Abstract. In this paper, a Jacobi-collocation spectral method is developed for Volterra integral equations of the second kind with a weakly singular kernel. We use some function transformations and variable transformations to change the equation into a new Volterra integral equation defined on the standard interval [−1, 1], so that the solution of the new equation possesses better regularity and the Jacobi orthogonal polynomial theory can be applied conveniently. In order to obtain high-order accuracy for the approximation, the integral term in the resulting equation is approximated by using Jacobi spectral quadrature rules. The convergence analysis of this novel method is based on the Lebesgue constants corresponding to the Lagrange interpolation polynomials, polynomial approximation theory for orthogonal polynomials and operator theory. The spectral rate of convergence for the proposed method is established in the L ∞ -norm and the weighted L 2 -norm. Numerical results are presented to demonstrate the effectiveness of the proposed method.

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Thermal transport properties of isotopic-superlattice graphene nanoribbons with zigzag edge (IS-ZGNRs) are investigated. We find that by isotopic superlattice modulation the thermal conductivity of a graphene nanoribbon can be reduced significantly. The thermal transport property of the IS-ZGNRs strongly depends on the superlattice period length and the isotopic mass. As the superlattice period length decreases, the thermal conductivity undergoes a transition from decreasing to increasing. This unique phenomenon is explained by analyzing the phonon transmission coefficient. While the effect of isotopic mass on the conductivity is monotone. Larger mass difference induces smaller thermal conductivity. In addition, the influence of the geometry size is also discussed. The results indicate that isotopic superlattice modulation offers an available way for improving the thermoelectric performance of graphene nanoribbons.

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In this paper, we study the Legendre-Galerkin spectral approximation for a constrained optimal control problem. We first derive a priori error estimates for the spectral approximation of optimal control problems. Then a posteriori error estimates are obtained for both the state and the control approximation. A preconditioning projection algorithm is proposed with some numerical tests. Introduction.The spectral method employs global polynomials as the trial functions for the discretization of PDEs. It provides very accurate approximations with a relatively small number of unknowns when the solutions are smooth. Recently the spectral method has been extended to approximate unconstrained optimal control problems; see, for example, [10].However spectral accuracy generally cannot be achieved when the approximated solutions have lower regularities, and this is typically the case when, for example, there exist control constraints in an optimal control problem (the so-called constrained optimal control problem). Thus the spectral method is not widely used in solving constrained distributed optimal control problems where the solutions often have singularities at the boundary of constraints even though all the initial data are smooth. Thus although there has been much work on the finite element method for numerically solving constrained optimal control problems, it seems that there has not been much work on the spectral method. Furthermore, the optimality conditions, which are normally the starting point of spectral approximation, are just PDE systems for unconstrained control problems, while those for constrained control problems contain variational inequalities, as shown later on. This also raises new issues in analyzing and solving the systems discretized using the spectral method.In this work we study a constrained optimal control problem. It in fact represents a class of useful optimal control problems. We show that the solution of this particular control problem can be infinitely smooth if the initial data are so. Then we propose to use the Galerkin spectral method to approximate the solution. As there