Results in Applied Mathematics最新文献

{"title":"Pseudo-time finite differences for two-dimensional nonlocal integro-HJB equations","authors":"Zhuang Zou,&nbsp;Sufang Han","doi":"10.1016/j.rinam.2026.100743","DOIUrl":"10.1016/j.rinam.2026.100743","url":null,"abstract":"<div><div>We develop a pseudo-time finite-difference framework for computing stationary value functions and feedback controls of two-dimensional nonlinear nonlocal integro-Hamilton–Jacobi–Bellman (integro-HJB) equations arising in jump–diffusion stochastic control. The scheme combines monotone upwinding for drift, central differencing for diffusion, and a consistent quadrature discretization of the nonlocal jump operator. Feedback controls are recovered from value gradients using boundary-aware smoothing and control clipping, yielding stable banded control fields on truncated domains. Numerical reliability is documented by pseudo-time residual stabilization and grid-refinement sensitivity, with discrepancies largely confined to thin boundary saturation layers. As a fully reproducible testbed, we apply the method to a dual-ratio state <span><math><mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow></math></span> constructed from provincial GDP and population series reported in the <em>China Statistical Yearbook</em> (1998–2024) under a refined Qinling–Huaihe split. The resulting policy partitions the state space into negative, neutral, and positive intervention bands and supports distributional first-passage timing diagnostics. The implementation is readily transferable to other controlled jump–diffusion systems governed by nonlocal HJB equations.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100743"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148636098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A first-moment improvement for unshifted rank-one lattice rules in unanchored Sobolev spaces","authors":"Yaoran Yang,&nbsp;Yutong Zhang","doi":"10.1016/j.rinam.2026.100750","DOIUrl":"10.1016/j.rinam.2026.100750","url":null,"abstract":"&lt;div&gt;&lt;div&gt;We study deterministic unshifted rank-one lattice rules in weighted unanchored Sobolev spaces of smoothness one. Let &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; denote the average, over all non-zero generating vectors, of the squared worst-case error of an &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-point rule. For odd prime &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, the averaged Kazashi–Sloan–Goda inequality reduces the analysis to integer moments of an arithmetic quantity &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. The finite-field computation giving the first moment of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is implicit in earlier work; the new step here is to retain that first moment in exact &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; form and combine it with the uniform cap &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;κ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;π&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. This gives an &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;-&lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℓ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; conversion controlling all integer moments required by the averaged inequality. Consequently, for every odd prime &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and finite dimension &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mover&gt;&lt;mrow&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;¯&lt;/mo&gt;&lt;/mover&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; is bounded by &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mo&gt;log&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; times an explicit weighted subset sum. For product weights &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∏&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;∞&lt;/mi&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, this subset sum is bounded independently of &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Hence there exists an unshifted rank-one lattice rule with worst-case error &lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;m","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100750"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148776286","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A ℙ1 conforming virtual element method for elliptic problems with singular loads using meshfree function","authors":"Jiayu Fang,&nbsp;Mohammad Z. Al-Hamdan,&nbsp;Yaoxin Zhang,&nbsp;Xiaobo Chao","doi":"10.1016/j.rinam.2026.100742","DOIUrl":"10.1016/j.rinam.2026.100742","url":null,"abstract":"<div><div>This paper presents a novel approach for handling the singular loads, i.e., <em>H</em>⁻¹(Ω) loads, in the <em>ℙ</em>₁ conforming virtual element method (VEM) using a meshfree function. While VEM can accurately and efficiently solve elliptic problems with loads in <em>L</em>²(Ω), its implicit construction of shape functions prevents it from addressing singular