A Chebyshev pseudospectral–two-step three-order boundary value coupled method is proposed and presented for handling the issue associated with complicated calculation, low precision, and poor stability in the process of transient response of transmission line. The first order differential equation in time domain is obtained via dispersing the telegraph equation in space domain by utilizing the pseudospectral method (PSM) based on Chebyshev polynomial. Then the two-step three-order boundary value method (BVM3) is presented and employed to resolve the obtained differential equation, so the numerical solution of the space discrete points can be obtained. Furthermore, the Chebyshev pseudospectral–two-step three-order boundary value coupled method (PSM-BVM3) is presented and compared with the Chebyshev pseudospectral–two-step two order boundary value coupled method (PSM-BVM2), the pseudospectral–differential quadrature method (PSM-DQM), and the pseudospectral method–trapezoid rule (PSM-TR) to validate the feasibility of the new proposed method. Theoretical analysis and numerical simulation reveal that the proposed Chebyshev PSM-BVM3 has a higher performance than the conventional method. For the proposed Chebyshev PSM-BVM3, the higher precision, efficiency, and numerical stability can be obtained and achieved only with fewer discrete points in the space domain, which is suitable for solving the transient response of transmission line. The proposed PSM-BVM3 can improve the drawback of numerical instability of the PSM and can also improve the disadvantage of the BVM as it is not easy to change the latter’s timestep size.

We consider the problem of recovering the initial value, from the trace on the light cone, of the solution of an initial value problem for the wave equation. When the space is odd dimensional, we show that the map from the initial value to the traces of the (even or odd in time) solutions on the light cone is an isometry and we characterize the range of this map and construct its inverse. We do this by relating the problem to the recovery of a function from its spherical means over all spheres through the origin, which in turn is related to the Radon transform inversion via the inversion map on R n .