The aim of this paper is to obtain the numerical solutions of fractional Volterra integro-differential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points. We convert the fractional order integro-differential equation into integral equation by fractional order integral, and transfer the integro equations into a system of linear equations by the Gausssian quadrature. We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in $L^{\infty }$ norm. Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.
In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the $P_k$-harmonic polynomial space instead of the full polynomial space $P_{k}$ is to use a much smaller number of basis functions to achieve the same accuracy when $k\geqslant 2$. The optimal rate of convergence is derived in both $H^1$ and $L^2$ norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the $P_{2}$-harmonic polynomial space and using the standard $P_2$ polynomial space are presented.
In the present paper, we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of orders $\alpha ,1+\alpha ,2+\alpha ,3+\alpha $ and $4+\alpha $. The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.