We propose two novel multiagent systems characterized by discontinuous vector fields, designed to track a given moving or stationary target while ensuring collision avoidance. Unlike traditional approaches that employ continuous vector fields, we adopt Filippov's differential inclusion framework to model the dynamics at discontinuity points. For each model, we present an admissible set of initial conditions, system parameters, kernel functions, and a target configuration that guarantees both qualitative and quantitative asymptotic tracking of a prescribed target while avoiding collisions.
{"title":"Asymptotic Target Tracking in Discontinuous Dynamical Systems: Applications of Filippov's Inclusion Framework","authors":"Woojoo Shim, Hyunjin Ahn","doi":"10.1111/sapm.70142","DOIUrl":"https://doi.org/10.1111/sapm.70142","url":null,"abstract":"<div>\u0000 \u0000 <p>We propose two novel multiagent systems characterized by discontinuous vector fields, designed to track a given moving or stationary target while ensuring collision avoidance. Unlike traditional approaches that employ continuous vector fields, we adopt Filippov's differential inclusion framework to model the dynamics at discontinuity points. For each model, we present an admissible set of initial conditions, system parameters, kernel functions, and a target configuration that guarantees both qualitative and quantitative asymptotic tracking of a prescribed target while avoiding collisions.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145521856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots—closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots. In the rigid case, evaluating a bonded knot in the basis of this module yields an bonded knot invariant closely related to the APS bracket and the Simplified RNA polynomial.
{"title":"The Kauffman Bracket Skein Module of Bonded Knots and Applications to Entangled Proteins","authors":"Boštjan Gabrovšek, Matic Simonič","doi":"10.1111/sapm.70123","DOIUrl":"https://doi.org/10.1111/sapm.70123","url":null,"abstract":"<p>We model proteins with intramolecular bonds, such as disulfide bridges, using the concept of bonded knots—closed loops in three-dimensional space equipped with additional bonds that connect different segments of the knot. We extend the Kauffman bracket polynomial (which is closely related to the Jones polynomial) to bonded knots through the introduction of the bonded version of the Kauffman bracket skein module. We show that this module is infinitely generated and torsion-free for both the rigid and topological case of bonded knots. In the rigid case, evaluating a bonded knot in the basis of this module yields an bonded knot invariant closely related to the APS bracket and the Simplified RNA polynomial.</p>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/sapm.70123","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145470009","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the dynamics of the modified Korteweg-de Vries (mKdV) equation from the perspective of the weak Kolmogorov-Arnold-Moser (KAM) theory, and obtain that the associated Hamiltonian system admits a weak KAM solution. This solution satisfies the Hamilton–Jacobi equation in the sense of the minimal measure, which allows the system to be reduced to a class of integrable Hamiltonian systems. The minimal measure is closely related to the Aubry–Mather theory. Consequently, we establish the existence of a weak solution of the perturbed mKdV equation, despite the presence of system resonance. Finally, we extend this method to quasi-periodic perturbed mKdV equations and coupled mKdV equations, proving the existence of weak KAM solutions.
{"title":"Weak KAM Theory for Modified Korteweg-de Vries Equations","authors":"Xun Niu, Yong Li, Kaizhi Wang","doi":"10.1111/sapm.70136","DOIUrl":"https://doi.org/10.1111/sapm.70136","url":null,"abstract":"<div>\u0000 \u0000 <p>We investigate the dynamics of the modified Korteweg-de Vries (mKdV) equation from the perspective of the weak Kolmogorov-Arnold-Moser (KAM) theory, and obtain that the associated Hamiltonian system admits a weak KAM solution. This solution satisfies the Hamilton–Jacobi equation in the sense of the minimal measure, which allows the system to be reduced to a class of integrable Hamiltonian systems. The minimal measure is closely related to the Aubry–Mather theory. Consequently, we establish the existence of a weak solution of the perturbed mKdV equation, despite the presence of system resonance. Finally, we extend this method to quasi-periodic perturbed mKdV equations and coupled mKdV equations, proving the existence of weak KAM solutions.