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It is challenging to use the piecewise polynomial collocation method numerically solving the Caputo fractional differential equations (FDEs), since it is related to the well-known Conjecture 6.3.5 in Brunner’s 2004 monograph on the convergence of ...

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform ...

This note addresses the well-posedness of weak solutions for a general linear evolution problem on a separable Hilbert space. For this classical problem there is a well known challenge of obtaining a priori estimates, as a constructed weak ...${}_{\mathrm{}}$${}_{\mathrm{}}$

The present article introduces, mathematically analyzes, and numerically validates a new weak Galerkin mixed finite element method based on Banach spaces for the stationary Navier–Stokes equation in pseudostress–velocity formulation. Specifically, ...

A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary ...${}^{\mathrm{}}$${{}^{\mathrm{}}}^{\mathrm{}}$${{}^{\mathrm{}}}^{}$

Variable-exponent weakly singular Volterra integral equations of the second kind with integral kernels of the form ${(t-s)}^{-\alpha (t)}$ are considered. In Liang and Stynes (IMA J Numer Anal 19:drad072, 2023) it is shown that a typical solution of such an ...$\mathrm{}$$$

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform ...$\mathbf{}\mathit{}\mathbf{}$$\mathit{}\mathbf{}$

In this paper, we propose an effective Chebyshev–Picard iteration (CPI) method for solving delay differential equations with a constant delay. This approach adopts the Chebyshev series to represent the solution and improves the accuracy of the ...

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different ...

A collocation method for the numerical solution of Volterra integro-differential equations with weakly singular kernels, based on piecewise polynomials of fractional order, is constructed and analysed. Typical exact solutions of this class of ...$\mathrm{}$

In this paper, we extend the utilization of pseudostress-based formulation, recently employed for solving diverse linear and nonlinear problems in continuum mechanics via mixed finite element methods, to the weak Galerkin method (WG) framework and ...

We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally, we show ...

The purpose of this paper is to develop and analyze an adaptive collocation method for Fredholm integral equations of the second kind, even if the equation exhibits localized rapid variations, steep gradients, or steep front. The strategy of the ...${}^{}$${}^{\mathrm{}}$

We propose a very simple but efficient postprocessing technique for improving the global accuracy of the discontinuous Galerkin (DG) time stepping method for solving nonlinear Volterra integro-differential equations. The key idea of the ...

• In this paper, the weak Galerkin finite element method is studied for a singularly perturbed problem with two parameters. On a piecewise uniform Shishkin ...

A weak Galerkin finite element method is proposed for solving singularly perturbed problems with two parameters. A robust uniform optimal convergence has been proved in the corresponding energy and a stronger balanced norms using ...

In this work, we will propose and analyse a novel reproducing kernel function-based piecewise approach for solving oscillatory systems of second-order initial value problems (IVPs). Also, the approach can be used to effectively solve wave ...

In this article, we propose and analyze a fully coupled, nonlinear, and energy-stable virtual element method (VEM) for solving the coupled Poisson–Nernst–Planck (PNP) and Navier–Stokes (NS) equations. These equations model microfluidic and ...${}^{\mathrm{}}$${}^{\mathrm{}}$${\mathbf{}}^{\mathrm{}}$${\mathbf{}}^{\mathrm{}}$${}^{\mathrm{}}$

We introduce and analyze a very simple but efficient postprocessing technique for improving the accuracy of the continuous Galerkin (CG) time stepping method for nonlinear first-order initial value problems with smooth and singular solutions. The ...$\mathrm{}$${}^{\mathrm{}}$${}^{\mathrm{}}$${}^{}$${}^{\mathrm{}}$${}^{\mathrm{}}$${}^{}$

In Mu and Zhang (2019), an immersed weak Galerkin finite element method (IWG-FEM) is developed for solving elliptic interface problems and it is proved that this method has optimal a-priori error estimate in an energy norm under ...

An efficient numerical scheme based on exponential B-spline (EBS) functions is developed in this work for numerical solution of Asian option pricing (AOP) problem and dealing with delta values. Convergence and stability of proposed ...