2026-02-23
Yucheng Song, Tingting Li, Jiu Ding, Congming Jin
Approximate solution of Fredholm integral equations with certain types of weakly singular algebraic kernels is obtained by the piecewise linear maximum entropy method. During implementation of the method, two-dimension numerical integration is reduced to one-dimension numerical integration, which allows to lower the computational cost. The results of numerical experiments are consistent with theoretical findings and show the effectiveness of the method.
2026-02-14
Subham Jangid, Kaladhar Kolla
This study investigates the natural convection flow of Williamson fluid between two concentric cylinders while affected by the radiation effect and magnetic field. The inner cylinder remains fixed while the outer cylinder rotates. Additionally, magnetic field is oriented radially, which influences the flow of the fluid. Applying a proper transformation, one transform the non-linear partial differential equations of the Williamson fluid model into ordinary differential equations. Artificial neural networks (ANN) facilitate the computation of solutions to these nonlinear ordinary differential equations. Trial functions employ a multilayer perceptron neural network with tunable parameters, including weights and biases. The governing equations are satisfied by determining the trial solution’s changeable parameters by applying the Adam (adaptive moment estimation algorithm) optimization technique. Compared to the analytical solutions, the ANN’s result demonstrates good accuracy. Moreover, graphs show how pertinent parameters affect the velocity and temperature profiles. The temperature and velocity profiles get smaller as the magnetic parameter value increases. Furthermore, the temperature and velocity profiles increase as the Hall parameter value rises.
2026-01-12
Hao-Qing Chen, Guang-Mei Wei, Yu-Xin Song
A variable-coefficient Sawada-Kotera system is investigated that models the nonlinear behaviors of waves in shallow water, ion-acoustic waves in plasma environments and fluid flow dynamics. The Painlevé integrability is tested by the WTC method with the simplified form of Krustal. The Hirota bilinear method is employed to derive the bilinear form. Consequently, we obtain a variety of analytic solutions, including soliton, lump, and breather solutions. In addition, the interactions between the lump soliton and one stripe soliton, among with the breather soliton and one stripe soliton are discussed.
2025-11-19
Lechuan Gu, Yihang Guo, Meng Li
In this paper, we extend the BGN formulation [J.W. Barrett, H. Garcke and R. Nürnberg, J. Comput. Phys. 222 (2007)] by incorporating the $k$-order backward differentiation formulae (BDFk) for time discretization. This allows us to develop high-order temporal parametric finite element methods for simulating anisotropic surface diffusion flows and solid-state dewetting problems, achieving accuracy levels from second-order to fourth-order. We prove the well-posedness of the constructed high-order schemes. The proposed schemes maintain good mesh quality characteristic of the classical first-order BGN scheme. Finally, we present several numerical simulations to demonstrate the high-order temporal accuracy and verify the preservation of good mesh quality and energy stability throughout the evolution.
2025-11-13
Shitao Liu, Mingkang Ni
Internal layer phenomena are often appear in various piecewise smooth nonlinear singular singularly perturbed problems of natural sciences. To characterize these special structures, we study piecewise smooth $n$-dimensional nonlinear singular singularly perturbed boundary value problems. In particular, we show the existence of solutions with an internal layer and construct their asymptotic expansions. The remainder estimations of the approximate solutions are also given. Finally, an example aimed to verify the correctness of the developed theory is presented.
2025-11-13
Eunwoo Heo, Kwanghyuk Park, Jae-Hun Jung
Singularly perturbed differential equations with the Dirac delta function usually yield discontinuous solutions. Therefore, careful consideration is required when using numerical methods to solve these equations because of the Gibbs phenomenon. A remedy based on the Schwartz duality has been previously proposed, yielding superior results without oscillations. However, this approach has primarily been applied to linear problems and still exhibits the Gibbs phenomenon when extended to nonlinear or higher-dimensional problems. In this paper, we propose a consistent yet simple approach based on Schwartz duality that can handle such problems. Our proposed approach utilizes a modified direct projection method with a consistent discrete derivative of the Heaviside function, which directly approximates the Dirac delta function. As numerical examples, we consider several problems, including the Burgers’ equation and the two-dimensional time-dependent advection equation. The proposed method effectively eliminates Gibbs oscillations without the need for traditional regularization and demonstrates uniform error reduction for the problems considered.
2025-11-13
Yueran Wang, Wenli Yang, Zhongyi Huang
An energy stable tailored finite point method based Petrov-Galerkin scheme to solve Allen-Cahn equation is proposed. In time discretization, we present both first-order and second-order semi-discrete schemes based on stabilized and convex-splitting techniques, which satisfy unconditional energy stability. We prove the maximum bound preserving principle for first-order schemes. Due to nonlinearity, the well-posedness of weak formulations based on semi-discrete schemes are demonstrated. As the nature of singularly perturbation in semi-discrete level remains when $ε$ is extremely small, we establish a specified Petrov-Galerkin scheme which leads to a unified way for space discretization. To this end, we set up nonlinear solvers which are proved to be stable and convergent. Then we construct our Petrov-Galerkin scheme, which is built upon problem-dependent test function space. The stability and second-order convergence of this scheme are rigorously proved in one dimension. In order to compute test functions, specialized TFPM schemes are incorporated into the scheme. Numerical experiments show the accuracy, efficiency, and the good performance of the method on uniform meshes even when mesh size $h$ is much larger than $ε$.
2025-11-09
Xinyuan Wu
This paper compares the performance of the conjugate gradient method and geometric approach in the case of symmetric positive definite (SPD) linear systems. This approach is based on the geometric theory of ODEs which was effectively initiated by Poncaré and Liapunov. The simplest and most obvious advantage of the geometric approach over the conjugate gradient method (the MATLAB code pcg) is that this approach can find the inverse of the underlying positive definite matrix and the solution. We present various numerical examples, which demonstrate the superiority of the geometric approach. For SPD linear systems, this approach provides much higher accuracy than the conjugate gradient method. In particular, since it is a one-stop procedure, it can avoid the growth of accumulated round-off errors to some extent.
2025-10-22
Ting Wei, Jianming Xu, Xi Yue
This paper is devoted to identifying the order of time fractional derivative and a space-dependent source term in a time fractional diffusion-wave equation from some additional measured data in a subdomain or on a subboundary with a small time period. The Lipschitz continuity of forward operators mapping the unknown order and source term into the given data are established based on the stability estimates of solution for the direct problem. We prove the uniqueness of the considered inverse problems by using the asymptotic behavior of the solution at $t$ = 0, the Titchmarsh convolution theorem and the Duhamel principle. Moreover, a Tikhonov-type regularization method is proposed with $H^1$-norm as a penalty term. The existence of the regularized solution and its convergence to the exact solution under a suitable regularization parameter choice are obtained. Then we employ a linearized iteration algorithm combined with the piecewise linear finite element approximation to find simultaneously the approximate order and space source term. Three numerical examples for one- and two-dimensional cases are tested and the numerical results demonstrate the effectiveness of the proposed method.