Mathematical Methods in the Applied Sciences最新文献

{"title":"Stability analysis and error estimation based on difference spectral approximation for Allen–Cahn equation in a circular domain","authors":"Zhenlan Pan, Jihui Zheng, Jing An","doi":"10.1002/mma.10481","DOIUrl":"https://doi.org/10.1002/mma.10481","url":null,"abstract":"For the first time, we propose an efficient difference spectral approximation for Allen–Cahn equation in a circular domain. Firstly, we introduce the polar coordinate transformation and derive the equivalent form of Allen–Cahn equation under this coordinate system, as well as the corresponding essential polar condition. Then, by using first‐order Euler and second‐order backward difference methods in the temporal direction, we deduce the first‐order and second‐order semi‐implicit schemes, based on which the first‐order and second‐order fully discrete schemes are established by employing Legendre‐Fourier spectral approximation in the spatial direction. In addition, the energy stability and error estimations for the two types of numerical schemes are theoretically proved. Finally, we provide some numerical examples, the results of which demonstrate the stability and convergence of the algorithm.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"19 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220215","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 new representation for the solution of the Richards-type fractional differential equation","authors":"Iz-iddine EL-Fassi, Juan J. Nieto, Masakazu Onitsuka","doi":"10.1002/mma.10394","DOIUrl":"10.1002/mma.10394","url":null,"abstract":"<p>Richards in [35] proposed a modification of the logistic model to model growth of biological populations. In this paper, we give a new representation (or characterization) of the solution to the Richards-type fractional differential equation \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mi>y</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>y</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <mo>·</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>+</mo>\u0000 <mi>a</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>y</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {mathcal{D}}^{alpha }y(t)=y(t)cdotp left(1+a(t){y}^{beta }(t)right) $$</annotation>\u0000 </semantics></math> for \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>≥</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$$ tge 0 $$</annotation>\u0000 </semantics></math>, where \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>:</mo>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>)</mo>\u0000 <mo>→</mo>\u0000 <mi>ℝ</mi>\u0000 </mrow>\u0000 <annotation>$$ a:left[0,infty right)to mathrm{mathbb{R}} $$</annotation>\u0000 </semantics></math> is a continuously differentiable function on \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>)</mo>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>0,1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ left[0,infty right),alpha in left(0,1right) $$</annotation>\u0000 </seman","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 2","pages":"1519-1529"},"PeriodicalIF":2.1,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258435","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":"Exponential stability of a class of quaternion‐valued memristor‐based neural network with time‐varying delay via M‐matrix","authors":"Shengye Wang, Yanchao Shi, Jun Guo","doi":"10.1002/mma.10486","DOIUrl":"https://doi.org/10.1002/mma.10486","url":null,"abstract":"This paper investigates the problems of exponential stability for a class of quaternion‐valued memristor‐based neural networks. By using M‐matrix theory and fixed point theorem, the existence and uniqueness of the equilibrium point of quaternion‐valued neural network are proved, respectively. Then, by combining M‐matrix with exponential stability theory, a non‐factorization method is obtained by using some inequality techniques to give the effective conditions of global exponential stability of quaternion‐valued memristor‐based neural network with time‐varying delay. Finally, numerical examples are given to demonstrate the validity of the derived results.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"57 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220238","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":"Bayesian inversion of a fractional elliptic system derived from seismic exploration","authors":"Yujiao Li","doi":"10.1002/mma.10474","DOIUrl":"https://doi.org/10.1002/mma.10474","url":null,"abstract":"In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"19 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220237","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":"Propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence","authors":"Juan He, Guo‐Bao Zhang","doi":"10.1002/mma.10466","DOIUrl":"https://doi.org/10.1002/mma.10466","url":null,"abstract":"In this paper, we are interested in propagation dynamics of a nonlocal dispersal Zika transmission model with general incidence. When the threshold is greater than one, we prove that there is a wave speed such that the model has a traveling wave solution with speed , and there is no traveling wave solution with speed less than . When the threshold is less than or equal to one, we show that there is no nontrivial traveling wave solution. The approaches we use here are Schauder's fixed point theorem with an explicit construction of a pair of upper and lower solutions, the contradictory approach, and the two‐sided Laplace transform.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"86 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220241","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":"Product integration techniques for fractional integro‐differential equations","authors":"Sunil Kumar, Poonam Yadav, Vineet Kumar Singh","doi":"10.1002/mma.10464","DOIUrl":"https://doi.org/10.1002/mma.10464","url":null,"abstract":"This article presents an application of approximate product integration (API) to find the numerical solution of fractional order Volterra integro‐differential equation based on Caputo non‐integer derivative of order , where . Also, the idea is extended to a class of fractional order Volterra integro‐differential equation with a weakly singular kernel. For this purpose, two numerical schemes are established by utilizing the concept of the API method, and L1 and L1‐2 formulae. We applied L1 and L1‐2 discretization to approximate the Caputo non‐integer derivative. At the same time, Taylor's series expansion of an unknown function is taken into consideration when approximating the Volterra part in the considered mathematical model using the API method. Combination of API method with L1 and L1‐2 formula provided the order of convergence and for Scheme‐I and Scheme‐II, respectively. The derived techniques reduced the proposed model to a set of algebraic equations that can be resolved using well‐known numerical algorithms. Furthermore, the unconditional stability, convergence, and numerical stability of the formulated schemes have been rigorously investigated. Finally, we conducted some numerical experiments to validate our theoretical findings and guarantee the accuracy and efficiency of the recommended schemes. The comparison between the numerical outcomes obtained by proposed schemes and existing numerical techniques has also been provided through tables and graphs.","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"93 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220239","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":"Solvability of a Hadamard fractional boundary value problem with multi-term integral and Hadamard fractional derivative boundary conditions","authors":"Tugba Senlik Cerdik","doi":"10.1002/mma.10475","DOIUrl":"10.1002/mma.10475","url":null,"abstract":"<p>In the present paper, we construct the existence of nontrivial solutions to a new kind of Hadamard fractional boundary value problem on an unbounded domain. With the contribution of some fixed point theorems in cone and the corresponding Green function, we ensure sufficient conditions for the Hadamard fractional boundary value problem. Also, the paper concludes with two examples to demonstrate our results.