In this paper, we propose an improved learning scheme-interface operator network based on extreme learning machines (ELM-IONet) to solve elliptic interface problems, where different coefficients, source terms, and other parameters are treated as input features. Here the extension-and-projection method is used in ELM-IONet to capture the discontinuities of functions. Meanwhile, the idea of the Gram-Schmidt orthogonalization is adopted to handle the output matrices of the trunk and branch networks of ELM-IONet, thereby improving its stability and generalization ability. Additionally, we integrate ELM-IONet with physics-informed neural networks, enabling it to remain effective even in the absence of paired input-output data within the computational domain. Numerous experiments are conducted to demonstrate that ELM-IONet significantly outperforms the traditional operator neural networks in terms of both accuracy and efficiency.
This paper demonstrates the solvability and trajectory controllability of a multi-term fractional stochastic differential equation driven by Poisson jumps in the pth-moment. In general, an integer order differential equation with n derivatives can be turned into an abstract form of the first order differential system, but fractional differential equation does not have this property. This study introduces a method to investigate fractional differential equation with more than one fractional derivatives. The key findings of this work include the existence and trajectory control of the proposed model without imposing the compactness condition on the resolvent operators. The solvability of the proposed system is addressed by employing fractional calculus, semigroup theory, and the stochastic version of Banach fixed point theorem. The existence of trajectory controllability is established using Gronwall's inequality. Finally, a multi-term time-fractional wave equation with stochastic diffusion is provided to illustrate and justify the importance of the proposed study.
This paper introduces a class of Time-Accurate and highly-Stable Explicit Runge-Kutta schemes for solving delay differential equations with distributed delays. The delay terms are approximated using Lagrange interpolation. Then, by applying the argument principle, the delay-dependent stability criteria of the numerical methods are derived. Two numerical examples are given to demonstrate that the proposed methods exhibit larger stability regions compared to explicit Runge-Kutta methods. The validity of the stability criteria is also verified.
In this study, the simulation and behaviour of motile microorganisms are examined in hydromagnetic Walters' B nanofluid with elastic deformation. The fluid radiates nonlinearly in the presence of Biot number and viscous dissipation. The mathematical formulations of the model, expressed as partial differential equations, were transformed into ordinary differential equations using similarity conversion variables. The converted equations are solved using the Galerkin Weighted Residual Method (GWRM), whose residual errors have been proven to be extremely minimal. The convergence of the iteration is also verified, and validation is provided against existing work in the literature. The GWRM outputs are further verified by the Spectral Collocation Method (SCM), whose differences are very minimal (negligible). The novel result of these findings demonstrates that the motile microorganisms can easily survive when nanoparticles are diluted, and the Weissenberg number intensifies the dominance of tensile stresses via viscoelasticity of the fluid, which is crucial for polymer processing.
For nonlinear partial differential equations on the whole line, where the solution exhibits Gaussian-like characteristics and severe oscillations, we propose a spectral element method based on non-uniform meshes. To approximate the highly oscillatory part of the solution on bounded subdomains, we employ Legendre spectral elements, while for the part of the solution that decays at infinity on unbounded subdomains, we utilize Laguerre functions. A novel composite spectral element scheme is developed, featuring movable common boundaries between adjacent subdomains. This design makes the scheme particularly effective for Gaussian-type solutions governed by nonlinear partial differential equations. The convergence and stability of the proposed algorithm are established through a newly constructed composite quasi-orthogonal approximation framework over the entire real line. Numerical results confirm the effectiveness of the algorithm and are in excellent agreement with the theoretical analysis.
