فن حدیث میں مولانا عبد الرحمن مبارک پوری کی خدمات کا جائزہ

Sūnan-ul-Tirmizi is an encyclopedia of Ahādith-ul-Ahkām. Imam Tirmizi is the Mohadith who divided hadiths into Sahih and Zaeef for the first time. He accepts or rejects a hadith on the base of Taāmul-e-Ummah. He is only the Mohadith who established the terminology of “Filbāb” in which he gives the words of hadith from a Sahabi and mentions the names of all other sahabies who are rawi of the same hadith. There are many sharh of Tirmizi written by Muhadiseen. Among them Tuhfat ul Ahwazi is famously written by Molana Abdul Rahman Mubarakpuri. He explores the terminologies of Sonan-e-Tirmizi. He discussed uloom ul hadith, books of hadith, Shoroh-ul-hadith, Asma-ul-rejal and ilm ul ansab etc. He mentions tafsiri aqwal, fiqhi problems and usool-e-hadith. He also solved the Tasaholat-e-Tirmizi in validity (sihat) and unvalidity (zouf). He is the first mohadith who tried to find the words of hadith from other sahabies whose names are given in “Filbab”. He did it but could not find the words of 87 ahadith for which he used the term “Lam aqif alaih” and 417 ahadith for which he used the term “Le Yonzar man akhraja haza ul hadith”. This thing makes it distinct from other shoroh of Sūnan-e-Tirmizi. He depends on the usool-e-hadith of forefather Muhadiseen and he did not establish his own usool hadith.

Localized Approximation of Time Dependent Partial Differential Equations Using Radial Kernels

Time dependent partial differential equations (PDEs) model systems that experience change as a function of time. Time dependent PDEs have numerous applications such as diffusion, heat transfer, thermodynamics, population dynamics and wave phenomena. They are naturally parabolic or hyperbolic. Meshless methods have large advantages in accuracy over other methods, such as finite difference method (FDM), finite volume method (FVM), finite element method (FEM). The main features of the meshless methods are its simplicity, efficiency and invariance under euclidian transformation and can handle problems defined on complex shape domains. Meshless methods have some serious drawbacks as well. When the nodes are increased the method solve comparatively large system, and the ill-conditioning of the system matrix causes instability. Due to which it becomes difficult to achieve spectral convergence. This thesis is concerned with two issues that is to solve the ill- conditioning problem of the interpolation matrix by radial kernels in local setting and to replace the time marching scheme with the numerical inversion of Laplace transforms which eliminates temporal truncation errors and the need for many time integration steps. The method is applied to solve fractional and integer order time dependent PDEs. The method comprises of three steps. First the Laplace transform is applied to the partial differential equation and boundary conditions in a given differential system. Second, the kernel based method is employed to solve the transformed differential system. Third, the solution is represented as a contour integral evaluated to high accuracy by trapezoidal rule.