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The authors derive a series of explicit formulas to approximate the Riemann-Liouville derivative and integral of arbitrary order by shifted Chebyshev polynomials. This is then applied to solve boundary value problems involving Riemann-Liouville derivatives.

We employ the truncated Chebyshev series to approximate the Bessel function of the second kind Y"n(z) for |z| @? 8. Detailed manipulations and discussions for Y"0(z) and Y"1(z) are given. Results of numerical experiments are presents are presented to demonstrate the computed accuracy by using the Chebyshev series approximations. Advantages and disadvantages of the Chebyshev series approximation compared with other polynomial approximation methods, e.g., the tau-method approximations, are discussed.

When a function is singular at the ends of its expansion interval, its Chebyshev coefficients a n converge very poorly. We analyze three numerical strategies for coping with such singularities of the form (1± x ) k log(1± x ), and in the process make some modest additions to the theory of Chebyshev expansions. The first two numerical methods are the convergence-improving changes of coordinate x =sin[( π /2) y ] and x= tanh [Ly⧸(1-y 2 ) 1 2 ] . We derive the asymptotic Chebyshev coefficients in the limit n → ∞ for both mappings and for the original, untransformed Chebyshev series. For the original function, the asymptotic approximation for general k is augmented by the exact Chebyshev coefficients for integer k . Numerical tests show that the sine mapping is excellent for k ⩾1, increasing the rate of convergence to b n = O (1⧸ n 4 k +1 ). Although the tanh transformation is guaranteed to be better for sufficiently large n , we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice: “sufficiently large n ” is usually huge. Instead of mapping, one may use a third strategy: supplementing the Chebyshev polynomials with singular basis functions. Simple experiments show that this approach is also successful.