Department of Mathematics

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Author: search.filters.author.Dr. Saad Ihsan ButtSubject: search.filters.subject.FA23

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    New Generalized Ostrowski, Simpson and Boole’s Type Inequalities
    (Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Muhammad Mehtab (FA23-RMT-028); Dr. Saad Ihsan Butt; LHR TP 9774
    In this proposal firstly, we introduce a parametric identity for generalized differentiable functions using a generalized fractal-fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski and Simpson type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical H¨older and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of H¨older and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis.
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    New Perspectives of Solution of Heat Equation Using Neural Networks and Jensen’s Inequality
    (Library Information Services COMSATS University Islamabad Lahore Campus, 2025) Rafay Ahmed (FA23-RMT-031); Dr. Saad Ihsan Butt; LHR TP 9777
    In this study, we used Physics-Informed Neural Networks (PINNs) to solve the 1D, 2D, and 3D heat equation. To enhance training stability and accuracy, we replaced Jensen’s inequality as the loss function with more traditional methods such as Mean Square Error (MSE). We aimed to demonstrate that it is possible for PINNs to efficiently solve the heat equation with minimal data. In order to implement the model, we used a number of libraries, such as TensorFlow and Keras for creating the neural networks and NumPy, SciPy, and Matplotlib for managing the data and displaying the outcomes.