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Purpose <p>The aim of this article is to study the solution of time-fractional Bloch equations, a system of differential equations arising in magnetic resonance, using Caputo fractional derivative in a fuzzy environment. The spin-lattice relaxation time <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{1}}\)</EquationSource> </InlineEquation> and spin-spin relaxation time <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{2}}\)</EquationSource> </InlineEquation> are treated as triangular fuzzy numbers to account for uncertainty.</p> Methods <p>A hybrid method called ARA-Residual power series method is utilized to solve the model in both crisp and fuzzy cases. It is the incorporation of residual power series method with ARA transform.</p> Results <p>The solutions are analyzed at different fractional orders, and the corresponding solution bounds are obtained. The results demonstrate the adaptability of ARA-residual power series method in dealing time-fractional Bloch equations in both crisp and fuzzy cases. Additionally, treating <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{1}}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{2}}\)</EquationSource> </InlineEquation> as TFNs improves the modeling of relaxation process in diagnostics and imaging.</p> Conclusion <p>According to the results, the present approach is an efficient method to handle the fuzzy fractional models. The computational approach highlights the importance of capturing the uncertainty in the relaxation times <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{1}}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\varvec{T}_{\varvec{2}}\)</EquationSource> </InlineEquation> for better modeling of tissue contrast, signal dynamics and characterization in NMR-based diagnostics and MRI applications.</p>

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Computational Technique for the Solution of Time-fractional Bloch Equations with Uncertain Relaxation Times

  • 期刊论文

  • 发表日期：

    2025年10月20日

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  • B. Bala Sai Sankar,
  • P. Karunakar

摘要

Purpose

The aim of this article is to study the solution of time-fractional Bloch equations, a system of differential equations arising in magnetic resonance, using Caputo fractional derivative in a fuzzy environment. The spin-lattice relaxation time \(\varvec{T}_{\varvec{1}}\) and spin-spin relaxation time \(\varvec{T}_{\varvec{2}}\) are treated as triangular fuzzy numbers to account for uncertainty.

Methods

A hybrid method called ARA-Residual power series method is utilized to solve the model in both crisp and fuzzy cases. It is the incorporation of residual power series method with ARA transform.

Results

The solutions are analyzed at different fractional orders, and the corresponding solution bounds are obtained. The results demonstrate the adaptability of ARA-residual power series method in dealing time-fractional Bloch equations in both crisp and fuzzy cases. Additionally, treating \(\varvec{T}_{\varvec{1}}\) and \(\varvec{T}_{\varvec{2}}\) as TFNs improves the modeling of relaxation process in diagnostics and imaging.

Conclusion

According to the results, the present approach is an efficient method to handle the fuzzy fractional models. The computational approach highlights the importance of capturing the uncertainty in the relaxation times \(\varvec{T}_{\varvec{1}}\) and \(\varvec{T}_{\varvec{2}}\) for better modeling of tissue contrast, signal dynamics and characterization in NMR-based diagnostics and MRI applications.