Inverse Problem for the Inhomogeneous One-Dimensional Euler–Poisson–Darboux Equation with Variable Potential

Abstract The direct initial–boundary value and inverse coefficient problems for the one-dimensional Euler–Poisson–Darboux equation with the Bessel differential operator with respect to time is considered. The Fourier method is applied for the direct problem, and the Fourier coefficients of the unknown function are shown to satisfy Volterra integral equations of the second kind. A priori estimates for these coefficients are obtained in terms of the norms of the unknown inverse problem. To study the inverse problem, a nonlinear integral equation is constructed, and local existence and uniqueness of the solution to the inverse problem are proved using the fixed-point principle. An estimate of the conditional stability of the solution is also established

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