An inverse problem for computing a leading coefficient in the Sturm-Liouville operator by using the boundary data

Keywords: Inverse coefficient problem (ICP) Lie-group adaptive method (LGAM) Leading coefficient Inverse Sturm–Liouville operator Iterative method a b s t r a c t We consider an inverse problem for identifying a leading coefficient a(x) in À(a(x)y 0 (x)) 0 + q(x)y(x) = H(x), which is known as an inverse coefficient problem for the Sturm–Liouville operator. We transform y(x) to u(x, t) = (1 + t)y(x) and derive a parabolic type PDE in a fictitious time domain of t. Then we develop a Lie-group adaptive method (LGAM) to find the coefficient function a(x). When a(x) is a continuous function of x, we can identify it very well, by giving boundary data of y, y 0 and a. The efficiency of LGAM is confirmed by comparing the numerical results with exact solutions. Although the data used in the identification are limited, we can provide a rather accurate solution of a(x). To motivate the present study, we consider the longitudinal wave motion of a one-dimensional rod with a variable Young's modulus E(x): 1 A @ @x EðxÞA @uðx; tÞ @x ¼ q @ 2 uðx; tÞ @t 2 ; ð1Þ where A is a constant cross-sectional area of the rod, q is a constant mass-density, and u(x, t) is the axial displacement. Let u(x, t) = e ixt y(x); Eq. (1) can be simplified to À d dx EðxÞA dyðxÞ dx ¼ qAx 2 yðxÞ; ð2Þ where x is the vibrational frequency. In the inverse problem, it is technically important to identify the material property E(x) for a rod made of non-homogeneous material. This problem is known as an inverse problem for identifying the rigidity function E(x)A of the rod, which falls into a category of the parameter identification problems of differential operators. In this paper we consider a mathematical modeling of these problems by the following Inverse coefficient problem: Find the pair of unknown functions hy(x), a(x)i in the problem.