{"title":"Special Affine Fourier Transform on Tempered Distribution and Its Application","authors":"Manish Kumar, Bhawna","doi":"10.1002/mma.10657","DOIUrl":"https://doi.org/10.1002/mma.10657","url":null,"abstract":"<div>\u0000 \u0000 <p>The main aim of this work is to develop a theoretical framework for generalized pseudo-differential operators involving the special affine Fourier transform (SAFT), associated with a symbol \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>δ</mi>\u0000 <mo>(</mo>\u0000 <mi>μ</mi>\u0000 <mo>,</mo>\u0000 <mi>η</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ delta left(mu, eta right) $$</annotation>\u0000 </semantics></math>. Some important properties of the SAFT are established, and it is proved that the product of two generalized pseudo-differential operators is shown to be a generalized pseudo-differential operator. Further, we explore the practical applications of the SAFT in solving generalized partial differential equations, such as the generalized telegraph and wave equations, providing closed-form solutions. Furthermore, graphical visualizations for these solutions are illustrated via MATLAB R2023b.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 5","pages":"6092-6102"},"PeriodicalIF":2.1,"publicationDate":"2025-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143595523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Radial Basis Function-Hermite Finite Difference Method for the Two-Dimensional Distributed-Order Time-Fractional Cable Equation","authors":"Majid Haghi, Mohammad Ilati","doi":"10.1002/mma.10696","DOIUrl":"https://doi.org/10.1002/mma.10696","url":null,"abstract":"<div>\u0000 \u0000 <p>In this article, our main objective is to propose a high-order local meshless method for numerical solution of two-dimensional distributed-order time-fractional cable equation on both regular and irregular domains. First, the distribution-order integral is approximated by the Gauss-Legendre quadrature formula, and then a second-order weighted and shifted Grünwald difference (WSGD) scheme is applied to approximate the time Riemann-Liouville derivatives. The stability and convergence analysis of the time-discrete outline are investigated by the energy approach. The spatial dimension of the model is discretized by the fourth-order local radial basis function-Hermite finite difference (RBF-HFD) method. Some numerical experiments are performed on regular and irregular computational domains to verify the ability, efficiency, and accuracy of the proposed numerical procedure. The numerical simulations clearly demonstrate the high accuracy of the provided numerical process in comparison to existing procedures. Finally, it can be concluded that the presented technique is a suitable alternative to the existing numerical techniques for the distributed-order time-fractional cable equation.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6573-6585"},"PeriodicalIF":2.1,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Effect of Nonlinearities With Arbitrary-Order Derivatives on Dynamic Transitions","authors":"Taylan Şengül, Burhan Tiryakioglu","doi":"10.1002/mma.10709","DOIUrl":"https://doi.org/10.1002/mma.10709","url":null,"abstract":"<div>\u0000 \u0000 <p>The primary objective of this paper is to classify the first transitions of a general class of one spatial dimensional nonlinear partial differential equations on a bounded interval. The linear part of the equation is assumed to have a real discrete spectrum with a complete set of eigenfunctions, which are of the form \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>sin</mi>\u0000 <mi>k</mi>\u0000 <mi>x</mi>\u0000 </mrow>\u0000 <annotation>$$ sin kx $$</annotation>\u0000 </semantics></math> or \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>cos</mi>\u0000 <mi>k</mi>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>k</mi>\u0000 <mo>∈</mo>\u0000 <mi>ℕ</mi>\u0000 </mrow>\u0000 <annotation>$$ cos kx,kern0.3em kin mathbb{N} $$</annotation>\u0000 </semantics></math>. The nonlinear operator consists of arbitrary finite products and sums of the unknown function and its derivatives of arbitrary order. The equations allow for a trivial steady-state solution that becomes unstable when a control parameter exceeds a certain threshold. Unlike most of the previous research in this direction that considers specific equations, this general framework is suitable for extension in several directions such as the higher spatial dimensions and different basis vectors. Under a set of assumptions that are often valid in many interesting applications, we derive two numbers called the transition number and the critical index which completely describe the first dynamic transition. We make detailed numerical computations that reveal the properties of the transition numbers. To show the applicability of our theoretical results, we determine the first transitions of several well-known equations including the Cahn–Hilliard, thin film, Harry Dym, Kawamoto, and Rosenau–Hyman equations.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 6","pages":"6704-6716"},"PeriodicalIF":2.1,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143622557","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}