loads. Although this issue can be mitigated by computing VEM shape functions explicitly through companion operators, doing so requires decomposing polygonal meshes into multiple simplices, which is challenging, particularly for concave meshes and higher dimensional cases. On the other hand, the principle of VEM decomposition can be applied to meshfree functions, where the polynomial part contributes to the consistency term, and the non-polynomial part is approximated by the stability term. Consequently, we propose using the meshfree function as the VEM shape function, enabling the duality pairing with <em>H</em>⁻¹(Ω) loads to be computed easily, while the calculation of the local VEM stiffness matrix remains unchanged. The partition of unity method, coupled with the Gaussian radial basis function (RBF), is used to construct the meshfree shape function, which is then tested on two cases. Good agreement between numerical and analytical solutions is observed in all cases, indicating the accuracy of the approach. Moreover, the model accuracy is found to be insensitive to the shape parameter of the Gaussian RBF, demonstrating its robustness.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100742"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148636095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical analysis and exact solutions of a coupled Maccari system: Dynamical insights into chaos, multistability, and nonlinear wave evolution","authors":"Muhammad Iqbal,&nbsp;Muhammad Aziz ur Rehman,&nbsp;Muhammad Bilal Riaz,&nbsp;Zaheer Hussain Shah","doi":"10.1016/j.rinam.2026.100740","DOIUrl":"10.1016/j.rinam.2026.100740","url":null,"abstract":"<div><div>The nonlinear complex coupled Maccari system (CCMS) is a significant model in optics, quantum mechanics, plasma physics, hydrodynamics and some other fields. This paper focuses the analytical and dynamical properties of a nonlinear CCMS. For analytical insight, check the nonlinear behavior and closed-form solutions are obtained using the modified Khater method. These solutions applied in the explanation of the stationary and propagating wave motions. The deeper dynamical analysis is performed with the time-series trajectories, Poincaré sections, bifurcation diagrams, and Lyapunov exponents to classify the dynamic complex states of the system. The system exhibits transitions from periodic to quasi-periodic and chaotic responses with parameter variation, along with multistability and coexisting attractors due to softening and hardening nonlinearities. In addition, advanced diagnostic tools such as power spectrum analysis, fractal dimension, return maps, and recurrence plots are employed to rigorously characterize the system behavior. These techniques confirm the emergence of chaos and provide deeper insight into the geometric and spectral properties of the underlying attractors. The graphical representations clearly illustrate the transitions between regular and chaotic dynamics, as well as regions of multistability. Overall, this work establishes a unified analytical–computational framework for the investigation of higher-dimensional nonlinear evolution equations. The findings offer significant theoretical contributions and hold potential applications in fluid dynamics, nonlinear optics, and plasma physics.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100740"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148636096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability analysis of a stochastic logistic growth model driven by multiplicative α-stable Lévy noise","authors":"Bikram Bhusal,&nbsp;Hongwei Long","doi":"10.1016/j.rinam.2026.100739","DOIUrl":"10.1016/j.rinam.2026.100739","url":null,"abstract":"<div><div>We present stochastic processes defined as solution to one dimensional stochastic logistic differential equations driven by Brownian motion and <span><math><mi>α</mi></math></span>-stable Lévy motion that are appropriate to study the behavior of non-Gaussian processes having infinite variance. We establish the existence of a unique positive global solution of the model and find a sufficient condition for the almost sure exponential stability of the trivial solution of the model. We also present the explicit formula of the solution. Finally, numerical simulations are presented to illustrate the theoretical findings.