Fractional Bloch equation with delay. (English) Zbl 1217.34123

Summary: We investigate a fractional generalization of the Bloch equation that includes both fractional derivatives and time delays. The appearance of the fractional derivative on the left side of the Bloch equation encodes a degree of system memory in the dynamic model for magnetization. The introduction of a time delay on the right side of the equation balances the equation by also adding a degree of system memory on the right side of the equation. The analysis of this system shows different stability behavior for the \(T_{1}\) and the \(T_{2}\) relaxation processes. The \(T_{1}\) decay is stable for the range of delays tested (\(1-100 \mu s\)), while the \(T_{2}\) relaxation in this model exhibited a critical delay (typically \(6 \mu s\)) above which the system was unstable. Delays are expected to appear in NMR systems, in both the system model and in the signal excitation and detection processes. Therefore, by including both the fractional derivative and finite time delays in the Bloch equation, we believe that we have established a more complete and more realistic model for NMR resonance and relaxation.


34K37 Functional-differential equations with fractional derivatives
34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
45J05 Integro-ordinary differential equations


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