Abstract
In this note, we solve the inverse nodal problem for Bessel-type p-Laplacian problem \t\t\t−(y′(p−1))′=(p−1)(λ−ω(x))y(p−1),1≤x≤a,y(1)=y(a)=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& - \\bigl( y^{{\\prime} (p-1)} \\bigr) ^{\\prime} = ( p-1 ) \\bigl( \\lambda- \\omega(x) \\bigr) y^{(p-1)},\\quad1\\leq x\\leq a, \\\\& y(1) =y(a)=0, \\end{aligned}$$ \\end{document} on a special interval. We obtain some nodal parameters like nodal points and nodal lengths. In addition, we reconstruct the potential function by nodal points. Results obtained in this paper are similar to the classical Sturm–Liouville problem. However, equations of this type are considered with the condition defined at the origin. We solve the problem on the interval [1,a], that problem is not singular.
Highlights
By using separation of variables, the wave equation can be written with spherical symmetry–y (x) + ω(x)y = λy, (1.1)where λ is a constant referring to the eigenvalue of the problem, and ω(x) = ωo(x) + l(l+1) x2[1], where l is a positive integer or zero and ωo(x) will be defined in what follows.Let us take into account the eigenvalue problem– y (p–1) = (p – 1) λ – ω(x) y(p–1), 1 ≤ x ≤ a, y(1) = y(a) = 0
An inverse nodal problem means finding the potential function through the nodal points without any other spectral data
1.1 Results and discussion In the present paper, we find the potential function by using nodal parameters for the p-Laplacian Bessel operator on a regular interval [1, a]
Summary
Note that for p = 2, the inverse problem for the Bessel operator has been studied by [2]. – y (p–2)y = (p – 1) y (p–2)y, y(0) = 0, y (0) = 1, has a solution Sp(x), where Sp(x) is called the sine function for any p, and they defined inversion of the integral An inverse nodal problem means finding the potential function through the nodal points (zeros of eigenfunctions) without any other spectral data. Given nodal points, one can find the potential function in a general case.
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