The Aims of the Prophetic Hijrah and its Teaching and Guidance

The aim of Hijrah is not to run away from problem that occurred in the process of giving da'wah, but rather to establish the resolve in solving the problem. Hijrah as a movement carried out by the Prophet Muhammad (PBUH) and his companions from Makkah to Madinah, aimed to keep, maintain and uphold the message of Allah, in the form of Islamic aqeedah and shari’a, in order to achieve the mercy and pleasure of Allah SWT. This move, as we can see in the seerah, later brought a great impact to the success of the Islamic da’wah which was increasingly evolving. Hence the fundamental problem that has led to the migration has been solved. In fact, the Hijrah brought a significant impact not just to the Islamic world but also to world civilisation. The story and background of the migration of Rasulullah (PBUH) from Makkah to Madinah is touched in this writing to show that there were a lot of lessons and guidance that can be inferred. It did not merely signify a final destination for Makkah Muslims, but was also the beginning of a continuous effort to establish a strong and resilient ummah. The event of Hijrah in fact had changed the world after that forever. It highlighted the perseverance of the Rasulullah (PBUH) and the early Muslims through the difficult times. Hijrah demonstrates that for people with faith, there is hope for ease after difficulties. There was also a great unity model among Muslims of different backgrounds. The Hijrah had also marked the beginning of the Islamic State under the leadership of Rasulullah (PBUH) which then became a reference for all state governance everywhere in the future

Approximate Solutions of Higher Dimensional Flow Models of Non-Newtonian Fluids

This thesis deals with the application of a new hybrid scheme to find the approximate solutions of non-Newtonian fluids under different circumstances involving fractional derivatives. The hybrid technique we are employing has less computational effort and time cost as compared to other schemes presents in the literature. In starting, some preliminaries and basic concepts related to non-Newtonian fluids, constitutive equations, fractional calculus and hybrid scheme have been presented. Then in the next chapters the new hybrid scheme has been successfully applied to find the approximate solutions of second grade fluid with fractional derivatives, heat transfer in fractional Walters’-B fluid, Maxwell fluid with fractional derivatives, heat and mass transfer in fractional Jeffrey’s fluid and blood flow having magnetic particles with fractional derivatives. In Chapter 2, approximate solution of unsteady flow of second grade fluid with non-integer order derivatives through a circular cylinder has been procured. The flow of the fluid is produced due to stress applied on the surface of cylinder. The methodology adopted is the use of Laplace transform with numerical inverse Laplace algorithms to solve the governing equations. The inverse Laplace transformation has been procured through Talbot’s algorithm using Matlab software. The validation of numerical results of inverse Laplace transform is performed by employing two other numerical inverse Laplace algorithms namely as Stehfest’s and Tzou’s. The comparison between existing exact solution and our approximate solution is also presented in tabular form. Towards the end, the velocity field and shear stress graphs are depicted to understand the response of physical parameters. The aim of Chapter 3 is to study the heat transfer in Walters’-B fluid involving Caputo-Fabrizio fractional derivatives through an infinite oscillating vertical plate with Newtonian heating in the presence of magnetic field. The x governing equations of velocity and temperature fields are converted first in non-dimensional form by using dimensionless variables and then solved by employing Laplace transformation. The inverse Laplace transform has been simulated by using Stehfest’s inverse Laplace algorithm. The validation of numerical results is provided by employing Tzou’s algorithm. Mathcad software is used for all numerical calculations. The graphical illustrations represent the behavior of material parameters on the solutions. The variation in Nusselt number with the change in fractional and physical parameters is also presented. The goal of Chapter 4 is to examined the flow characteristics of a Maxwell fluid involving fractional derivatives in an infinitely long circular cylinder. A new hybrid scheme is applied to achieve semi-analytical solutions. The fluid is lying inside the cylinder. The approximate solutions for the velocity field and the time dependent shear stress have been established. The approximate solutions are procured by employing Laplace transform. The inverse Laplace transformation has been calculated with Talbot’s algorithm using Matlab software. At the end, velocity and time dependent shear stress graphs are plotted to see the behavior of physical parameters. In Chapter 5, the approximate solution of heat and mass transfer in magneto hydro dynamics (MHD) Jeffrey’s fluid is presented. The fluid is lying over a vertical plate with exponentially heating and constant mass dispersion. The Caputo-Fabrizio fractional operator has been utilized to build up the fractional model. Laplace transformation has been applied to find the approximate solutions of concentration, temperature and velocity fields. In this hybrid technique we have employed Laplace transform together with Stehfest’s inverse Laplace numerical algorithm. The physical effect of material parameters on velocity, concentration and temperature fields are delineated graphically. The impact of non-integer order parameter on solution is also displayed in tabular form. The reason for Chapter 6 is to study the magneto hydrodynamic flow of a viscous fluid having magnetic particles in a cylinder. The fluid is initially electrically charged in the presence of a uniform transverse external magnetic field. To obtain the flow model involving non-integer order derivatives, the fractional calculus approach is used. The solution of the flow model is obtained using Laplace transformation. Simon’s numerical algorithm is employed to obtain inverse Laplace transform. Similar solutions of flow model with ordinary derivatives and flow without magnetic particles has been procured as limiting case. At the end, the impact of non-integer order parameter, Reynolds number and Hartmann number on flow and magnetic particles velocity is analyzed and depicted by graphs.