Abstract This study retrieves some novel exact solutions to the family of 3D space–time fractional Wazwaz–Benjamin–Bona–Mahony (WBBM) equations in the context of diverse nonlinear physical phenomena ...resulting from water wave mechanics. The family of WBBM equations is transformed for this purpose using a space and time fractional transformation into an ordinary differential equation (ODE). The ODE then uses a strong method, namely the Unified Method. Consequently, lump solutions, dark-bright soliton, singular and multiple soliton solutions, and periodic solutions are investigated. The disparities between the current study's conclusions and previously acquired solutions via other approaches are examined. All wave solutions produced are determined to be novel in terms of fractionality, unrestricted parameters, and implemented technique sense. The impact of unrestricted parameters and fractionality on the obtained solutions are visually presented, along with physical explanations. It is observed that the wave portents are varied with the increase of unrestricted parameters as well as fractionality. We dynamically showed that the appropriate transformation and the applied Unified approach more proficient in the study of water wave dynamics and might be used in future researches to clarify the many physical phenomena. The novelty of this work validate that the proposed method seem simple and useful tools for obtaining the solutions in PDEs and it is expected to use in mathematical physics and optical engineering.
This paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations-hyperbolic equations with boundary, trace, and integral nonlocal terms. Using ...the backstepping approach, the controller is designed by formulating an integral operator, whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation. By applying the method of successive approximations, the kernel's well-posedness is given. We prove the exponential stability of the origin of the system in a suitable Hilbert space. Moreover, a wave system with nonlocal terms is stabilized by applying the above result.
Significant advancements have been made in hyperspectral image (HSI) super-resolution with the development of deep-learning techniques. However, the current application of deep neural network ...architectures to HSI super-resolution heavily relies on empirical design strategies, which can potentially impede the improvement of image reconstruction performance and introduce distortions in the results. To address this, we propose an innovative HSI super-resolution network called dual ordinary differential equations (Dual ODEs). Drawing inspiration from ordinary differential equations (ODEs), our approach offers reliable guidelines for the design of HSI super-resolution networks. The Dual ODE model leverages a spatial ODE block to extract spatial information and a spectral ODE block to capture internal spectral features. This is accomplished by redefining the conventional residual module using the multiple ODE functions method. To evaluate the performance of our model, we conducted extensive experiments on four benchmark HSI datasets. The results conclusively demonstrate the superiority of our Dual ODE approach over state-of-the-art models. Moreover, our approach incorporates a small number of parameters while maintaining an interpretable model design, thereby reducing model complexity.
Fueled by breakthrough technology developments, the biological, biomedical, and behavioral sciences are now collecting more data than ever before. There is a critical need for time- and ...cost-efficient strategies to analyze and interpret these data to advance human health. The recent rise of machine learning as a powerful technique to integrate multimodality, multifidelity data, and reveal correlations between intertwined phenomena presents a special opportunity in this regard. However, machine learning alone ignores the fundamental laws of physics and can result in ill-posed problems or non-physical solutions. Multiscale modeling is a successful strategy to integrate multiscale, multiphysics data and uncover mechanisms that explain the emergence of function. However, multiscale modeling alone often fails to efficiently combine large datasets from different sources and different levels of resolution. Here we demonstrate that machine learning and multiscale modeling can naturally complement each other to create robust predictive models that integrate the underlying physics to manage ill-posed problems and explore massive design spaces. We review the current literature, highlight applications and opportunities, address open questions, and discuss potential challenges and limitations in four overarching topical areas: ordinary differential equations, partial differential equations, data-driven approaches, and theory-driven approaches. Towards these goals, we leverage expertise in applied mathematics, computer science, computational biology, biophysics, biomechanics, engineering mechanics, experimentation, and medicine. Our multidisciplinary perspective suggests that integrating machine learning and multiscale modeling can provide new insights into disease mechanisms, help identify new targets and treatment strategies, and inform decision making for the benefit of human health.
A small-gain approach is presented for analyzing exponential stability of a class of (dynamical) hybrid systems. The systems considered in the paper are composed of finite-dimensional dynamics, ...represented by a linear ordinary differential equation (ODE), and infinite-dimensional dynamics described by a parabolic partial differential equation (PDE). Exponential stability is established under conditions involving the maximum allowable sampling period (MASP). This new stability result is shown to be useful in the design of sampled-output exponentially convergent observers for linear systems that are described by an ODE-PDE cascade. The new stability result also proves to be useful in designing practical approximate observers involving no PDEs.
