**1. Introduction**

The theory of fluid flow on a shrinking surface has numerous applications in real-life problems, such as shrinking film. Additionally, it has capillary effects in small pores, the shrinking-swell behavior of a rising shrinking balloon, and hydraulic properties of agricultural clay soils [1], fuel-cells [2,3], porous materials [4,5], and petroleum engineering [6,7]. Viscous fluid on a shrinking surface has been examined for the first time by Miklavˇciˇc and Wang [8], and they discovered that the flow over a shrinking surface did not exist unless sufficient mass suction was applied. It is worth mentioning that the fluid flow on shrinking and stretching surfaces have different characteristics. Gupta et al. [9] examined the magnetohydrodynamic (MHD) flow of micropolar fluid on a shrinking surface with the effect of mixed convection parameter. Meanwhile, Naveed et al. [10] considered the MHD flow of viscous fluid on a curved shrinking sheet. In order to model this problem, a curvilinear coordinates system was employed and the dual solutions were obtained. The MHD flow of nanofluid over a nonlinear stretching/shrinking wedge was considered by Khan et al. [11]. Soid et al. [12] investigated the unsteady MHD stagnation point flow over a shrinking surface and found dual solutions. Likewise,

Zaib et al. [13] considered the unsteady flow of the Williamson nanofluid over a shrinking surface and found dual solutions by using the shooting method. Lund et al. [14] analyzed the Darcy–Forchheimer flow of the Casson nanofluid with the impact of the slip condition on the shrinking surface and expressed that the existence of dual solutions relies upon the suction parameter. The slip effects on the nanofluid by utilizing the Buongiorno model has been examined by Dero et al. [15]. They found dual solutions by implementations of the shooting method and stated that it was due to the unsteadiness of the parameter. Similarly, Alarifi et al. [16] considered the stagnation point flow and found dual solutions for an opposing case. Triple solutions of micropolar nanofluid over a shrinking surface have been obtained by employing the shooting method [17]. Moreover, Lund et al. [18] performed a stability analysis by using the three-stage Lobatto IIIa formula and concluded only first solution to be stable. To the best of our knowledge, most of the studies and investigations of fluid flow have not used the Caputo fractional derivatives for multiple solutions. Therefore, the main objective of this work is to consider Caputo fractional derivatives, solve the governing equations by using the Adams-type predictor-corrector method, and find multiple solutions.

From published literature, it can be concluded that the possibility of the existence of multiple solutions of boundary layer flow on a shrinking surface is greater than on a stretching sheet. It is also discovered that the solution of fluid flow over a shrinking surface is possible only in the presence of high suction [19]. In other words, the solution is possible only on permeable surfaces. According to Mishra et al. [20], multiple solutions depend on the non-linearity in governing equations of fluid flow and other factors. Moreover, the existence of multiple solutions also depends on the values of different physical parameters such as magnetic, Reynold numbers, Prandtl numbers, and suction parameters, as claimed by Schlichting [21]. This claim complies with the findings of other researchers who discovered that the ranges of multiple solutions, single solutions, and no solutions depend on the values, such as the magnetic parameter [18], suction parameter [14], and surface velocity parameter [22]. Fang and Zhang [23] examined the steady MHD flow of viscous fluid over a shrinking surface and found dual solutions analytically. They concluded that dual and single solutions exist when 0 < M < 1 and M ≥ 1, respectively. Previous researchers attempted to determine multiple solutions using various analytical and numerical methods. Rana et al. [24] used the homotopy analysis method to find the multiple solutions. Rohni et al. [25] and Ishak et al. [26] found multiple solutions by using the Keller-box method. Fang et al. [27] employed an analytical approach to find multiple solutions of viscous fluid in exact form and Raza et al. [28] considered the shooting method with the Runge–Kutta of the fourth order method to find the multiple solutions of fluid flow. The objective of this paper is to extend the work of Fang and Zhang [23] under the consideration of unsteady flow and heat equation with viscous dissipation using the new approach with the Caputo derivative to reduce the governing equations to the first order ordinary differential equations, which are then solved by the Adams-type predictor-corrector method.

The MHD field was initiated by Hannes Alfvén (1908–1995) who was a famous Swedish physicist. Interest in the MHD flow started to gain attention when Hartmann invented the electromagnetic pump in 1918 [29]. In recent years, the study of non-uniform transverse-magnetic field effects is applied in many engineering problems. For example, electrically-conducting fluids that flow along with magnetic field have significant applications in oil exploration, cooling nuclear reactors, boundary layer control in the aerodynamics field, extraction of geothermal energy, and MHD generators and plasma studies. Due to the important applications of MHD flow, many researchers, mathematicians, and engineers considered MHD flow-related problems in their studies [30–32]. Ellahi et al. [33] considered the effect of MHD on Couple Stress Fluid. Makinde et al. [34] examined nanofluid under the influence of MHD and found that the hydrodynamic boundary layer is a decreasing function for higher values of magnetic parameters. This article is presented as follows: Section 2 discusses the problem formulation, in which governing equations are derived, and also gives some useful definitions and properties using solution methodology. In Section 3, numerical are presented numerically and graphically. Finally, Section 4 concludes this study by giving key findings and remarks.
