An inverse polynomial method for the identification of the leading coefficient in the Sturm-Liouville operator from boundary measurements
An inverse polynomial method of determining the unknown leading coefficient k=k(x) of the linear Sturm-Liouville operator Au=-(k(x)u^'(x))^'+q(x)u(x), x@?(0,1), is presented. As an additional condition only two measured data at the boundary (x=0,x=1) are used. In absence of a singular point (u^'(x) 0,u^''(x) 0,@?x@?[0,1]) the inverse problem is classified as a well-conditioned . If there exists at least one singular point, then the inverse problem is classified as moderately ill-conditioned (u^'(x"0)=0,x"0@?(0,1);u^'(x) 0,@?x x"0;u^''(x) 0,@?x@?[0,1]) and severely ill-conditioned (u^'(x"0)=u^''(x"0)=0,x"0@?(0,1);u^'(x) 0,u^''(x) 0,@?x x"0). For each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a Cauchy problem for to the first order differential equation with respect the unknown function k=k(x). An approximate analytical solution of the each Cauchy problems are derived in explicit form. Numerical simulations all the above cases are given for noise free and noisy data. An accuracy of the presented approach is demonstrated on numerical test solutions.
Determination of leading coefficients in Sturm-Liouville operator from boundary measurements. I. A stripping algorithm
We present a stripping algorithm for determination of the unknown coefficient k=k(x) in the Sturm-Liouville operator Au=(k(x)u^'(x))^'+q(x)u(x), [email protected]?(a,b), form boundary measurements. Due to the only two physically possible measured data at the boundary, the problem is of strong unstable. The formulation of the problem based on the Tikhonov's quasisolution approach. The coefficient k(x)@?L"2[a,b] is assumed to be a monotone and uniform bounded function. This class of functions K"c is compact in L"2[a,b] and hence the inverse problem has at least one quasisolution in K"c. The stripping algorithm is implemented for the cases, when the unknown function k(x) is interpolated by the first- and second-order polynomials. Effectiveness of the method is demonstrated on concrete numerical examples with exact and noisy data.
Identification of parameters in a linear equation of groundwater flow
An inductive method is presented for evaluating parameters in a two-dimensional linear equation describing groundwater flow. The approach employs finite difference approximations, which can be simply programed for calculation by computer. Before illustrating the method, which is applicable to both stationary and time-dependent problems, the various types of data required for evaluation in general are systematically enumerated. An assumption is introduced limiting the local variability of T, the transmissivity, and its relation to apparently similar assumptions used elsewhere is discussed. The particular aquifer chosen for illustration is the chalk of the South Downs between the rivers Adur and Ouse. The parameters are calculated on the basis of cyclic data in which annual abstraction is about 30% of the total infiltration for the year. The method gives a match to the minimum and maximum water levels during the year to within 7%.