Magnetic resonance imaging (MRI) is used in physics, chemistry, engineering and medicine to study complex materials. In this paper, numerical solution of fractional Bloch equations in MRI is obtained using fractional variation iteration method (FVIM) and fractional homotopy perturbation transform method (FHPTM). Sufficient conditions for the convergence of FVIM and its error estimate are established. The obtained results are comparedwith the existing as well as recently developed methods and with the exact solution. The obtained numerical results for different fractional values of time derivative are discussed with the help of figures and tables. Figures are drawn using the Maple package. Test examples are provided to illustrate the accuracy and competency of the proposed schemes.

The objective of this paper is to study the nonlinear coupled dynamical fractional model of romantic and interpersonal relationships using fractional variation iteration method (FVIM) and fractional homotopy perturbation transform method (FHPTM). These procedures inspect the dynamics of love affairs among couples. Sufficient conditions for their convergence and error estimates are established. Obtained results are compared with the existing and recently developed methods. It is interesting to observe that these methods also work for those fractional models that do not have an exact solution. Results for different fractional values of time derivative are discussed with the help of figures and tables. Figures are drawn using Maple package. Test examples are provided to illustrate the accuracy and competency of the proposed schemes. Results divulge those schemes that are attractive, accurate, easy to use and highly effective.

In this paper, a solution of coupled fractional Navier–Stokes equation is computed numerically using the proposed q-homotopy analysis transform method (q-HATM), and the solution is found in fast convergent series.The given test examples illustrate the leverage and effectiveness of the proposed technique. The obtained results are demonstrated graphically. The present method handles the series solution in a large admissible domain in an extreme manner. It offers us a modest way to adjust the convergence region of the solution. Results with graphs explicitly reveal the efficiency and capability of the proposed algorithm.

In this paper, a reliable numerical scheme, the q-fractional homotopy analysis transform method (q-FHATM), is proposed to examine the Helmholtz equation of fractional order arising in seismic wave propagation, imaging and inversion. Sufficient conditions for its convergence and error estimates are established. The q-FHATMprovides a solution in a rapidly convergent series. Results for different fractional values of space derivatives are compared with the existing methods and discussed with the help of figures. A proper selection of parameters yields approximations identical to the exact solution. Parameter $\bar{h}$ offers an expedient way of controlling the region of convergence of the solution. Test examples are provided to illustrate the accuracy and competency of the proposed scheme. The outcomes divulge that our scheme is attractive, user-friendly, reliable and highly effective.

The main purpose of this work is to suggest an efficient hybrid computational technique, namely the $q$-homotopy analysis transform method ($q$-HATM) to find the solution of the nonlinear time-fractional Zakharov–Kuznetsov (FZK) equation in two dimensions. The uniqueness and convergence analysis of the nonlinear time-FZK equation is presented. The Laplace decomposition method (LDM) is also employed to get the approximate solution of the nonlinear FZK equation. We implemented these techniques on two numerical examples, plotted the solution and compared the absolute error with the variational iteration technique and homotopy perturbation transform technique to show the efficiency of these techniques.