عناية أئمة النقد الحديثي بفقه الحديث

It is generally perceived in contemporary intellectual movements that canonical Traditionalists did not take had฀th text into consideration as their scholarly efforts were limited to the evaluation of had฀th chains. Aforementioned notion - in my opinion - originates from shallow study of methodology adopted by canonical had฀th critics، as a deeper look into their scholarly works reveals that sciences of had฀th includes the authentication and disparagement of traditions as well as comprehension and deduction from had฀th content. The sole objective of early Traditionalists from transmission، collection of had฀th، its evaluation، authentication and disparagement was to safeguard the true meaning of Sunnah and to transmit it in its pure form to the successors. In fact the peculiarity of their work is that they exert all efforts in order to deal with had฀th as a single undivided whole، where examination of content was not irrelevant to the evaluation of chain، their conscientious efforts recorded in major works of had฀th show how they evaluated content of had฀th to determine that it was not contradictory to Shari‘ah، or with another sound tradition، as there was a possibility that a certain reliable reporter made mistake or speculated in transmitting the meaning of had฀th. Therefore we witness them disparaging a certain transmitter for his negligence and errors whereas his had฀th is forsaken، moreover they would not consider him a Traditionalists or muhaddith฀n if excessive speculations were found in his report. This research paper aims at investigating the aforementioned hypothesis.

Meshfree Methods for the Numerical Solutions of Partial Differential Equations

In this dissertation, meshfree (meshless) methods using meshless shape functions are proposed for the numerical solutions of partial differential equations (PDEs). These PDEs have either integer or fractional order time derivatives. Weighted θ-scheme (0≤θ ≤1) is used for time discretization of integer case, whereas, for fractional case, the same discretization scheme is combined with a simple quadrature formula. For space (spatial) discretization we used meshless shape functions owing Kronecker delta function property. These shape functions are obtained viapointinterpolationapproachandradialbasisfunctions(RBFs). Finallywiththehelpofcollocationmethodthe given PDE reduces to system of algebraic equations, which are then solved via LU decomposition in iterations. For the proposed numerical scheme, stability analysis is carried out theoretically and computational examples are provided to support the analysis. The proposed scheme has been tested via application to several concrete and benchmark problems of engineering interest. ApproximationqualityandaccuracyofcomputedsolutionsaremeasuredusingL∞, L2 andLrms discrete errornorms. Efficiencyandorderofapproximationoftheproposedschemeinspaceandtimeareanalyzedthrough variation of number of nodal points N and time step-size δt. The documented results, in the form of tables and figures, reveal very good agreement to true solutions as well high accuracy to earlier proposed technique available in the literature. In RBFs, the presence of shape (support) parameter c∗ plays a crucial role. Accuracy of the RBFs based scheme can be improved via proper selection of this parameter. For this purpose, an automatic optimal shape parameter selection algorithm is proposed. To check effectiveness and automatic (adaptive) nature of this algorithm in RBFs approximation method, time fractional Black-Scholes models have been solved. It has been noted that the proposed algorithm worked well and gives excellent accurate solutions for various fractional order time derivatives. The RBFs approximation (Kansa) method results in dense ill-conditioned matrix. For the treatment of this issue weproposeahybridRBFs(HRBFs)approximationmethod. Byextendingthisidea, anadaptive(automatic)algorithm is proposed for optimal parameters selection in HRBFs. For validation, again time fractional Black-Scholes models are reconsidered. Simulations revealed acceptable accurate solutions in hybrid RBFs method too. Along with that significant reduction in condition number of the resultant matrix is observed up to several manifold. Hence, HRBFs method can be seen as an alternative remedy for curing ill-conditioning in usual RBFs method. Computer simulations have been carried out via MATLAB R2013a on a personal laptop with configuration, Processor: Intel(R) Core(TM) i5-5200U CPU @ 2.20GHz 2.20GHz, RAM: 4.00 GB, System type: 64-bit Operating System, x64-based processor.