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To analyze a wire antenna excited by a time varying voltage source or a wire scatterer excitated by transient electromagnetic incident wave, the problem is formulated in terms of a time-domain integral equation for the induced current. To solve the integral equation, we reduce it to matrix equation via the method of moments using the known-to-be-stable implicit scheme. However, rather than directly constructing and solving the relatively large matrix equation, we propose an iterative procedure which allows us to gradually obtain a solution of refined accuracy both everywhere and simultaneously at any time instance. To render this procedure rapidly converging, we use a basis of spatio-temporal wavelet functions. This basis facilitates a good approximation of the induced current using far less basis functions than would be needed if other expansions, such as standard-pulse or Fourier basis functions were chosen. The use of this basis further enables the iterative procedure to increase the temporal and spatial resolutions where required without unnecessarily affecting their levels elsewhere.