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

Boundary Value Problems for Fractional Differential Equations: Existence Theory and Numerical Solutions

Fractional calculus can be considered as supper set of conventional calculus in the sense that it extends the concepts of integer order differentiation and integration to an arbitrary (real or complex) order. This thesis aims at existence theory and numerical solutions to fractional differential equations. Particular focus of interest are the boundary value problems for fractional order differential equations. This thesis begins with the introduction to some basic concepts, notations and definitions from fractional calculus, functional analysis and the theory of wavelets. Existence and uniqueness results are established for boundary value problems that include, two–point, three–point and multi–point problems. Sufficient conditions for the existence of positive solutions and multiple positive solutions to scalar and systems of fractional differential equations are established using the Guo–Krasnoselskii cone expansion and compression theorems. Owning to the increasing use of fractional differential equations in basic sciences and engineering, there exists strong motivation to develop efficient, reliable numerical methods. In this work wavelets are used to develop a numerical scheme for solution of the boundary value problems for fractional ordinary and partial differential equations. Some new operational matrices are developed and used to reduce the boundary value problems to system of algebraic equations. Matlab programmes are developed to compute the operational matrices. The simplicity and efficiency of the wavelet method is demonstrated by aid of several examples and comparisons are made between exact and numerical solutions.