A recurrence relation for solution of singular Volterra integral equations using Chebyshev polynomials

Abstract

We obtain a recurrence relation for the numerical evaluation of integrals $$\int_0^x {(T_k^* (t)/(x^p - t^p )^\alpha ) } dt, 0 \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} x \mathbin{\lower.3ex\hbox{$\buildrel<\over{\smash{\scriptstyle=}\vphantom{_x}}$}} 1,$$ wherep is a positive integer, 0≦α<1, andT*k(t)=Tk (2t-1), 0≦t≦1,Tk(t)=cos (k arccost) the usual Chebyshev polynomials. This relation can be used in the Chebyshev series method for approximate solution of first and second kind Volterra integral equations with singular and non-singular kernels.