In these lectures I present a review of non‐perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. I first consider the structure of these ...effects in the case of ordinary differential equations, which provide a model for more complicated theories, and I introduce in a pedagogical way some technology from resurgent analysis, like trans‐series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, I address general aspects of large N instantons, and then present a detailed review of non‐perturbative effects in matrix models. Finally, I consider two applications of these techniques in string theory.
These lectures review non‐perturbative instanton effects in quantum theories, with a focus on large N gauge theories and matrix models. First the structure of these effects in the case of ordinary differential equations is considered providing a model for more complicated theories and leading to a pedagogical approach to some technology from resurgent analysis like trans‐series and the resurgent version of the Stokes phenomenon. After reviewing instanton effects in quantum mechanics and quantum field theory, general aspects of large N instantons are addressed. A detailed review of non‐perturbative effects in matrix models including two applications of these techniques in string theory is presented.
In this paper, we survey selected software packages for the numerical solution of boundary value ODEs (BVODEs), time-dependent PDEs in one spatial dimension (1DPDEs), and initial value ODEs (IVODEs).
...A unifying theme of this paper is our focus on software packages for these problem classes that compute error-controlled, continuous numerical solutions.
A continuous numerical solution can be accessed by the user at any point in the domain. We focus on error-control software; this means that the software adapts the computation until it obtains a
continuous approximate solution with a corresponding error estimate
that satisfies the user tolerance. The second section of the paper will provide an overview of recent work on the development of COLNEWSC, an updated version of the widely used collocation BVODE solver, COLNEW, that returns an error-controlled continuous approximate solution based on the use of a superconvergent interpolant to the underlying collocation solution. The third section of the paper gives a brief review of recent work on the development of a new 1DPDE solver, BACOLIKR, that provides time- and space-dependent event detection for an error-controlled continuous numerical solution. In the fourth section of the paper, we briefly review the state of the art in IVODE software for the computation of error-controlled continuous numerical solutions.
Sequential recommendation aims at understanding user preference by capturing successive behavior correlations, which are usually represented as the item purchasing sequences based on their past ...interactions. Existing efforts generally predict the next item via modeling the sequential patterns. Despite effectiveness, there exist two natural deficiencies: (i) user preference is dynamic in nature, and the evolution of collaborative signals is often ignored; and (ii) the observed interactions are often irregularly-sampled, while existing methods model item transitions assuming uniform intervals. Thus, how to effectively model and predict the underlying dynamics for user preference becomes a critical research problem. To tackle the above challenges, in this paper, we focus on continuous-time sequential recommendation and propose a principled graph ordinary differential equation framework named GDERec. Technically, GDERec is characterized by an autoregressive graph ordinary differential equation consisting of two components, which are parameterized by two tailored graph neural networks (GNNs) respectively to capture user preference from the perspective of hybrid dynamical systems. On the one hand, we introduce a novel ordinary differential equation based GNN to implicitly model the temporal evolution of the user-item interaction graph. On the other hand, an attention-based GNN is proposed to explicitly incorporate collaborative attention to interaction signals when the interaction graph evolves over time. The two customized GNNs are trained alternately in an autoregressive manner to track the evolution of the underlying system from irregular observations, and thus learn effective representations of users and items beneficial to the sequential recommendation. Extensive experiments on five benchmark datasets demonstrate the superiority of our model over various state-of-the-art recommendation methods.
The Riccati equation method is used to establish oscillation and non-oscillation criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the ...obtained oscillation criterion is a generalization of J.S.W. Wong's oscillation criterion.
Abstract
While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. ...In our work, based on the homotopy perturbation method, we propose a quantum algorithm for solving
n
-dimensional nonlinear dissipative ordinary differential equations (ODEs). Our algorithm first converts the original nonlinear ODEs into the other nonlinear ODEs which can be embedded into finite-dimensional linear ODEs. Then we solve the embedded linear ODEs with quantum linear ODEs algorithm and obtain a state
ϵ
-close to the normalized exact solution of the original nonlinear ODEs with success probability Ω(1). The complexity of our algorithm is
O
(
gηT
poly(log(
nT
/
ϵ
))), where
η
,
g
measure the decay of the solution. Our algorithm provides exponential improvement over the best classical algorithms or previous quantum algorithms in
n
or
ϵ
.
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