Browsing by Author "Dr. Kashif Nazar"

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Analysis of the Formulations and Continuities of Kernels of Integral Equations for Ahlfors Map
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Muhammad Amin Khan; FA22-RMT-017; Dr. Kashif Nazar; LHR TP 9380
This study aims to examine and analyze the boundary integral equations of the Ahlfors map specifically for multiply connected regions. The Ahlfors map, known for mapping the boundary of a domain to the unit circle, helps in understanding the domain’s structure. Our goal is to rederive boundary integral equations that are more effective in determining the Ahlfors map and its unknown zeros. Identifying these zeros is crucial as they provide important information about the map’s behavior and the domain’s geometry. By develop ing these new equations, we hope to address existing challenges and solve long-standing problems in the field, ultimately advancing mathematical understanding and applications of the Ahlfors map in complex analysis and related disciplines.
Analyzing the Nystrom Method for Second Kind Fredholm Integral Equations
(Library Information Services COMSATS University Islamabad Lahore Campus, 2023-03-12) Kainat Tahir; SP22-RMT-017; LHR TP 8724; Dr. Kashif Nazar
This research proposal aims to study if second-kind Fredholm Integral Equations can possibly be solved using the Nystrom Method, which are common in many different fields and which require for the development of effective numerical solutions. The Nystrom Method will be thoroughly examined, along with an explanation of how it applies to second-kind Fredholm Integral Equations. Additionally, it will look at how different parameters influence the convergence of the Nystrom Method, such as the number of discretization points and the choice of numerical quadrature. The expected outcomes are meant to deepen our understanding and facilitate the advancement of numerical algorithms, leading to a more reliable and efficient solutions for integral equations. This discovery holds importance for numerous scientific and engineering domains, and it can lead in advancements in computer methodologies
Comparisons of the Computation of Zeros of Ahlfors Map in Bounded Doubly Connected Regions
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Mohsin Khan; FA23-RMT-023; Dr. Kashif Nazar; LHR TP 9769
This thesis compares and analyzes various methods for determining the zeros of the Ahlfors map in bounded doubly connected regions. The Riemann mapping theorem’s extension to multiply connected domains, known as the Ahlfors map, is crucial for complex analysis and has important uses in engineering and science fields like fluid dynamics. The two special points, or zeros, of the Ahlfors map in doubly connected regions can be found using a variety of methods described in the literature. The Ahlfors map is a fundamental mathematical tool that enables us to project complex structures onto a unit disk while preserving the same angles. The second zero of the Ahlfors map is more difficult to locate than the first. Several mathematical and numerical techniques, including boundary integral equations, and specific kernels like the Szego˝ and Neumann kernels, are used and compared in the thesis. The research compares these methods to see which ones are more accurate, faster, and more stable when applied to different types of doubly connected regions. The results help identifies the best approaches for calculating the Ahlfors map’s zeros and can be useful in practical fields like fluid dynamics, electricity, and image processing. This work also helps future studies on more complex shapes and higher connectivity regions.
discussions on ahlfors mapping funetion and its zeros
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Talha Rafiq (FA20-BSM-048) : Syeda Rubab Fatima (FA20-BSM-061); Dr. Kashif Nazar; LHR TP 9925
Conformal mapping is very usefull technique complex analysis to solve
Discussions on Solutions of Fredholm Integral Equations
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2024) Muhammad Adeel Khan; CIIT/SP21-BSM-022/LHR; Dr. Kashif Nazar; LHR TP 9888
This thesis investigates the analytical and numerical solutions of Fredholm integral equa- tions of the second kind, focusing on both homogeneous and non-homogeneous cases. In the analytical framework, we explore solutions with degenerate kernels and methods to ap- proximate complex kernels by degenerate forms, providing efficient pathways for problem simplification. The study covers both theoretical and applied aspects, emphasizing cases where the kernel exhibits specific structures that facilitate closed-form solutions. Numer- ical methods are also developed and analyzed for solving these integral equations in both homogeneous and non-homogeneous settings. The numerical approaches are evaluated for their accuracy and computational efficiency, providing insights into practical implemen- tations and approximations for real-world applications. Through comprehensive analysis and comparison of methods, this research contributes to the effective solution of Fredholm integral equations, offering new perspectives for both theoretical exploration and applied problem-solving.
Formation and Analysis of Ahlfors Map Integral Equations for the Regions with Multiple
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2021) Usman Ashraf; Sp19-RMT-005; LHR TP 7410; Dr. Kashif Nazar
This research work elaborates the detailed numerical analysis of the integr equation related to Szegö kernel and Ahlfors map with the assistance Mathematica 12.0. The Ahlfors map and Szegö kernel are both associated to ea other. This is a Fredholm integral equation of second kind via Kerzman-Stein kern where Szegö kernel is the solution of this integral equation. The solution of integr equation is obtained by transforming the integral equations into matrix form aft discretizing. This work presents the computation techniques of Ahlfors map, zeros, boundary values and the interior points of a region under Ahlfors map. Th research construct one new Ahlfors map integral equation with Neumann-ty kernel for the regions with multiple connectivity onto a unit disk. The kernel an non-homogeneous term of this equation contain zeros of Ahlfors map. This integr equations is constructed by a non-homogeneous boundary relationship for t regions with multiple connectivity. The new Ahlfors map integral equation analyzed by parameterizing and discretizing. Discretized integral equations lead
Integral Equations Method for Ahlfors Map and its Zeros
(Library Information Services COMSATS University Islamabad Lahore Campus, 2021) Muhammad Ilyas; SP19-RMT-023; LHR TP 7409; Dr. Kashif Nazar
The purposeofthisstudyistoexamineandanalyzetheboundaryintegral equations ofanAhlforsmapformultiplyconnectedregions.Alsotoderive some newboundaryintegralequationsrelatedtoanAhlforsmapwhich maymorebesuitableforfindingtheAhlforsmapanditsunknownzeros. The Nystr¨om methodwiththetrapezoidalrulewillbeusedtoparametrize and discretizetheseintegralequationsintosystemsofequations.
Studies On Inner Product Spaces And Hilbert Spaces
(Library Information Services COMSATS University Lahore Campus, 2024-03-18) Laraib Akram; SP23-RMT-013; Dr. Kashif Nazar; LHR TP 9578
Studies On Inner Product Spaces And Hilbert Spaces