Home
Add
Get on Google Play
Home
> Edit
Add/Update Thesis
Title*
Author's Name*
Supervisor's Name
Abstract
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.
Subject/Specialization
Language
Program
Faculty/Department's Name
Institute Name
Univeristy Type
Public
Private
Campus (if any)
Institute Affiliation Inforamtion (if any)
City where institute is located
Province
Country
Degree Starting Year
Degree Completion Year
Year of Viva Voce Exam
Thesis Completion Year
Thesis Status
Completed
Incomplete
Number of Pages
Urdu Keywords
English Keywords
Link
Select Category
Religious Studies
Social Sciences & Humanities
Science
Technology
Any other inforamtion you want to share such as Table of Contents, Conclusion.
Your email address*