Multidomain Spectral Collocation Method And Chebyshev-Legendre Spectral Collocation Method For Volterra Integral Equations

As an important method for solving partial diferential equations, the spectral methodhas been widely used due to its high order of convergence rate for problems with smooth so-lutions. In this thesis, the spectral collocation method for solving the second kind Volterraintegral equations is investigated. The numerical errors decay exponentially in conditionthat the kernel function and the source function are sufciently smooth. The Chebyshevmethod is easier to be implemented and needs less computing time with the help of the FastFourier Transfer method. That is why the Chebyshev spectral collocation methods are morepopular in engineering computation. The domain decomposition Chebyshev spectral col-location method and the Chebyshev-Legendre spectral collocation method are investigatedfor the Volterra integral equations of the second kind in this thesis.Firstly, a domain decomposition Chebyshev collocation method is proposed for thelinear and nonlinear Volterra integral equations of the second kind. That is, the wholeinterval is partitioned into several non-overlapping subintervals and the Chebyshev spectralcollocation method is applied in each subinterval. So, the scale of the integral interval canbe converted to a small integral interval. Generally, the domain decomposition method is anefcient technique to improve the operating rate because we can make parallel computation.For Chebyshev spectral collocation method, the function in the integrand is expanded byusing Lagrange interpolation basis function, and the Lagrange interpolation basis functionis expanded in term of the Chebyshev polynomials. Thus, the appearance of the Chebyshevweakly singular weight function can be avoided. The integral term of the equation is dealtwith by using the Chebyshev spectral collocation method. The convergence of the linear andnonlinear Volterra integral equations of the second kind is proved. And the error estimationis obtained in L∞-norm. Numerical results are provided to verify the proposed method.A comparision is made between the domain decomposition method and the single domainmethod. The validity of the domain decomposition Chebyshev collocation method and thecorrectness of the theoretical analysis are proved.Secondly, a Chebyshev-Legendre spectral collocation scheme is constructed for thesecond-kind Volterra integral equations. For the integral term of the equation, the ker-nel function and the unknown function are approximated by the Chebyshev-Gauss-Lobattointerpolation. Then, the form of the Legendre polynomials is obtained by using theChebyshev-Legendre transforms. Therefore the integral form can be written into inner-product form. The computation can also be simplified due to the orthogonality of the Legendre polynomials. The convergence analysis of the scheme of one-dimensional lin-ear equations is given, and the error estimation is obtained in L∞-norm. Numerical resultsare given to compare with the results of Legendre spectral collocation method. The spectralaccuracy is observed by using the Chebyshev-Legendre spectral collocation method.