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145470252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work studies the global well-posedness of the Oldroyd-B model with anisotropic viscosity. While global existence and uniqueness of strong solutions for the fully dissipative Oldroyd-B model were established in Constantin–Kliegl [Archive for Rational Mechanics and Analysis, 206 (2012): 725–740], under initial data, the horizontally viscous counterpart—where dissipation of velocity acts only along the horizontal direction—remains unexplored. We establish the global existence and uniqueness of strong solutions to the initial-boundary value problem for the horizontally viscous Oldroyd-B model with arbitrary large initial data in the vertical strip domain and the periodic channel , where represents the one-dimensional torus. To the best of our knowledge, this work provides the first rigorous proof of global-in-time existence and uniqueness of strong solutions for the partially dissipative Oldroyd-B model. In addition, we investigate the long-time asymptotic behavior of solutions for small initial data in the periodic channel . Our results extend the current analytical understanding of visco
本文研究了具有各向异性黏度的Oldroyd-B模型的全局适定性。在Constantin-Kliegl中建立了全耗散Oldroyd-B模型强解的整体存在唯一性[j] .力学与分析学报，206 (2012)：[725-740]，在H 2(R 2)$ H^2(mathbb {R}^2)$初始数据下，水平粘性对应部分（速度耗散仅沿水平方向起作用）仍未探索。本文建立了具有任意大初始数据的水平粘性Oldroyd-B模型初边值问题强解的全局存在唯一性[0]。1] × R $[0,1]乘以mathbb {R}$和周期通道[0,1]× T $[0,1]乘以mathcal {T}$，其中T $mathcal {T}$表示一维环面。据我们所知，这项工作首次提供了部分耗散的oldyd - b模型强解的全局时间存在性和唯一性的严格证明。此外，我们研究了周期通道[0,1]× T $[0,1]times mathcal {T}$中小初始数据解的长时间渐近行为。我们的结果扩展了目前对部分耗散约束下粘弹性流体动力学的分析理解。
{"title":"On the Initial-Boundary Value Problem for the 2D Partially Dissipative Oldroyd-B Model: Global Well-Posedness and Large Time Stability","authors":"Zhenrong Nong, Yinghui Wang, Huancheng Yao, Shihao Zhang","doi":"10.1111/sapm.70139","DOIUrl":"https://doi.org/10.1111/sapm.70139","url":null,"abstract":"<div>\u0000 \u0000 <p>This work studies the global well-posedness of the Oldroyd-B model with anisotropic viscosity. While global existence and uniqueness of strong solutions for the fully dissipative Oldroyd-B model were established in Constantin–Kliegl [<i>Archive for Rational Mechanics and Analysis</i>, 206 (2012): 725–740], under <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>H</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H^2(mathbb {R}^2)$</annotation>\u0000 </semantics></math> initial data, the horizontally viscous counterpart—where dissipation of velocity acts only along the horizontal direction—remains unexplored. We establish the global existence and uniqueness of strong solutions to the initial-boundary value problem for the horizontally viscous Oldroyd-B model with arbitrary large initial data in the vertical strip domain <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>]</mo>\u0000 <mo>×</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$[0,1]times mathbb {R}$</annotation>\u0000 </semantics></math> and the periodic channel <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>]</mo>\u0000 <mo>×</mo>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation>$[0,1]times mathcal {T}$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mi>T</mi>\u0000 <annotation>$mathcal {T}$</annotation>\u0000 </semantics></math> represents the one-dimensional torus. To the best of our knowledge, this work provides the first rigorous proof of global-in-time existence and uniqueness of strong solutions for the partially dissipative Oldroyd-B model. In addition, we investigate the long-time asymptotic behavior of solutions for small initial data in the periodic channel <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>]</mo>\u0000 <mo>×</mo>\u0000 <mi>T</mi>\u0000 </mrow>\u0000 <annotation>$[0,1]times mathcal {T}$</annotation>\u0000 </semantics></math>. Our results extend the current analytical understanding of visco","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145470253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study an initial-boundary value problem for a novel phase-field model describing the evolution of spinodal decomposition, a typical example of solid-phase separation. The nonuniform, degenerate parabolic evolution equation in the model differs from the classical Cahn–Hilliard equation by a nonsmooth gradient term of the order parameter. The existing results are mostly limited to one-dimensional cases, and this paper aims to extend the results to two-dimensional spaces. We prove the global existence of weak solutions to this model and use the model to simulate the microstructural evolution of phase separation.
