Department of Mathematics
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Item New Multiplicative Mid-Point and Trapezoidal Fractional Integral Inequalities(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Huma Tanveer; SP23-RMT-011; Dr. Saad Ihsan Butt; LHR TP 9584In this thesis the mid-point fractional integral inequalities and multiplicative trapezoidal inequalities are thoroughly examined. We construct new lemmas and provide their pre cise estimations for differentiable and twice differentiable functions using fractional and k-fractional integrals. Furthermore, new bounds via Holder inequalities are obtained to ¨ demonstrate the strength, flexibility and usefulness of the suggested conclusions in the context of fractional calculus. A variety of examples are analyzed and their graphical rep resentations drawn for easier interpretation in order to validate the theoretical findings. In addition to contributing to our understanding of fractional integral inequalities, this re search provides new opportunities for future investigation into fractional calculus and its uses in science and engineeringItem Hybrid Fractional Estimations for Quadrature Inequalities(Library Information Services COMSATS University Lahore Campus, 2024-03-17) Saira Arif; FA22-RMT-012; Dr. Saad Ihsan Butt; LHR TP 9338We study new fractional integral operators that involve linear combinations of Riemann and Caputo fractional integral operators. Thus, under the fundamentals of generalized con vexities, new fractional integral inequalities are explored and investigated for such hybrid fractional integral operators. This enables us to obtain several interfusing cases for frac tional parameter α ≥ 0. In order to derive fractional quadrature-type inequalities, some hybrid quadrature-type integral identities i.e. Simpson’s and Newton’s type in fractional calculus are derived for differentiable functions. Thus, by employing convexities of first and twice differentiable functions, several estimations of quadrature fractional integral in equalities are obtained. Finally, a number of fractional outcomes are provided related to special mean quadrature inequalities, q-digamma functions and Bessel functionsItem Generalized Fractal–Fractional Integral Inequalities on Fractal Sets with Applications(Library Information Services COMSATS University Lahore Campus, 2023-03-13) Hafiz Muhammad Umer Yasin; SP22-RMT-026; LHR TP 8731; Dr. Saad Ihsan ButtIn this study, we establish a set of novel Bullen-type inequalities applicable to differentiable convex functions within the framework of extended fractional integrals in a fractal domain. The key benefit of employing these inequalities and associated operators lies in their ver satility, allowing the conversion of these inequalities into established results for Riemann integrals. Additionally, they give rise to new inequalities applicable to Riemann-Liouville fractional integral inequalities, as well as generalized Riemann-Liouville fractional integral inequalities. To bolster the relevance of the conclusions, we also present the applications of recently developed results regarding the probability density functions, the quadrature formulae and the special meansItem New Generalized Fractional Variants of Integral Inequalities(Library Information Services, COMSATS University Islamabad, Lahore Campus, 2022) Iram Javed; FA20-RMT-041; LHR TP 7942; Dr. Saad Ihsan ButtThe graphical elegance of fractal theory takes into account the development’s achievability and exceptionalism. Due to its fascinating existence in the mathematical fields of sci- ences, there is a clear association between fractal sets and convexity. In this proposal, we will present generalized convexity and related integral inequalities on a fractal set Rv ( 0 < v 1). In the context of the Beta function, this research presents a new class of gener- alized Hermite-Hadamard type inequalities. This research contributes significant results of novel versions of fractal H¨ older’s and Young’s inequalities. We derive some general con- clusions that capture novel results under investigation. One more remarkable contribution of the study is that two novel auxiliary results along with Trapezoidal and Midpoint type inequalities are provided. Hence, these new results will lead us to generalization of prior results.Item 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 9774In 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.Item 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 9777In 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.