The Distinction Between Noncausal and Nonlocal Behavior in a Time-Fractional Wave Equation

摘要

There is some confusion in the ultrasound literature with regards to the distinction between noncausal and nonlocal behavior in fractional wave equations that describe power law attenuation. In a causal system, the effect follows the cause, and in a noncausal system, some portion of the response occurs prior to the application of the input. In contrast, local operators only relate values at a single point, and nonlocal operators require information from distant sources in time and/or space. Causality strictly describes the time response, whereas the absence or presence of locality applies to both the temporal and the spatial response of a system. Thus, causality and locality are two completely different concepts, which we will reinforce through the presentation of several different examples. The distinction between causality and locality is demonstrated through analysis of the power law wave equation, which exhibits four different combinations of causality and locality, namely causal/local, noncausalonlocal, causalonlocal, and noncausal/local for different values of the power law exponent y. The power law wave equation is causal for power law exponents 0 ≤ y <; 1, and the power law wave equation is noncausal for 1 <; y ≤ 2. Also, the power law wave equation is local for y = 0 and y = 2, while the power law wave equation is nonlocal for 0 <; y <; 1 and 1 <; y <; 2. Causal and noncausal responses are described through calculation of the time-domain Green's function with the Stable Toolbox, whereas local and nonlocal behavior is demonstrated for the integer and fractional derivatives. When y = 0, the power law wave equation simultaneously describes local and causal behavior, and when y = 2, the power law wave equation is noncausal and local. When 0 <; y <; 1, the power law wave equation is causal and nonlocal, and when 1 <; y <; 2, the power law wave equation is noncausal and nonlocal. Numerical results show an example of a causal response for a nonlocal system with power law exponent y = 1/2, a noncausal response for a nonlocal system with power law exponent y = 3/2, and a noncausal response for a local system with power law exponent y = 2. From these results, the causal and noncausal responses are easily identified, and then additional mathematical analysis of the operators is required to distinguish between local and nonlocal behavior.