{"title":"Global Solutions in the 2-D Case of a Degenerate Phase-Field Model for Spinodal Decomposition","authors":"Xingzhi Bian, Yangxin Tang, Lixian Zhao","doi":"10.1111/sapm.70132","DOIUrl":"https://doi.org/10.1111/sapm.70132","url":null,"abstract":"<div>\u0000 \u0000 <p>We study an initial-boundary value problem for a novel phase-field model describing the evolution of spinodal decomposition, a typical example of solid-phase separation. The nonuniform, degenerate parabolic evolution equation in the model differs from the classical Cahn–Hilliard equation by a nonsmooth gradient term of the order parameter. The existing results are mostly limited to one-dimensional cases, and this paper aims to extend the results to two-dimensional spaces. We prove the global existence of weak solutions to this model and use the model to simulate the microstructural evolution of phase separation.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145407260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given that relativistic density and velocity propagate at the same speed, the Cauchy problem with smooth and compactly supported initial data for relativistic dust in Schwarzschild–de Sitter (SdS) and Schwarzschild–anti-de Sitter (SAdS) spacetimes without black hole has been investigated using the characteristic method. Based on the explicit solution formulas under the spherically symmetric assumption, a precise classification of the initial data is provided, elucidating whether the classical solution for relativistic dust will persist globally or encounter a finite-time blowup. Moreover, the paper offers a further analysis of the exact blowup phenomenon, including detailed insights into the blowup rate related to the blowup time.
{"title":"On the Evolution of Relativistic Dust in Schwarzschild–de Sitter and Schwarzschild–Anti-de Sitter Spacetimes. Part I: The Vanishing Mass Case","authors":"Yifan Liu, Xianshu Qu, Yuzhu Wang, Changhua Wei","doi":"10.1111/sapm.70134","DOIUrl":"https://doi.org/10.1111/sapm.70134","url":null,"abstract":"<div>\u0000 \u0000 <p>Given that relativistic density and velocity propagate at the same speed, the Cauchy problem with smooth and compactly supported initial data for relativistic dust in Schwarzschild–de Sitter (SdS) and Schwarzschild–anti-de Sitter (SAdS) spacetimes without black hole has been investigated using the characteristic method. Based on the explicit solution formulas under the spherically symmetric assumption, a precise classification of the initial data is provided, elucidating whether the classical solution for relativistic dust will persist globally or encounter a finite-time blowup. Moreover, the paper offers a further analysis of the exact blowup phenomenon, including detailed insights into the blowup rate related to the blowup time.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 5","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145407261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
我们考虑趋化系统t = Δ的经典解U−χ∇·（U∇v）+ a u−μ u α，τ v t = Δ非线性Neumann边界条件下的逻辑源v−v + u $$begin{equation*} {leftlbrace defeqcellsep{&}begin{array}{l} u_t=Delta u-chi nabla cdot (u nabla v) + a u- mu u^{alpha }, [3pt] tau v_t=Delta v-v +u end{array} right.} end{equation*}$$∇u·ν = | u | r $nabla ucdot nu = |u|^r$ （r ＆gt; 1 $r>1$）在光滑有界域中Ω∧R N $Omega subset mathbb {R}^N$，其中N≥2 $Nge 2$, α≥2 $alpha ge 2$。在一定条件下，我们证明了上述系统具有非线性Neumann边界条件和边界的亚临界增长率的经典解的整体存在性。特别地，我们去掉了定义域的凸性假设，放宽了r $r$的取值范围。
{"title":"Global Existence of Solutions to a Chemotaxis System With Nonlinear Neumann Boundary Condition","authors":"Juan Yang, Chunyou Sun","doi":"10.1111/sapm.70133","DOIUrl":"https://doi.org/10.1111/sapm.70133","url":null,"abstract":"<div>\u0000 \u0000 <p>We consider the classical solutions to the chemotaxis system\u0000\u0000 </p><div><span><span><!--FIGURE--><span></span><math>\u0000 <semantics>\u0000 <mfenced>\u0000 <mtable>\u0000 <mtr>\u0000 <mtd>\u0000 <mrow>\u0000 <msub>\u0000 <mi>u</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mi>Δ</mi>\u0000 <mi>u</mi>\u0000 <mo>−</mo>\u0000 <mi>χ</mi>\u0000 <mo>∇</mo>\u0000 <mo>·</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>u</mi>\u0000 <mo>∇</mo>\u0000 <mi>v</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 <mi>a</mi>\u0000 <mi>u</mi>\u0000 <mo>−</mo>\u0000 <mi>μ</mi>\u0000 <msup>\u0000 <mi>u</mi>\u0000 <mi>α</mi>\u0000 </msup>\u0000 <mo>,</mo>\u0000 </mrow>\u0000 </mtd>\u0000 </mtr>\u0000 <mtr>\u0000 <mtd>\u0000 <mrow>\u0000 <mi>τ</mi>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mi>Δ</mi>\u0000 <mi>v</mi>\u0000 <mo>−</mo>\u0000 <mi>v</mi>\u0000 <mo>+</mo>\u0000 <mi>u</mi>\u0000 </mrow>\u0000 </mtd>\u0000 </mtr>\u0000 </mtable>\u0000 </mfenced>\u0000 <annotation>$$begin{equation*} {leftlbrace defeqcellsep{&}begin{array}{l} u_t=Delta u-chi nabla cdot (u nabla v) + a u- mu u^{alpha }, [3pt] tau v_t=Delta v-v","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 4","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145407082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the formation of singularities in Bernoulli-type free boundary problems for steady axially symmetric inviscid incompressible flows with general vorticity and swirl. We focus on the asymptotic behavior near degenerate points and provide a complete characterization of singular wave profiles on the free boundary. Degenerate points are classified into two types: Type 1, which occur away from the axis of symmetry, and Type 2, which occur on the axis.
