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1.
We apply Jacobi spectral collocation approximation to a two-dimensional nonlinear weakly singular Volterra integral equation with smooth solutions. Under reasonable assumptions on the nonlinearity, we carry out complete convergence analysis of the numerical approximation in the L-norm and weighted L2-norm. The provided numerical examples show that the proposed spectral method enjoys spectral accuracy.  相似文献   

2.
We consider an h-p version of the continuous Petrov-Galerkin time stepping method for Volterra integro-differential equations with proportional delays. We derive a priori error bounds in the L 2-, H 1- and L -norm that are explicit in the local time steps, the local approximation orders, and the local regularity of the exact solution. Numerical experiments are presented to illustrate the theoretical results.  相似文献   

3.
We propose and analyze a spectral Jacobi-collocation approximation for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collection method, which shows that the errors of the approximate solution decay exponentially in L norm and weighted L2-norm. The numerical examples are given to illustrate the theoretical results.  相似文献   

4.
We introduce the notion of bounded variation in the sense ofL 1-norm for periodic functions and prove a version of the classical Dirichlet-Jordan test for the convergence of Fourier series inL 1-norm. We also give an estimate of the rate of convergence.  相似文献   

5.
In this paper, the convergence of solutions for incompressible dipolar viscous non-Newtonian fluids is investigated. We obtain the conclusion that the solutions of non-Newtonian fluids converge to the solutions of Navier-Stokes equations in the sense of L2-norm (resp. H1-norm), as the viscosities tend to zero and the initial data belong to H1(Ω) (resp. H2(Ω)). Moreover, we obtain L-norm convergence of solutions if the initial data belong to H2(Ω).  相似文献   

6.
The subject of this paper is the analytic approximation method for solving stochastic differential equations with time-dependent delay. Approximate equations are defined on equidistant partitions of the time interval, and their coefficients are Taylor approximations of the coefficients of the initial equation. It will be shown, without making any restrictive assumption for the delay function, that the approximate solutions converge in Lp-norm and with probability 1 to the solution of the initial equation. Also, the rate of the Lp convergence increases when the degrees in the Taylor approximations increase, analogously to what is found in real analysis. At the end, a procedure will be presented which allows the application of this method, with the assumption of continuity of the delay function.  相似文献   

7.
Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L2-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L2-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.  相似文献   

8.
In this paper, we investigate the stability and convergence of some fully discrete finite element schemes for solving the acoustic wave equation where a discontinuous Galerkin discretization in space is used. We first review and compare conventional time-stepping methods for solving the acoustic wave equation. We identify their main properties and investigate their relationship. The study includes the Newmark algorithm which has been used extensively in applications. We present a rigorous stability analysis based on the energy method and derive sharp stability results covering some well-known CFL conditions. A convergence analysis is carried out and optimal a priori error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the L 2-norm error over a finite time interval converges optimally as O(h p+1+??t s ), where p denotes the polynomial degree, s=1 or 2, h the mesh size, and ??t the time step.  相似文献   

9.
We give a complete proof of Morrey’s estimate for the W 1,p -norm of a solution of a second-order elliptic equation on a domain in terms of the L 1-norm of this solution. The dependence of the constant in this estimate on the coefficients of the equation is investigated.  相似文献   

10.
In this article, we are concerned with the numerical analysis of a nonlinear implicit difference scheme for Burgers' equation. A priori estimation of the analytical solution is provided in the sense of L -norm when the initial value is bounded in H1-norm. Conservation, boundedness, and unique solvability are proved at length. Inspired by the method of the priori estimation for the analytical solution, we prove the convergence and stability of the difference scheme in L -norm. Finally, numerical examples are carried out to verify our theoretical results.  相似文献   

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