1 Introduction
2 Preliminaries
3 Conformable derivative characterizations of magnetic timelike curves
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Conformable derivatives of normalizing function for \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}} _{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) of timelike \(T{\textbf{t}}\)-magnetic curves are obtained as followsThen, normalizing functions of \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}}_{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) are$$\begin{aligned} D_{\theta }\Gamma \left( {\textbf{t}}\right)= & {} \left( (k_{1}\varsigma ^{1-\theta })^{2}+(k_{2}\varsigma ^{1-\theta })^{2}\right) \textbf{t}+\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta } +\varsigma ^{1-2\theta }(1-\theta )k_{1}\right) {\textbf{m}}_{1} \\{} & {} +\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }+(-\theta +1)\varsigma ^{1-2\theta }k_{2}\right) {\textbf{m}}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right)= & {} \left( k_{1}(-\theta +1)\varsigma ^{1-2\theta }+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }\rho k_{2}\right) {\textbf{t}} \\{} & {} + (k_{1}\varsigma ^{1-\theta })^{2}{\textbf{m}}_{1}+\left( \dfrac{d\rho }{ d\varsigma }\varsigma ^{1-\theta }+k_{1}\varsigma ^{2-2\theta }k_{2}\right) \textbf{ m}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right)= & {} \left( (1-\theta )\varsigma ^{1-2\theta }k_{2}+\varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }-\rho \varsigma ^{1-\theta }k_{1}\right) {\textbf{t}} \\{} & {} + \left( \varsigma ^{2-2\theta }k_{1}k_{2}-\varsigma ^{1-\theta }\dfrac{ d\rho }{d\varsigma }\right) {\textbf{m}}_{1}+(\varsigma ^{1-\theta }k_{2})^{2}\textbf{ m}_{2}. \end{aligned}$$and$$\begin{aligned} {{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} -{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2}+(k_{2}\varsigma ^{1-\theta })^{2}){\textbf{t}} \\{} & {} +\varsigma ^{1-\theta }k_{1}{\textbf{m}}_{1}+\varsigma ^{1-\theta }k_{2}{\textbf{m}}_{2}, \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{\textbf{I}}_{\theta }^{x}(\rho k_{2}\varsigma ^{1-\theta }){\textbf{t}}+\rho {\textbf{m}}_{2}\textbf{,} \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} {\textbf{I}}_{\theta }^{x}(\rho k_{1}\varsigma ^{1-\theta })\mathbf {t-}\rho {\textbf{m}}_{1}, \end{aligned}$$Moreover, we get recursions of \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}}_{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) vector fields$$\begin{aligned} {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right)= & {} -{\textbf{I}}_{\theta }^{x}\left( -k_{1}\varsigma ^{1-\theta }\left( (1-\theta )k_{2}\varsigma ^{1-2\theta }+\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) \right. \\{} & {} \left. +k_{2}\varsigma ^{1-\theta }\left( \varsigma ^{1-2\theta }k_{1}(1-\theta ) +\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }\right) \right) {\textbf{t}}\\{} & {} -\left( (1-\theta )k_{2}\varsigma ^{1-2\theta }+\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) {\textbf{m}}_{1}\\{} & {} +\left( \varsigma ^{1-2\theta }k_{1}(1-\theta )+\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }\right) {\textbf{m}}_{2}, \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{1}\right) \right)= & {} {\textbf{I}}_{\theta }^{x}\left( \dfrac{d\rho }{d\varsigma } \varsigma ^{2-2\theta }k_{1}\right) {\textbf{t}}-\left( k_{1}\varsigma ^{2-2\theta }k_{2}+\varsigma ^{1-\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{m}} _{1}+(\varsigma ^{1-\theta }k_{1})^{2}{\textbf{m}}_{2}, \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{2}\right) \right)= & {} {\textbf{I}}_{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta } \dfrac{d\rho }{d\varsigma }\right) {\textbf{t}}-(\varsigma ^{1-\theta }k_{2})^{2} {\textbf{m}}_{1}+\left( k_{1}k_{2}\varsigma ^{2-2\theta }-\dfrac{d\rho }{d\varsigma } \varsigma ^{1-\theta }\right) {\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {\textbf{I}} _{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( (1-\theta )k_{2}\varsigma ^{-2\theta +1}+\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) \right. \\{} & {} \left. +\varsigma ^{-\theta +1}k_{2}\left( k_{1}\varsigma ^{-2\theta +1}(1-\theta )+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) \right) {\textbf{t}}\\{} & {} +\left( k_{2}(1-\theta )\varsigma ^{1-2\theta }+\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) {\textbf{m}}_{1}-\left( k_{1}(1-\theta )\varsigma ^{1-2\theta }+\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }\right) {\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) =-{\textbf{I}} _{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{t}} \\{} & {} +\left( k_{1}k_{2}\varsigma ^{2-2\theta }+\dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }\right) {\textbf{m}}_{1}-(\varsigma ^{1-\theta }k_{1})^{2}{\textbf{m}}_{2},\\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) =-{\textbf{I}} _{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{t}} \\{} & {} +(\varsigma ^{1-\theta }k_{2})^{2}{\textbf{m}}_{1}-\left( k_{1}k_{2}\varsigma ^{2-2\theta }-\dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }\right) {\textbf{m}} _{2}. \end{aligned}$$
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Conformable derivatives of normalizing function for \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}} _{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) of timelike \(T{\textbf{m}}_{1}\)-magnetic curves are obtainedwhere \(d_{0}\) and \(d_{1}\) are recursion constants and moreover, derivatives of \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}}_{1}\right) , \) \(\Gamma \left( {\textbf{m}}_{2}\right) \) are$$\begin{aligned} {{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} -{\textbf{I}}_{\theta }^{x}(\varsigma ^{2-2\theta }k_{1}{}^{2}+\varsigma ^{1-\theta }\mu k_{2}) {\textbf{t}} \\{} & {} +\varsigma ^{1-\theta }k_{1}{\textbf{m}}_{1} + \mu {\textbf{m}}_{2}, \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} d_{0}\textbf{t,} \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} d_{1}{\textbf{t}}, \end{aligned}$$We have that$$\begin{aligned} D_{\theta }\Gamma \left( {\textbf{t}}\right)= & {} ((k_{1}\varsigma ^{1-\theta })^{2}+\varsigma ^{1-\theta }\mu k_{2})\textbf{t}+\left( \varsigma ^{2-2\theta } \dfrac{dk_{1}}{d\varsigma } +(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) {\textbf{m}}_{1}+\varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma }{\textbf{m}}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right)= & {} \left( (1-\theta )\varsigma ^{1-2\theta }k_{1}+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) {\textbf{t}} + k_{1}^{2}\varsigma ^{2-2\theta }{\textbf{m}}_{1}+k_{1}k_{2}\varsigma ^{2-2\theta }{\textbf{m}}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right)= & {} \varsigma ^{1-\theta }\dfrac{ d\mu }{d\varsigma }\textbf{t}+\mu \varsigma ^{1-\theta }k_{1}{\textbf{m}} _{1}+k_{2}\varsigma ^{1-\theta }\mu {\textbf{m}}_{2}. \end{aligned}$$Thus, we get$$\begin{aligned} {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right)= & {} -{{\textbf{I}}}_{\theta }^{x}\left( -\varsigma ^{2-2\theta }\dfrac{ d\mu }{d\varsigma }+\varsigma ^{1-\theta }k_{2}(\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma } +(1-\theta )\varsigma ^{1-2\theta }k_{1})\right) {{\textbf{t}}}\\{} & {} -\varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma }{\textbf{m}}_{1}+\left( \varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) {\textbf{m}}_{2}, \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{1}\right) \right)= & {} -d_{2}{{\textbf{t}}}-k_{1}\varsigma ^{-2\theta +2}k_{2}{\textbf{m}}_{1}+(k_{1}\varsigma ^{1-\theta })^{2}{\textbf{m}}_{2}, \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{2}\right) \right)= & {} -d_{3}{{\textbf{t}}}-\mu \varsigma ^{1-\theta }k_{2}{\textbf{m}}_{1}+\mu k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}. \end{aligned}$$where \(d_{2}\) and \(d_{3}\) are recursion constants.