In this paper the differential equation y″ + (ρ2φ2(x) –q(x))y = 0 is considered on a finite interval I, say I = [0, 1], where q is a positive sufficiently smooth function and ρ2 is a real parameter. Also, [0, 1] contains a finite number of zeros of φ2, the so called turning points, 0 < x1 < x2 < … < xm < 1. First we obtain the infinite product representation of the solution. The product representation, satisfies in the original equation. As a result the associated dual equation is derived and then we proceed with the solution of the inverse problem.
Marasi, H., & Akbarfam, A. J. (2012). Dual equation and inverse problem for an indefinite Sturm–Liouville problem with m turning points of even order. Mathematical Modelling and Analysis, 17(5), 618-629. https://doi.org/10.3846/13926292.2012.732972
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