Article
Computer Science, Interdisciplinary Applications
Ali C. Bekar, Erdogan Madenci
Summary: This study introduces an approach to identify significant terms in PDEs based on measured data using linear regression model, PDDO, and sparse linear regression learning algorithm. The solution is achieved through Douglas-Rachford algorithm with regularization, demonstrating effectiveness in handling challenging nonlinear PDEs.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Davide Palitta
Summary: This study presents a novel solution strategy for addressing the discrete operator issues arising from the time-space discretization of evolutionary partial differential equations, efficiently solving problems with a large number of degrees of freedom while maintaining low storage demand.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
S. S. Santra, S. Priyadharshini, V. Sadhasivam, J. Kavitha, U. Fernandez-Gamiz, S. Noeiaghdam, K. M. Khedher
Summary: This article investigates the oscillatory behavior of solutions to conformable elliptic partial differential equations of the Emden-Fowler type. By employing the Riccati method, new necessary conditions for the oscillation of all solutions are established. These findings build upon previous results for integer order equations and are further illustrated through an example.
Article
Mathematics
Antoine Hocquet
Summary: The study explores the Cauchy problem for a quasilinear equation with transport rough input, providing sufficient conditions for existence in any dimension and uniqueness in the case of divergence-free X. It focuses on one-dimensional scenario with slightly more regular coefficients, proving existence of a class of solutions satisfying a specific condition, and uniqueness by obtaining an L-infinity(L-1) estimate on the difference of two solutions. This is achieved by establishing a link with a backward dual equation and a (rough) iteration lemma akin to Moser.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Mathematics, Applied
Ahmad Qazza, Rania Saadeh, Emad Salah
Summary: In this paper, we introduce a new technique, called the direct power series method, to solve various types of time-fractional partial differential equations and systems using the Caputo derivative. We present a simple algorithm to demonstrate the versatility of the method in solving different time-fractional partial problems. We also introduce a new theorem to explain the necessary substitutions in the proposed method and provide convergence analysis conditions. Furthermore, we discuss illustrative examples of time-fractional partial differential equations and systems to showcase the practicality and simplicity of the new approach.
Article
Computer Science, Artificial Intelligence
Somdatta Goswami, Katiana Kontolati, Michael D. D. Shields, George Em Karniadakis
Summary: This paper presents a transfer learning framework for task-specific learning based on Deep Operator Network (DeepONet), addressing conditional shifts in transfer learning scenarios such as nonlinear partial differential equations.
NATURE MACHINE INTELLIGENCE
(2022)
Article
Computer Science, Interdisciplinary Applications
Daniel A. Messenger, David M. Bortz
Summary: Sparse Identification of Nonlinear Dynamics (SINDy) has been extended to the weak formulation for partial differential equations (PDEs) to improve accuracy in model coefficient recovery and robustness in identifying PDEs in large noise regime. The implementation utilizes Fast Fourier Transform for efficient model identification and reveals a connection between noise robustness and spectra of test functions. Additionally, a learning algorithm for threshold in sequential thresholding least-squares (STLS) is introduced for model identification from large libraries.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Review
Mathematics, Applied
Duy Phan, Alexander Ostermann
Summary: This paper considers two types of second-order in time partial differential equations, namely semilinear wave equations and semilinear beam equations. To solve these equations with exponential integrators, an approach to efficiently compute the action of the matrix exponential and related matrix functions is presented. Various numerical simulations are provided to illustrate the effectiveness of this approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics
M. C. Carbinatto, K. P. Rybakowski
Summary: In this paper, a Conley index framework is introduced for partial differential equations with delay terms. The existence of special nonconstant full solutions for parametrized classes of such equations, which are close to PDEs without delays, is then proven. The proof utilizes a tubular Conley index continuation principle established in a previous work by Carbinatto and Rybakowski (2013) [1].
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Diego Catalano Ferraioli, Tarcisio Castro Silva
Summary: This paper investigates third order equations describing spherical surfaces and pseudospherical surfaces. By classifying and providing examples of these equations, the problem of determining sequences of conservation laws in both cases is discussed.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2024)
Article
Mathematics
Mawia Osman, Yonghui Xia, Omer Abdalrhman Omer, Ahmed Hamoud
Summary: In this article, the fuzzy Adomian decomposition method and fuzzy modified Laplace decomposition method are presented to solve the fuzzy fractional Navier-Stokes equations. The fuzzy Elzaki transform and fuzzy Elzaki decomposition method are also investigated for solving fuzzy linear-nonlinear Schrodinger differential equations. Comparisons are made with other methods, and the proposed methods are found to be simpler and consistent with analytical and numerical results.
Article
Computer Science, Interdisciplinary Applications
Alec J. Linot, Joshua W. Burby, Qi Tang, Prasanna Balaprakash, Michael D. Graham, Romit Maulik
Summary: Careful consideration is needed in capturing the dynamics of high wavenumbers in data-driven modeling of spatiotemporal phenomena, especially when shocks or chaotic dynamics are present. To address this challenge, a new architecture called stabilized neural ordinary differential equation (ODE) is proposed, which accurately captures shocks and chaotic dynamics. By combining the outputs of two neural networks, one learning the linear term and the other the nonlinear term, the proposed architecture learns the right-hand-side (RHS) of the ODE. Experimental results on the viscous Burgers equation and the Kuramoto-Sivashinsky equation demonstrate that stabilized neural ODEs outperform standard neural ODEs in short-time tracking, prediction of energy spectrum, robustness to noisy initial conditions, and long-time trajectory keeping on the attractor.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Wenbo Zhang, Wei Gu
Summary: This paper investigates the parameter estimation problem for various types of differential equations controlled by linear operators. Data-driven algorithms based on Gaussian processes are employed to solve the inverse problem and estimate the unknown parameters of the partial differential equations. Numerical tests demonstrate that the data-driven methods based on Gaussian processes can accurately estimate the parameters and approximate the solutions and inhomogeneous terms of the considered partial differential equations simultaneously.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Interdisciplinary Applications
Muhammad Bhatti, Md Habibur Rahman, Nicholas Dimakis
Summary: A multivariable technique using B-polynomials is employed to estimate solutions of NPDE, with coefficients determined using the Galerkin method before conversion to an operational matrix equation. The method provides higher-order precision compared to finite difference in solving NPDE equations and has potential for solving complex partial differential equations in multivariable problems.
FRACTAL AND FRACTIONAL
(2021)