$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {{\textbf{I}} }_{\theta }^{x}\left( -\dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }k_{2}\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta } +(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) {{\textbf{t}}}\\{} & {} +\varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma }{\textbf{m}}_{1}-\left( \varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) {\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) =d_{2} {{\textbf{t}}}+\varsigma ^{2-2\theta }k_{1}k_{2}{\textbf{m}} _{1}-(\varsigma ^{1-\theta }k_{1})^{2}{\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) =d_{3} {{\textbf{t}}}+\mu \varsigma ^{1-\theta }k_{2}{\textbf{m}}_{1}-\mu k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \end{aligned}$$
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Conformable derivatives of normalizing function for timelike \(T{\textbf{m}}_{2}\)-magnetic curves are obtainedThen, normalizing functions of \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}}_{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) are$$\begin{aligned} D_{\theta }\Gamma \left( {\textbf{t}}\right)= & {} \left( (\varsigma ^{1-\theta }k_{2})^{2}+\gamma \varsigma ^{1-\theta }k_{1}\right) \textbf{t}+\varsigma ^{1-\theta }\dfrac{d\gamma }{d\varsigma }{\textbf{m}}_{1} \\{} & {} +\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) {\textbf{m}}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right)= & {} \varsigma ^{1-\theta }\dfrac{ d\gamma }{d\varsigma }\textbf{t}+\gamma k_{1}\varsigma ^{1-\theta }{\textbf{m}} _{1}+\gamma k_{2}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right)= & {} \left( (1-\theta )\varsigma ^{1-2\theta }k_{2}+\varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }\right) {\textbf{t}} \\{} & {} + \varsigma ^{2-2\theta }k_{1}k_{2}{\textbf{m}}_{1}+(\varsigma ^{1-\theta }k_{2})^{2}{\textbf{m}}_{2}. \end{aligned}$$where \(d_{4}\) and \(d_{5}\) are recursion constants and moreover, we give that$$\begin{aligned} {{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} -{\textbf{I}}_{\theta }^{x}(\varsigma ^{2-2\theta }k_{2}^{2}+k_{1}\varsigma ^{1-\theta }\gamma ){\textbf{t}} \\{} & {} +\gamma {\textbf{m}}_{1} + \varsigma ^{-\theta +1}k_{2}{\textbf{m}}_{2}, \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} d_{4}\textbf{t,} \\ {{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} d_{5}{\textbf{t}}, \end{aligned}$$Recursions for \(\Gamma \left( {\textbf{t}}\right) \), \(\Gamma \left( {\textbf{m}} _{1}\right) \), \(\Gamma \left( {\textbf{m}}_{2}\right) \) are$$\begin{aligned} {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right)= & {} -{\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1} \left( \dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }+(-\theta +1)\varsigma ^{1-2\theta }k_{2}\right) +k_{2}\varsigma ^{2-2\theta }\dfrac{d\gamma }{d\varsigma }\right) \textbf{t}\\{} & {} -\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) {{\textbf{m}}_{1}+}\varsigma ^{1-\theta }\dfrac{ d\gamma }{d\varsigma }{{\textbf{m}}_{2},} \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{1}\right) \right)= & {} -d_{6}\mathbf {t-}\gamma k_{2}\varsigma ^{1-\theta } {\textbf{m}}_{1}-\gamma k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ {{\mathbb {N}}}\left( {\textbf{t}}\times D_{\theta }\Gamma \left( {\textbf{m}} _{2}\right) \right)= & {} -d_{7}\mathbf {t-}(\varsigma ^{1-\theta }k_{2})^{2} {{\textbf{m}}_{1}\mathbf {-}}\varsigma ^{2-2\theta }k_{1}k_{2}\mathbf { {\textbf{m}}_{2}.