Browsing by Author "Dr. Muhammad Rafiullah"

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On 4th Order Iterative Methods for Nonlinear Equations
(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Muhammad Nasir; SP23-RMT-020; Dr. Muhammad Rafiullah; LHR TP 9573
The main objective of this study is Fourth-order iterative techniques to solve nonlinear equations. In numerous scientific and engineering problems, the nonlinear equations are indispensable and their solution are based on suitable and accurate methods. Fourth-order iterative techniques have been used to solve equations due to their high computational and ideal convergence order. The fourth-order iterative techniques of Chun, Ren, et al., and Qureshi et al. are reviewed and analyzed with a focus on function per iteration, error term, and efficiency indices in this study. Using a comparative analysis, we have demonstrated a new technique from Noor which is computationally efficient and accurate. The results also show that better error correction and optimization techniques significantly contribute to the performance of these methods. In order of convergence and efficiency comparison of their performance, this work provides important understanding
Predictive Modeling of Diabetes Classification using Artificial Neural Networks
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Amin; FA23-RMT-017; Dr. Muhammad Rafiullah; LHR TP 9764
Diabetes mellitus is a chronic disease that has become a major global public health challenge. Timely and accurate prediction not only aids in immediate treatment but also plays a crucial role in formulating effective strategies. In this research, we compare the prediction of diabetes using two popular neural network models Multi-Layer Perceptron (MLP) and General Regression Neural Network (GRNN). This analysis is based on the PIMA Indian Diabetes Dataset, which contains medical information of female patients, including glucose levels, BMI, insulin amount, age, etc. In this research, the dataset underwent stages of cleaning, normalization, and division into training and testing sets. Then, the mathematical details of the MLP and GRNN models were described, which included forward propagation, activation functions, and loss formulas. The MLP model used two hidden layers with ReLU and Sigmoid activation functions, while the GRNN model used Gaussian radial basis functions and Euclidean distance. After training, the performance of both models was evaluated using metrics such as accuracy, confusion matrix, and ROC-AUC, revealing that both models proved effective in predicting diabetes. However, GRNN demonstrated better overall performance due to its non-repetitive structure and smooth results, while MLP exhibited more effective adaptation thanks to fast computation and deep network architecture, making it suitable for large and complex data. This research indicates that if ANN models are designed and configured correctly, they can help in the timely and effective diagnosis of diabetes. This analysis provides guidance to healthcare professionals and data scientists in selecting appropriate models.
Resolving Set of Grid Graph and its Variants
(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2025) Muhammad Awais; FA23-RMT-026; Dr. Muhammad Rafiullah; LHR TP 9772
If the distance in equation 𝑑𝐺 (𝑥, 𝑧) ≠ (𝑦, 𝑧) is valid for separate vertices 𝑥, 𝑦, 𝑧 in graph 𝐺 = (𝑉, � �), we say the pair of vertices 𝑥, 𝑦 can be discriminated by vertex 𝑧. If any two distinct vertices in graph G can be identified by at least one vertex in the vertex subset 𝑆 ⊂ 𝑉, then 𝑆 ⊂ 𝑉 is a resolving set of graph G. The one with the smallest number of nodes among all of the resolving sets is termed a metric basis of graph G, and its cardinality is called the metric dimension of graph G [1]. Let (𝑉, 𝐸) be a simple connected graph. We define the distance between an edge 𝑒 = 𝑥𝑦 and vertex 𝑣 as follows: � �(𝑒, 𝑣) = 𝑚𝑖𝑛{𝑑(𝑥, 𝑣), 𝑑(𝑦, 𝑣)}. Two edges 𝑒1 and 𝑒2 are distinguished by a vertex v if (𝑒1, 𝑣) ≠ (𝑒2, 𝑣). If there is a vertex 𝑠 ∈ 𝑆 such that s differentiates 𝑒1 and 𝑒2 for any two unique edges 𝑒1,2∈ 𝐸, then a set 𝑆 ⊆ 𝑉 is an edge metric generator of a graph (𝑉, 𝐸). An edge basis of G is an edge generating set with the fewest members; the edge dimension of G is the number of elements in an edge basis [2]. This study calculates the diagonally folded grid (DFG) graph's metric and edge metric dimensions.
Solution of Some Dynamical Models with Pade Approximation
(Library Information Servicers COMSATS University, Lahore Campus, 2025) Shehroz Ahmad FA23-RMT-048; Dr. Muhammad Rafiullah; LHR TP 9791
In this thesis we expressed the solution of some dynamical models with Pade Approximation. For this purpose, we take two dynamical Epidemic and SIRV Models. Firstly, we solve these models with the help of Taylor series solution. Then we apply Pade Approximation on the Taylor series solution. We obtained the unique rational functions for Epidemic and SIRV Models. We compare the results of Pade Approximation with series solution of Epidemic and SIRV model with the help of graph. After this, we also analyze these models by changing the values of their parameters. In last we present our conclusion about these Models.
Solutions of Some Mathematical Models by Homotopy Perturbation Method
(Library Information Services, CUI Lahore, 2022) Muhammad Asif; SP21-RMT-006/; Dr. Muhammad Rafiullah
The main objective of this study is to apply homotopy perturbation method to solve some nonlinear dynamical models, especially the models which lead to predict and control the pandemic diseases or SIR (S = susceptible, I = infected and R = recovered). We solve two models by using homotopy perturbation method (HPM), one is the “population dynamics” and the second is “novel coronavirus (COVID-19)” with six compartments which consist on susceptible, exposed, asymptomatic infected, symptomatic infected and recovered people and the concentration of the COVID-19 in the environment, denoted by S(t), E(t), A(t), I(t), R(t) and B(t) respectively. We provide the convergence order of HPM and show the convergence order of a series solution. Moreover, we analyze the series solution of the model of novel coronavirus (COVID-19) by taking the different values of controlling parameter and showing them in graphs.