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100739"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148636102","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A scalable stochastic physics-informed neural ODE framework for learning hidden constitutive equations in multivariate SDEs with multiplicative noise","authors":"Ruaa Qahtan Mohammed,&nbsp;Zahraa Ammory Ali,&nbsp;Huda Emad Aldeen Jameel,&nbsp;Mahdi Ghafourivayghan,&nbsp;Nouf Abd Elmunim,&nbsp;Peiman Parand,&nbsp;Muhammad Akmal Chaudhary,&nbsp;Mohammad Alibakhshikenari,&nbsp;Taha A. Elwi,&nbsp;Takfarinas Saber","doi":"10.1016/j.rinam.2026.100758","DOIUrl":"10.1016/j.rinam.2026.100758","url":null,"abstract":"<div><div>We propose a scalable stochastic physics-informed neural ordinary differential equation (SPINODE) framework for systematically unraveling the hidden physics of multivariate stochastic differential equations (SDEs) driven by <em>multiplicative noise</em>. The key methodological idea is to propagate stochasticity through the known structural physics of the SDE so as to obtain a coupled system of deterministic ordinary differential equations (ODEs) that describe the time evolution of statistical <em>moment trajectories</em> of the latent stochastic state. By matching predicted moments to data-estimated moments through differentiable ODE solvers and adjoint sensitivity analysis, we recast a difficult stochastic learning problem as a deterministic trajectory-matching problem. To delineate the present contribution: the moment-propagation principle, the adjoint-based training loop and the Lotka–Volterra–Eyring (LVE) benchmark are <em>adopted</em> from the original SPINODE framework of O’Leary et al.; what is <em>new here</em> is (a) a self-contained nonlinear-analysis theory for the closed moment ODE, which the original work did not provide, and (b) a systematic study of two families of algorithmic variants that generalise the single fixed configuration used previously. On the theoretical side we give a uniqueness theorem (Picard–Banach in Bielecki norm) for the moment ODE, a moment-closure consistency lemma quantifying truncation error, a quantitative adjoint-gradient error bound, and two existence theorems for the optimal neural-network parameter set, one via the Banach contraction principle and one via the Leray–Schauder alternative. On a multivariate LVE benchmark with multiplicative noise, we report numerical experiments over 10 random seeds and across 7 adaptive uncertainty propagation (AUP) variants and 7 adaptive informative candidate selection (AICS) variants. The vanilla SPINODE baseline attains a mean final RMSE of 0.0602 ± 0.0033; an unscented-transform 4-moment variant (<span>UT4M_only</span>) improves the RMSE to 0.0576 ± 0.0063 at <span><math><mrow><mo>≈</mo><mn>2.4</mn><mo>×</mo></mrow></math></span> lower wall-clock than the default high-order propagator, demonstrating favourable scaling and <em>numerical robustness</em> of the proposed moment-matching estimator.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100758"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148934333","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Kronecker-structured completely positive biquadratic tensors: Certified construction, spectral characterization, and a reproducible benchmark framework","authors":"Zhengbo Li,&nbsp;Qiong Liu,&nbsp;Dingtao Peng,&nbsp;Cuiyu Yang,&nbsp;Shuang Hu","doi":"10.1016/j.rinam.2026.100759","DOIUrl":"10.1016/j.rinam.2026.100759","url":null,"abstract":"<div><div>Completely positive biquadratic (CPB) tensors arise naturally in tensor optimization, polynomial positivity, and structured nonconvex modeling, but reproducible benchmark instances with exactly certifiable reference spectral quantities remain limited. In this paper, we study a structured subclass of fourth-order biquadratic tensors generated by Kronecker products of symmetric positive definite completely positive matrices. For tensors of the form <span><math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mi>j</mi><mi>k</mi><mi>l</mi></mrow></msub><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi><mi>k</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi><mi>l</mi></mrow></msub><mo>,</mo></mrow></math></span> we prove that the associated biquadratic form factors as <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>A</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>⊤</mo></mrow></msup><mi>B</mi><mi>x</mi><mo>)</mo></mrow><mrow><mo>(</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>⊤</mo></mrow></msup><mi>C</mi><mi>y</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> which yields exact expressions for the minimal and maximal M-eigenvalues, the spectral condition number of the associated matricized Gram representation, and an explicit sum-of-squares decomposition bound. Based on these identities, we construct a reproducible benchmark suite from retained structured factor families, including archived Cauchy-type benchmark matrices together with Pascal and seeded random completely positive factors equipped with stored construction data. Applying admissibility and nonredundancy rules produces a canonical set of 28 benchmark tensors in dimensions <span><math><mrow><mn>3</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>5</mn></mrow></math></span>. For each retained instance, the reference quantities are determined exactly from the stored factor matrices through the Kronecker spectral identities. The resulting framework provides certified small-scale test instances for validating algorithms for CPB tensor optimization and related polynomial problems. MATLAB codes and retained factor data are available from the authors upon reasonable request.