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Reflectionless boundary propagation formulas for partial wave solutions to the wave equation. (English) Zbl 0849.35065
Summary: We consider solutions to the wave equation in \(3+ 1\) spacetime dimensions whose data is compactly supported at some initial time. For points outside a ball containing the initial support, we develop an outgoing wave condition, and associated one-way propagation formula, for the partial waves in the spherical-harmonic decomposition of the solution. The propagation formula expresses the \(l\)-th partial wave at time \(t\) and radius \(a\) in terms of order-\(l\) radial derivatives of the partial wave at time \(t- \Delta t\) and radius \(a- \Delta t\).
The boundary propagation formula can be applied to any differential equation that is well-approximated by the wave equation outside a fixed ball.
MSC:
35L05 Wave equation
35L15 Initial value problems for second-order hyperbolic equations
35C10 Series solutions to PDEs
35A35 Theoretical approximation in context of PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
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