550 research outputs found
The Convergence of Expansion Method of Chebyshev Polynomials
In this paper, the weakly singular linear and nonlinear integro-differential equations are solved by using expansion method of Chebyshev polynomials of the first kind .The approximation solution of this equation is calculated in the form of a series which its components are computed easily .The existence and uniqueness of the solution and the convergence of the proposed method are proved. Numerical examples are studied to demonstrate the accuracy of the presented method. Keywords: China insurance industry, Volterra integral equations, Fredholm integral equations, Integro-differential equations, Singular integral equations, Chebyshev polynomials method
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel
The Jacobi pseudo-spectral Galerkin method for the Volterra integro-differential equations of the second kind with a weakly singular kernel is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the LĻĪ±,Ī²2-norm and the Lā-norm) will decay exponentially provided that the source function is sufficiently smooth. Numerical examples are given to illustrate the theoretical results
Spectral collocation method for compact integral operators
We propose and analyze a spectral collocation method for integral
equations with compact kernels, e.g. piecewise smooth kernels and
weakly singular kernels of the form We prove that 1) for integral equations, the convergence
rate depends on the smoothness of true solutions . If
satisfies condition (R): }, we obtain a geometric rate of convergence; if
satisfies condition (M): ,
we obtain supergeometric rate of convergence for both Volterra
equations and Fredholm equations and related integro differential
equations; 2) for eigenvalue problems, the convergence rate depends
on the smoothness of eigenfunctions. The same convergence rate for
the largest modulus eigenvalue approximation can be obtained.
Moreover, the convergence rate doubles for positive compact
operators. Our numerical experiments confirm our theoretical
results
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