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"47 16","pages":"12946-12960"},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220240","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":"On explicit ninth-order, two-step methods addressing \u0000y″=f(x,y)","authors":"Houssem Jerbi,&nbsp;Sondess Ben Aoun,&nbsp;Obaid Alshammari,&nbsp;Theodore E. Simos,&nbsp;Ch. Tsitouras,&nbsp;Mourad Kchaou","doi":"10.1002/mma.10448","DOIUrl":"10.1002/mma.10448","url":null,"abstract":"<p>We present a new family of ninth-order hybrid explicit Numerov-type methods, effectively utilizing only eight stages, for solving the special second-order initial value problem. After applying a number of simplifying assumptions, we arrive to a reduced set of order conditions. Then, we derive an optimal method with constant coefficients that requires one less stage than standard methods found in the literature that use nine stages at this moment. Numerical tests are conducted using quadruple precision arithmetic on several well-known problems and the superiority of the new method is clear. Finally, in Section 6, a Mathematica package is presented that implements the corresponding algorithm.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 2","pages":"2517-2528"},"PeriodicalIF":2.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220242","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":"Data-driven dynamical analysis of an age-structured model: A graph-theoretic approach","authors":"Preeti Deolia,&nbsp;Anuraj Singh","doi":"10.1002/mma.10445","DOIUrl":"10.1002/mma.10445","url":null,"abstract":"&lt;p&gt;The dynamics of the propagation and outspread of infectious diseases are eminently intricate, mainly due to the heterogeneity of the host individuals. In this paper, an age-stratified SEIR (susceptible-exposed-infected-recovered) epidemiological model incorporating saturated treatment function and heterogeneous contact rates is developed to study infectious disease transmission dynamics among various age groups. The expression for the basic reproduction number \u0000&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {R}_0 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and conditions for the global stability of the system have been derived by a recently developed graph-theoretic (GT) approach. Digraph reduction creates a GT version of the Gauss elimination method for computing the \u0000&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {R}_0 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The global dynamics results have been formed by constructing the Lyapunov function using a GT approach. The endemic equilibrium exists uniquely if \u0000&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;&gt;&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {R}_0&amp;amp;gt;1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, whereas the disease-free equilibrium is observed to be globally stable if \u0000&lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;≤&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$$ {R}_0le 1 $$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. The numerical simulations are demonstrated by extracting the daily reported COVID-19 cases for the second wave in Italy. The age-dependent contact matrix for the Republic of Italy (data sourced from the POLYMOD study) is computed via paper–diary methodology (PDM) grounded on a population-prospective survey in European ","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"48 2","pages":"2446-2473"},"PeriodicalIF":2.1,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220090","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":"Variations of heat equation on the half-line via the Fokas method","authors":"Andreas Chatziafratis,&nbsp;Athanasios S. Fokas,&nbsp;Elias C. Aifantis","doi":"10.1002/mma.10303","DOIUrl":"10.1002/mma.10303","url":null,"abstract":"<p>In this review paper, we discuss some of our recent results concerning the rigorous analysis of initial boundary value problems (IBVPs) and newly discovered effects for certain evolution partial differential equations (PDEs). These equations arise in the applied sciences as models of phenomena and processes pertaining, for example, to continuum mechanics, heat-mass transfer, solid–fluid dynamics, electron physics and radiation, chemical and petroleum engineering, and nanotechnology. The mathematical problems we address include certain well-known classical variations of the traditional heat (diffusion) equation, including (i) the Sobolev–Barenblatt pseudoparabolic PDE (or modified heat or second-order fluid equation), (ii) a fourth-order heat equation and the associated Cahn–Hilliard (or Kuramoto–Sivashinsky) model, and (iii) the Rubinshtein–Aifantis double-diffusion system. Our work is based on the synergy of (i) the celebrated Fokas unified transform method (UTM) and (ii) a new approach to the rigorous analysis of this method recently introduced by one of the authors. In recent works, we considered forced versions of the aforementioned PDEs posed in a spatiotemporal quarter-plane with arbitrary, fully non-homogeneous initial and boundary data, and we derived formally effective solution representations, for the first time in the history of the models, justifying a posteriori their validity. This included the reconstruction of the prescribed initial and boundary conditions, which required careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. In each IBVP, the novel formula was utilized to rigorously deduce the solution's regularity properties near the boundaries of the spatiotemporal domain. Importantly, this analysis is indispensable for proving (non)uniqueness of solution. These works extend previous investigations. The usefulness of our closed-form solutions will be demonstrated by studying their long-time asymptotics. Specifically, we will briefly review some asymptotic results about Barenblatt's equation.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"47 16","pages":"12541-12566"},"PeriodicalIF":2.1,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220244","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}