\beta,\gamma\leq 1\). The non-integer-order derivative is in the Liouville-Caputo (LC) sense. The LC non-integer-order derivative of order \(\beta\) is defined as follows [8]:  
\[D^{\beta}f(x)=l^{l-\beta}D^{l}f(x)=\frac{1}{\Gamma(l-\beta)}\int_{0}^{x}(x-t) ^{l-\beta-1}\frac{d^{l}}{dt^{l}}f(t)dt,\ l-1<\beta<l,\ x>0.\]  
In this paper, we are considering that \(\beta\in(0,\ 1)\); therefore, we will take \(l=1\). The time-fractional derivatives play a key role upsetting the spin dynamics defined by the Bloch equations in Equation (3) (see [9; 10]). The magnetic resonance components of the magnetisation are identified in the initial state of the system, and hence, these should be visibly predictable. The physical meaning of the non-integer order Bloch equations can be understood in the basic preparation of the non-integer-order Schrodinger equation. Bloch equations in NMR can be simulated numerically and analytically (see, for details, [11; 12; 13; 14; 15; 16]). The time-fractional order Bloch equation having fractional derivative in Caputo sense is solved in [17]. Recently, [PERSON] et al. [18] solved fractional-order Bloch equation by using homotopy perturbation method (HPM). Use has been made of operational matrix method with Legendre polynomials in [19] and with the Laguerre polynomials in [20] for the solution of this equation. In [21], this equation was solved numerically by using the iterative method.