Abstract : Since integral equations, integro-differential and nonlocal equations play an important role as mathematical models in science, engineering and finance, recent years have seen major developments in the design and analysis of efficient numerical methods for such equations. It is the aim of this minisymposium to bring together leading experts in these fields, in order to describe recent achievements and further communication between numerical analysts and computational scientists working on these problems.
[02677] An hp-version of the discontinuous Galerkin method for fractional integro-differential equations with weakly singular kernels
Format : Talk at Waseda University
Author(s) :
Yanping Chen (South China Normal University)
Abstract : In this talk, an hp-discontinuous Galerkin method is developed for the fractional integro-differential equations with weakly singular kernels. The key idea of our method is to first convert the fractional integro-differential equations into the Volterra integral equations, and then solve the equivalent integral equations using the hp-discontinuous Galerkin method. The prior error bounds for the proposed method are established in the L2-norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.
[03691] Implicitly Linear Jacobi Spectral-Collocation Methods for Weakly Singular Volterra-Hammerstein Integral Equations
Format : Talk at Waseda University
Author(s) :
Qiumei Huang (Beijing University of Technology)
Huitinh Yang (Beijing University of Technology)
Abstract : Weakly singular Volterra integral equations of the second kind typically have nonsmooth solutions near the initial point of the interval of integration, which seriously affects the accuracy of spectral methods. We present Jacobi spectral-collocation method to solve two-dimensional weakly singular Volterra-Hammerstein integral equations based on smoothing transformation and implicit linear method. The solution of the smoothed equation is much smoother than the original one after smoothing transformation and the spectral method can be used. For the Hammerstein nonlinear term, the implicitly linear method is applied to simplify the calculation and improve the accuracy. Convergence analysis in the L∞−norm is carried out and the exponential convergence rate is obtained. Finally, we demonstrate the efficiency of the proposed method by numerical examples.
[03565] A collocation based approach for the numerical solution of singular fractional integro-differential equations
Format : Talk at Waseda University
Author(s) :
Kaido Latt (University of Tartu)
Arvet Pedas (University of Tartu)
Abstract : We consider a class of fractional integro-differential equations with certain type of singularities at the origin. We reformulate the original problem as a cordial Volterra integral equation and study the existence, uniqueness, and regularity of the exact solution. We also construct a collocation based numerical method for finding the approximate solution of the original problem and present some numerical examples.
[02147] Solutions of second kind Fredholm integral equations by discrete projection methods
Format : Talk at Waseda University
Author(s) :
Gobinda Rakshit (Rajiv Gandhi Institute of Petroleum Technology, Jais Campus, Amethi, Uttar Pradesh 229304)
Abstract : We are interested in approximate solutions of the integral equation $x(s)−\int_{0}^{1} \kappa(s, t, (x(t)) dt =f(s)$, where $f$ and the kernel $\kappa$ are given. A class of projection methods are available for obtaining approximate solutions to the above integral equation. Modified projection method is recently proposed and it exhibits higher orders of convergence as compared to the Galerkin/collocation (projection) methods. Here, we define and analyze a discrete version of the above projection methods.
[03563] Numerical solution of fractional integro-differential equations
Format : Talk at Waseda University
Author(s) :
Arvet Pedas (University of Tartu)
Mikk Vikerpuur (University of Tartu)
Abstract : We consider a wide class of linear multi-term fractional integro-differential equations with Caputo derivatives and weakly singular kernels. First, we discuss the existence, uniqueness and smoothness of the exact solution. Then, using a suitable smoothing transformation and spline collocation techniques, we construct a high-order method for the numerical solution of the underlying problem. Finally, a numerical illustration of the proposed method is presented.
[04175] A new linearized maximum principle preserving and energy stability scheme for the space fractional Allen-Cahn equation
Format : Talk at Waseda University
Author(s) :
Yin Yang (Xiangtan University)
Biao Zhang (Xiangtan University)
Abstract : In this talk, we present a new linearized two-level second-order scheme for the space fractional Allen-Cahn equation, which is based on the Crank-Nicolson method in time, second-order weighted and shifted Gr\"{u}nwald difference formula in space and Newton linearized technology to deal with nonlinear term. And we only need to solve a linear system at each time level. Then, the unique solvability of the scheme is given. Under the reasonable time step constraint, the discrete maximum principle, energy stability and error analysis are also studied. At last, some numerical experiments show that the proposed method is reasonable and effective.
[03914] High accuracy analysis of FEMs for several time-fractional PDEs
Format : Talk at Waseda University
Author(s) :
Yanmin Zhao (Xuchang University)
Abstract : In this talk, convergence and superconvergence analysis for several kinds of time-fractional partial differential equations will be discussed by use of finite element methods and proper finite difference schemes. At the same time, unconditional stability properties of fully-discrete schemes are presented. Moreover, numerical experiments are provided to confirm the theoretical results. And, some relevant topics are included.
[02240] Discontinuous piecewise polynomial collocation methods for integral-algebraic equations of Hessenberg type
Format : Talk at Waseda University
Author(s) :
Hui Liang (Harbin Institute of Technology, Shenzhen)
Hecong Gao (Harbin Institute of Technology, Shenzhen)
Abstract : We mainly consider the integral-algebraic equations of Hessenberg type. The tractability index is investigated. The existence, uniqueness, and regularity are analyzed, and the resolvent representation is given. First, the convergence theory of perturbed collocation methods in discontinuous piecewise polynomial space is established for first-kind Volterra integral equations, then it is used to derive the optimal convergence properties of discontinuous piecewise polynomial collocation methods for Hessenberg-type integral-algebraic equations. Numerical examples illustrate the theoretical results.
[04726] Mean square exponential stability and practical mean square exponential stability of stochastic delay differential equations driven by G-Brownian motion and Euler-Maruyama approximations
Format : Online Talk on Zoom
Author(s) :
Haiyan Yuan (Heilongjiang institute of technology)
Abstract : This paper investigates the mean-square (MS) exponential stability and the practical mean square (PMS)
exponential stability of stochastic delay differential equations driven by G-Brownian motion (G-SDDEs)
and the numerical solution generated by Euler-Maruyama (EM) method. We present a weaker condition
to prove the MS exponential stability of G-SDDEs instead of choosing a Lyapunov function under the
case that the origin is an equilibrium point. In order to study whether the performance of G-SDDEs
near an unstable equilibrium point is acceptable, we introduce the practical stability and establish a new
generalized Gronwall inequality based on which we prove the PMS exponential stability of G-SDDEs.
We also study the numerical approximations for G-SDDEs. We first establish the stability equivalence
between the discrete EM method and the continuous EM method, then we prove that the continuous
EM method can reproduce the MS exponential stability and the PMS exponential stability of G-SDDEs
under some restrictions on the step size. Furthermore, two numerical experiments are conducted to con-
firm our theoretical results.
[02676] Superconvergent postprocessing of the continuous and discontinuous Galerkin methods for nonlinear Volterra integro-differential equations
Format : Online Talk on Zoom
Author(s) :
Lijun Yi (Shanghai Normal University)
Mingzhu Zhang (Shanghai Normal University)
Abstract : In this talk, we introduce novel postprocessing techniques for improving the accuracy of the CG and DG methods for nonlinear Volterra integro-differential equations. We first show that the CG and DG method superconverge at the nodal points of the time partition. We further prove that the postprocessed CG and DG approximations converge one order faster than the unprocessed CG and DG approximations in the $L^2$-, $H^1$- and $L^{\infty}$-norms. As a by-product of the postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact. Numerical examples are presented to verify the theoretical results.
