Pseudospectral methods for solving an equation of hypergeometric type with a perturbation

• Authors: H. Alıcı;H. Taşeli
• Venue: Journal of Computational and Applied Mathematics
• Year: 2010

Almost all, regular or singular, Sturm-Liouville eigenvalue problems in the Schrodinger form -@J^''(x)+V(x)@J(x)=E@J(x),x@?(a@?,b@?)@?R,@J(x)@?L^2(a@?,b@?) for a wide class of potentials V(x) may be transformed into the form @s(@x)y^''+@t(@x)y^'+Q(@x)y=-@ly,@x@?(a,b)@?R by means of intelligent transformations on both dependent and independent variables, where @s(@x) and @t(@x) are polynomials of degrees at most 2 and 1, respectively, and @l is a parameter. The last form is closely related to the equation of the hypergeometric type (EHT), in which Q(@x) is identically zero. It will be called here the equation of hypergeometric type with a perturbation (EHTP). The function Q(@x) may, therefore, be regarded as a perturbation. It is well known that the EHT has polynomial solutions of degree n for specific values of the parameter @l, i.e. @l:=@l"n^(^0^)=-n[@t^'+12(n-1)@s^''], which form a basis for the Hilbert space L^2(a,b) of square integrable functions. Pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of EHTP, and hence the energies E of the original Schrodinger equation. Specimen computations are performed to support the convergence numerically.