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100759"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148934991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Stability and dynamical analysis of the Davey–Stewartson model","authors":"Muhammad Moneeb Tariq,&nbsp;Muhammad Aziz ur Rehman,&nbsp;Muhammad Bilal Riaz","doi":"10.1016/j.rinam.2026.100738","DOIUrl":"10.1016/j.rinam.2026.100738","url":null,"abstract":"<div><div>This study investigates the nonlinear coupled Davey–Stewartson Fokas system, a mathematical model describing pulse propagation in monomode optical fibers. Employing a powerful analytical technique, we derive novel periodic and solitary wave solutions, including bright, dark, periodic and kink soliton. These solutions significantly advance our understanding of the precise dynamics of solitons in nonlinear optics. The proposed methods effectively discover and characterize new solitary wave solutions, providing deeper insights into the model’s behavior. These computational techniques hold enormous promise for various scientific applications and enhance our knowledge of fundamental dynamics. This paper visually represents soliton solutions in the form of 2D, 3D, and contour plots. By applying the Galilean transformation and using appropriate parameter values, we obtain the planar dynamical system, which aids in the dynamical analysis of the Davey–Stewartson Fokas system. Using the planer dynamical system to examine the phase-portrait analysis of the system. Also examine the sensitivity analysis of the system’s reaction to various initial conditions during both the initial and intransitive phases. Furthermore, we investigate the chaotic behavior by adding an external force, examining whether the planar dynamical system exhibits periodic, quasi-periodic and chaotic patterns.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100738"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148636101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Crank–Nicolson–Galerkin FEM for Darcy–Forchheimer problem coupled with time-dependent convection–diffusion–reaction equation","authors":"P. Njeutcha Methouchelah,&nbsp;G. Chendjou,&nbsp;M.S. Daoussa Haggar,&nbsp;M. Mbehou","doi":"10.1016/j.rinam.2026.100747","DOIUrl":"10.1016/j.rinam.2026.100747","url":null,"abstract":"<div><div>This work addresses a time-dependent convection–diffusion–reaction equation coupled with the Darcy–Forchheimer system. By combining the Crank–Nicolson scheme in time with a finite element discretization in space, we propose a linearized, fully discrete scheme that is unconditionally stable. We first establish the well-posedness of the continuous system, followed by the analysis of the semi-discrete solution. Optimal error estimates for the velocity field, pressure, and concentration are derived for both the semi-discrete and fully discrete schemes. Finally, numerical simulations are presented to validate the theoretical results and demonstrate the efficiency of the proposed method.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100747"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148657662","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A novel adaptive FVE reduction dimension algorithm based on POD for parabolic problem","authors":"Yuejie Li,&nbsp;Zhendong Luo","doi":"10.1016/j.rinam.2026.100748","DOIUrl":"10.1016/j.rinam.2026.100748","url":null,"abstract":"<div><div>This article mostly builds a novel model dimension reduction algorithm for the two dimensional (2D) parabolic problem based on the finite volume element (FVE) and proper orthogonal decomposition (POD) methods. To accomplish this aim, a new FVE Crank–Nicolson (CN) (FVECN) algorithm will first be constructed through utilizing the CN difference technique to discretize the temporal differential quotient and using the FVE technology to approximate the spacial variables, and the error estimates for the FVECN solutions will also be attested. Whereafter, the most important thing is to utilize the POD technology to lower the dimensionality of the vectors constituted with the unknown solution coefficients for the FVECN algorithm, thereby establishing a new FVECN model dimension reduction (FVECNMDR) algorithm, and use the matrix analysis to testify the existence, stability, as well as error estimates of the FVECNMDR solutions, as well as make use of a number of numerical simulations to confirm the correctness of the theoretical conclusions together with the effectiveness to the FVECNMDR algorithm. The results show that the FVECNMDR algorithm can not only greatly cut down the unknowns in the FVECN algorithm, but can also highly mitigate the computing load, lessen the accumulation of calculation error, enhance the computation efficiency. The FVECNMDR algorithm herein is completely different from the dimension reduction of the trial function spaces in the FVE methods and is a novel algorithm.</div></div>","PeriodicalId":36918,"journal":{"name":"Results in Applied Mathematics","volume":"31 ","pages":"Article 100748"},"PeriodicalIF":1.5,"publicationDate":"2026-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148736259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}