By developing a unified framework that combines variational methods for semilinear elliptic equations with novel analytical techniques for axially symmetric Euler systems, we show that the singular profiles at Type 1 degenerate points are fundamentally limited to three canonical forms: Stokes corners, cusps, or horizontal flat profiles. Our analysis employs monotonicity formulas to construct blow-up limits at degenerate points and utilizes concentration-compactness arguments. A key contribution is the derivation of a frequency formula that rigorously excludes the possibility of horizontal flat singularities. For Type 2 degenerate points, we prove that the singular profiles must be cusps.
{"title":"Singularities in Steady Axisymmetric Euler Flows With Swirl","authors":"Fan Zhang","doi":"10.1111/sapm.70130","DOIUrl":"https://doi.org/10.1111/sapm.70130","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper investigates the formation of singularities in Bernoulli-type free boundary problems for steady axially symmetric inviscid incompressible flows with general vorticity and swirl. We focus on the asymptotic behavior near degenerate points and provide a complete characterization of singular wave profiles on the free boundary. Degenerate points are classified into two types: Type 1, which occur away from the axis of symmetry, and Type 2, which occur on the axis.</p>\u0000 <p>By developing a unified framework that combines variational methods for semilinear elliptic equations with novel analytical techniques for axially symmetric Euler systems, we show that the singular profiles at Type 1 degenerate points are fundamentally limited to three canonical forms: Stokes corners, cusps, or horizontal flat profiles. Our analysis employs monotonicity formulas to construct blow-up limits at degenerate points and utilizes concentration-compactness arguments. A key contribution is the derivation of a frequency formula that rigorously excludes the possibility of horizontal flat singularities. For Type 2 degenerate points, we prove that the singular profiles must be cusps.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 4","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145407073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper introduces two new mathematical functions, the sine Krätzel function and the cosine Krätzel function, which extend the classical Krätzel function into the trigonometric domain. These functions are rigorously defined through integral representations and their fundamental properties, such as absolute convergence, generating functions, and derivative formulas, are investigated in detail. The integral transforms of these functions, including Mellin, Fourier, and Laplace transforms, are derived, highlighting their analytical flexibility. Furthermore, the study explores the applications of these functions in wave optics, specifically in the context of Fresnel and Fraunhofer diffraction patterns, demonstrating their utility in understanding light diffraction phenomena. Numerical examples and graphical visualizations are provided to illustrate the influence of key parameters on diffraction patterns, emphasizing the potential of these functions in applied mathematical and physical research.
{"title":"Trigonometric Krätzel Functions: Properties, Integral Transforms, and Applications in Diffraction Phenomena","authors":"Durmuş Albayrak","doi":"10.1111/sapm.70135","DOIUrl":"https://doi.org/10.1111/sapm.70135","url":null,"abstract":"<div>\u0000 \u0000 <p>This paper introduces two new mathematical functions, the sine Krätzel function and the cosine Krätzel function, which extend the classical Krätzel function into the trigonometric domain. These functions are rigorously defined through integral representations and their fundamental properties, such as absolute convergence, generating functions, and derivative formulas, are investigated in detail. The integral transforms of these functions, including Mellin, Fourier, and Laplace transforms, are derived, highlighting their analytical flexibility. Furthermore, the study explores the applications of these functions in wave optics, specifically in the context of Fresnel and Fraunhofer diffraction patterns, demonstrating their utility in understanding light diffraction phenomena. Numerical examples and graphical visualizations are provided to illustrate the influence of key parameters on diffraction patterns, emphasizing the potential of these functions in applied mathematical and physical research.</p></div>","PeriodicalId":51174,"journal":{"name":"Studies in Applied Mathematics","volume":"155 4","pages":""},"PeriodicalIF":2.3,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145406990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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