https://scholarworks.unist.ac.kr/handle/201301/55 2026-04-19T14:10:16Z https://scholarworks.unist.ac.kr/handle/201301/91244 Title: Some applications of the traces of Frobenius of elliptic curves in a certain family Author(s): Cho, Peter J.; Yoo, Jinjoo Abstract: Let J = {a, b} be an unordered pair of elements of Fq and EJ the associated elliptic curve of the form y 3 = (x − a)(x − b) over Fq. Using its trace of Frobenius, we obtain several applications. We first compute the average analytic rank of elliptic curves for our family and generate elliptic curves with designated extremal primes. Furthermore, we determine explicit and average values on class numbers of every constant field extension of KJ := Fq(T, p3 (T − a)(T − b)). Finally, we estimate the exact values and average values on Euler-Kronecker constants of KJ . 2026-07-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/91002 Title: Existence and Stability of Sadovskii Vortex Patch: an odd-symmetric touching pair of uniform vortices Author(s): Sim, Young-Jin Abstract: As a traveling solution of the two-dimensional incompressible Euler equations, the Sadovskii vortex patch takes the form of a counter-rotating pair of vortex patches that are in contact. This model was suggested by Sadovskii [J. Appl. Math. Mech., 1971] and has since attracted considerable attention due to its significance to the inviscid limit of planar flows via the Prandtl–Batchelor theory and its role as an asymptotic state of the dynamics of vortex rings. In this paper, we establish the existence and stability of the Sadovskii vortex patch. First, we show that a Sadovskii vortex patch arises as a maximizer of the kinetic energy under an exact impulse condition. By analyzing the fluid velocity along the symmetry axis and its relation to the vorticity of a dipole, we verify that two patches in the dipole are in contact. Second, we construct a subcollection of such energy maximizers that is stable in the following sense: if an initial vorticity is sufficiently close to a maximizer, then the corresponding solution remains close to this collection up to a translation and travels at a similar speed to the maximizers. This is achieved via a concentration–compactness argument together with the conservation of impulse, circulation, and kinetic energy in time. Furthermore, using uniform estimates of energy maximizers and a shift estimate obtained by estimating the center of mass of the solution, we show that the solution keeps its touching structure uniformly in time. Major: Department of Mathematical Sciences 2026-01-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/91001 Title: Efficient Numerical Schemes for Fractional Models and Their Applications to Heat Transfer and Nanofluid Flows Author(s): ISMAIL, MUHAMMAD Abstract: This thesis presents the development and applications of efficient numerical frameworks for solving fractional partial differential equations (PDEs) and their extension to realistic heat-transfer problems governed by fractional dynamics. The research integrates advanced wavelets based formulations with fast algorithm to overcome the challenges associated with the non-local and memory dependent behavior of fractional operators. In the mathematical component, two computationally efficient solvers are pro- posed, the fast Shifted Legendre Wavelets (SLWs) method and the fast Green–CAS wavelets method. The fast SLWs approach combines the orthogonality of Shifted Legendre Wavelets with the efficient sum-of-exponential (SOE) base approximation of Caputo fractional derivative to accurately and effi- ciently solve two-dimensional, multi-term, and nonlinear fractional PDEs. Convergence analyses and extensive numerical experiments confirm its robustness, demonstrating comparable accuracy but signif- icantly reduced computational cost relative to traditional L1-based and conventional operational matrix based wavelets schemes. The Green–CAS wavelets framework is further enhanced through the introduction of a Green’s- function formulation that completely eliminates the need for operational integration matrices, simpli- fying implementation for boundary value problems. The resulting fast Green–CAS method, designed for time fractional orders α ∈ (0,1), exhibits superior stability and precision. Nonlinear equations are linearized using iterative schemes, and convergence studies verify the reliability of the technique. Nu- merical results reveal that the fast Green–CAS method consistently outperforms classical operational wavelets based and other existing approaches in terms of both efficiency and accuracy. Building upon these methodological advances, the thesis extends fractional modeling to the physical domain of nanofluid convection and radiative heat transfer. A fractional formulation of the Tiwari-Das nanofluid model is developed, incorporating Caputo time fractional derivatives and solved via a ADI based schemes. Parametric analyses demonstrate that the fractional order (γ), Rayleigh number (Ra), nanoparticles volume fraction (φ ), and radiation parameter (Rd) collectively govern the evolution of flow and temperature fields. Lower fractional orders enhance nanofluid flow and improve heat transfer rates, while Multiple Linear Regression (MLR) analyses highlight the dominant influence of nanoparticles concentration in fractional regimes. Comparative results show that fractional models achieve higher thermal efficiency than their integer-order counterparts. Overall, the work establishes a unified, high-accuracy, and computationally efficient framework for fractional modeling, from fundamental numerical formulations to applied thermofluid systems. In future work, these methods will be extended to variable and distributed order fractional systems and adapted to complex, multi-dimensional geometries. Further studies will focus on integrating data driven approaches such as physics-informed and graph neural networks with fractional solvers to enhance predictive capa- bility and parameter estimation in buoyancy-driven nanofluid convection systems. Major: Department of Mathematical Sciences 2026-01-31T15:00:00Z https://scholarworks.unist.ac.kr/handle/201301/91000 Title: Efficient and High-Order Numerical Approaches for Variable-Order Fractional Sub-Diffusion Equations Author(s): Lee, Junseo Abstract: This thesis proposes a unified, high-order, and computationally efficient framework for solving multi- dimensional variable-order (VO) fractional sub-diffusion equations. The primary challenge in these problems is the O(N2 t ) computational cost and memory-driven nonlocality of fractional operators, cou- pled with complex accuracy issues. Our framework overcomes these challenges by integrating three key components: (1) a novel high-order temporal discretization based on a VO extension of the L2-1σ scheme, (2) a Shifted Binary Block Partition (SBBP) algorithm for fast computation, and (3) a Weighted Alternating-Direction Implicit (WADI) scheme for multi-dimensional efficiency. The key theoretical contributions of this work are three-fold. First, we establish a rigorous stability analysis for the proposed SBBP-based fast method, defining the necessary conditions on the polynomial approximation’s accuracy to ensure the stability of the full sub-diffusion scheme. Second, we identify and resolve a critical accuracy degradation issue inherent in the VO-L2-1σ method, where its non- uniform nodes can corrupt the global convergence order. We propose a novel approach, restricting this operator to the initial time steps, which is shown to maintain the operator’s high accuracy while restoring the desired global convergence of the full PDE solution. Third, the framework’s accuracy is further advanced to the fourth order in space using a Compact Finite Difference (CFD) scheme, which is efficiently coupled with a corresponding Compact ADI solver. Numerical experiments are conducted to validate the theoretical analysis. The results demonstrate that our proposed integrated framework (e.g., the Fast WADI scheme) achieves significantly higher accu- racy and possesses a superior computational efficiency, with lower CPU times compared to traditional L1-ADI and other benchmark methods. This work establishes a robust, fast, and high-order methodol- ogy, providing a valuable tool for accurately simulating complex physical processes governed by VO fractional dynamics. Major: Department of Mathematical Sciences 2026-01-31T15:00:00Z