{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T15:43:05Z","timestamp":1772293385178,"version":"3.50.1"},"reference-count":35,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2012,4]]},"abstract":"<jats:p> The fundamental description of relaxation (T<jats:sub>1<\/jats:sub> and T<jats:sub>2<\/jats:sub>) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T<jats:sub>1<\/jats:sub> and T<jats:sub>2<\/jats:sub> relaxation. The T<jats:sub>1<\/jats:sub> decay is stable for the range of delays tested (1 \u03bcsec to 200 \u03bcsec), while the T<jats:sub>2<\/jats:sub> relaxation in this extended model exhibits a critical delay (typically 100 \u03bcsec to 200 \u03bcsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation. <\/jats:p>","DOI":"10.1142\/s021812741250071x","type":"journal-article","created":{"date-parts":[[2012,3,14]],"date-time":"2012-03-14T07:14:38Z","timestamp":1331709278000},"page":"1250071","source":"Crossref","is-referenced-by-count":38,"title":["GENERALIZED FRACTIONAL ORDER BLOCH EQUATION WITH EXTENDED DELAY"],"prefix":"10.1142","volume":"22","author":[{"given":"SACHIN","family":"BHALEKAR","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Pune, Pune 411007, India"},{"name":"Department of Mathematics, Shivaji University, Kolhapur 416004, India"}]},{"given":"VARSHA","family":"DAFTARDAR-GEJJI","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Pune, Pune 411007, India"}]},{"given":"DUMITRU","family":"BALEANU","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Cankaya University, 06530 Ankara, Turkey"},{"name":"Institute of Space Sciences, P.O. 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