} \end{aligned}$$where \(d_{6}\) and \(d_{7}\) are recursion constants.$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {\textbf{I}} _{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}(\dfrac{dk_{2}}{ d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-2\theta }(1-\theta )k_{2}) +k_{2}\varsigma ^{2-2\theta }\dfrac{d\gamma }{d\varsigma }\right) \textbf{t}\\{} & {} +\left( \dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) {{\textbf{m}}_{1}-}\dfrac{d\gamma }{d\varsigma } \varsigma ^{1-\theta }{{\textbf{m}}_{2},} \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) =d_{6} \mathbf {t+}\gamma k_{2}\varsigma ^{1-\theta }{\textbf{m}}_{1}+\gamma k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) =d_{7} \mathbf {t+}(\varsigma ^{1-\theta }k_{2})^{2}{{\textbf{m}}_{1}+} \varsigma ^{2-2\theta }k_{1}k_{2}{{\textbf{m}}_{2},} \end{aligned}$$
4 Fermi–Walker conformable derivative
4.1 F–W conformable derivative for normalizing function
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Conformable derivative of normalizing functions of timelike \(T{\textbf{t}}\)-magnetic curves are given as followsMoreover, F–W conformable derivative for \({{\mathbb {N}}}\Gamma \left( {\textbf{t}} \right) \), \({{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right) \), \({{\mathbb {N}}} \Gamma \left( {\textbf{m}}_{2}\right) \) normalizing functions are obtained$$\begin{aligned} D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} \left( \dfrac{dk_{1}}{ d\varsigma }\varsigma ^{-2\theta +2}+k_{1}\varsigma ^{1-\theta }{\textbf{I}} _{\theta }^{x}((\varsigma ^{1-\theta }k_{1})^{2}+(\varsigma ^{1-\theta }k_{2})^{2})\right) {\textbf{m}}_{1} \\{} & {} +\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }+k_{2}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}((\varsigma ^{1-\theta }k_{1}) ^{2}+(\varsigma ^{1-\theta }k_{2})^{2}){\textbf{m}}_{2}, \right. \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -\varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}(\rho \varsigma ^{1-\theta }k_{2}) {\textbf{m}}_{1}+\left( \dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }-\varsigma ^{1-\theta }k_{2}{\textbf{I}}_{\theta }^{x}(k_{2}\rho \varsigma ^{1-\theta })\right) {\textbf{m}}_{2}, \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} \left( \varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}(k_{1}\rho \varsigma ^{1-\theta })-\varsigma ^{1-\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{m}} _{1}+\varsigma ^{1-\theta }k_{2}{\textbf{I}}_{\theta }^{x}(k_{1}\rho \varsigma ^{1-\theta }){\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} -(\varsigma ^{3-3\theta }k_{1}^{2}+\varsigma ^{3-3\theta }k_{2}^{2})\textbf{t}+\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta } \right. \\{} & {} +\varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}((\varsigma ^{1-\theta }k_{1})^{2}+(\varsigma ^{1-\theta }k_{2})^{2})+\varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}((\varsigma ^{1-\theta }k_{1} )^{2} \\{} & {} +(k_{2}\varsigma ^{1-\theta })^{2})){\textbf{m}}_{1}+\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }+k_{2}\varsigma ^{1-\theta }\textbf{I }_{\theta }^{x}((\varsigma ^{1-\theta }k_{1})^{2} \right. \\{} & {} +(k_{2}\varsigma ^{1-\theta })^{2})-\varsigma ^{2-2\theta }k_{2} {\textbf{I}}_{\theta }^{x}((\varsigma ^{1-\theta }k_{1}) ^{2}+(\varsigma ^{1-\theta }k_{2})^{2})){\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -k_{2}\rho \varsigma ^{2-2\theta }{\textbf{t}}+(-\varsigma ^{1-\theta }k_{1} {\textbf{I}}_{\theta }^{x}(k_{2}\rho \varsigma ^{1-\theta }) \\{} & {} +k_{1}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}(k_{2}\rho \varsigma ^{1-\theta })){\textbf{m}}_{1}+\left( \dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta } \right. \\{} & {} \left. \left. -k_{2}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}(\rho k_{2}\varsigma ^{1-\theta })+k_{2}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}(\rho k_{2}\varsigma ^{1-\theta }\right) \right) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} k_{1}\rho \varsigma ^{2-2\theta }\textbf{t}-(k_{1}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}(\rho k_{1}\varsigma ^{1-\theta }) \\{} & {} \left. \left. +\dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }+\varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}(\rho k_{1}\varsigma ^{1-\theta }\right) \right) {\textbf{m}} _{1} \\{} & {} +(\varsigma ^{2-2\theta }k_{2}{\textbf{I}}_{\theta }^{x}(k_{1}\rho \varsigma ^{1-\theta })+\varsigma ^{1-\theta }k_{2}{\textbf{I}}_{\theta }^{x}(k_{1}\rho \varsigma ^{1-\theta })){\textbf{m}}_{2}. \end{aligned}$$
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Conformable derivative of normalizing functions of timelike \(T{\textbf{m}}_{1}\)-magnetic curves are givenF–W conformable derivative for \({{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right) \), \({{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right) \), \({{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right) \) normalizing functions are obtained$$\begin{aligned} D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} (\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }-k_{1}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2} \\{} & {} +\mu k_{2}\varsigma ^{1-\theta })+\varsigma ^{1-2\theta }(1-\theta )k_{1}){\textbf{m}}_{1} \\{} & {} +(-k_{2}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2}+\mu k_{2}\varsigma ^{1-\theta })+\dfrac{d\mu }{ d\varsigma }\varsigma ^{1-\theta }){\textbf{m}}_{2}, \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} k_{1}d_{0}\varsigma ^{1-\theta }{\textbf{m}}_{1}+k_{2}d_{0}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} k_{1}d_{1}\varsigma ^{1-\theta }{\textbf{m}}_{1}+k_{2}d_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} (-\varsigma ^{3-3\theta }k_{1}^{2}-k_{2}\mu \varsigma ^{2-2\theta })\textbf{t} \\{} & {} +\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta }) ^{2}+k_{2}\mu \varsigma ^{1-\theta })+\varsigma ^{1-2\theta }(1-\theta )k_{1}\right. \\{} & {} +\varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2}+k_{2}\mu \varsigma ^{1-\theta })){\textbf{m}} _{1}+(\varsigma ^{1-\theta }k_{2}{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2} \\{} & {} +\mu k_{2}\varsigma ^{1-\theta })+\dfrac{d\mu }{d\varsigma }\varsigma ^{1-\theta }+k_{2}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}((k_{1}\varsigma ^{1-\theta })^{2}+\mu k_{2}\varsigma ^{1-\theta })){\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} (k_{1}d_{0}\varsigma ^{1-\theta }-k_{1}d_{0}\varsigma ^{2-2\theta })\textbf{ m}_{1}+(k_{2}d_{0}\varsigma ^{1-\theta }-k_{1}d_{0}\varsigma ^{2-2\theta }) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} (k_{1}d_{1}\varsigma ^{1-\theta }-k_{1}d_{1}\varsigma ^{2-2\theta })\textbf{ m}_{1}+(k_{2}d_{1}\varsigma ^{1-\theta }-k_{1}d_{1}\varsigma ^{2-2\theta }) {\textbf{m}}_{2}. \end{aligned}$$
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Conformable derivative of normalizing functions of timelike \(T{\textbf{m}}_{2}\)-magnetic curves are obtainedF–W conformable derivative for \({{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right) \), \({{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right) \), \({{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right) \) normalizing functions are obtained$$\begin{aligned} D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} \left( \dfrac{d\gamma }{ d\varsigma }\varsigma ^{1-\theta }-k_{1}\varsigma ^{1-\theta }{\textbf{I}} _{\theta }^{x}(\gamma k_{1}\varsigma ^{1-\theta } \right. \\{} & {} \left. +(k_{2}\varsigma ^{1-\theta })^{2})\right) {\textbf{m}}_{1}+\left( -k_{2}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}(\gamma k_{1}\varsigma ^{1-\theta }\right. \\{} & {} \left. +(k_{2}\varsigma ^{1-\theta })^{2})+\varsigma ^{1-2\theta }(1-\theta )k_{2}+\varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma }\right) {\textbf{m}}_{2}, \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} k_{1}d_{4}\varsigma ^{1-\theta }{\textbf{m}}_{1}+k_{2}d_{4}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ D_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} k_{1}d_{5}\varsigma ^{1-\theta }{\textbf{m}}_{1}+k_{2}d_{5}\varsigma ^{1-\theta }{\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{t}}\right)= & {} (k_{2}^{2}\varsigma ^{3-3\theta }+k_{1}\gamma \varsigma ^{2-2\theta }) {\textbf{t}}+\left( \dfrac{d\gamma }{d\varsigma }\varsigma ^{1-\theta }+k_{1}\varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}(\gamma k_{1}\varsigma ^{1-\theta }+(k_{2}\varsigma ^{1-\theta })^{2}\right) \\{} & {} +k_{1}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}(k_{1}\gamma \varsigma ^{1-\theta }+(\varsigma ^{1-\theta }k_{2})^{2}){\textbf{m}}_{1}+(-k_{2} \varsigma ^{1-\theta }{\textbf{I}}_{\theta }^{x}(\gamma k_{1}\varsigma ^{1-\theta }+(k_{2}\varsigma ^{1-\theta })^{2}) \\{} & {} +\varsigma ^{1-2\theta }(1-\theta )k_{2}+\varsigma ^{2-2\theta }\dfrac{dk_{2} }{d\varsigma }+k_{2}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}(\varsigma ^{2-2\theta }k_{2}^{2}+k_{1}\gamma \varsigma ^{1-\theta })) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} (k_{1}d_{4}\varsigma ^{1-\theta }-k_{1}d_{4}\varsigma ^{2-2\theta })\textbf{ m}_{1}+(k_{2}d_{4}\varsigma ^{1-\theta }-k_{1}d_{4}\varsigma ^{2-2\theta }) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{{\mathbb {N}}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} (k_{1}d_{5}\varsigma ^{1-\theta }-k_{1}d_{5}\varsigma ^{2-2\theta })\textbf{ m}_{1}+(k_{2}d_{5}\varsigma ^{1-\theta }-k_{1}d_{5}\varsigma ^{2-2\theta }) {\textbf{m}}_{2}. \end{aligned}$$
4.2 F–W conformable derivative for recursion function
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Recursion functions of \(\Gamma \left( {\textbf{t}}\right) ,\Gamma \left( {\textbf{m}}_{1}\right) ,\Gamma \left( {\textbf{m}}_{2}\right) \) of timelike \(T{\textbf{t}}\)-magnetic curves are givenF–W conformable derivative for \({\varvec{R}}\Gamma \left( {\textbf{t}} \right) \), \({\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right) \), \( {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right) \) recursion functions are obtained$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {\textbf{I}} _{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( (1-\theta )k_{2}\varsigma ^{-2\theta +1}+\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) +\varsigma ^{-\theta +1}k_{2}\left( k_{1}\varsigma ^{-2\theta +1}\right. \right. \\{} & {} \left. \left. (1-\theta )+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) \right) {\textbf{t}} +\left( k_{2}(1-\theta )\varsigma ^{1-2\theta }+\dfrac{dk_{2}}{d\varsigma } \varsigma ^{2-2\theta }\right) {\textbf{m}}_{1}\\{} & {} -\left( k_{1}(1-\theta )\varsigma ^{1-2\theta }+\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }\right) {\textbf{m}} _{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) \\= & {} -{\textbf{I}} _{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{t}}+\left( k_{1}k_{2}\varsigma ^{2-2\theta }+\dfrac{d\rho }{d\varsigma } \varsigma ^{1-\theta }\right) {\textbf{m}}_{1}-(\varsigma ^{1-\theta }k_{1})^{2} {\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) \\= & {} -{\textbf{I}} _{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) {\textbf{t}}+(\varsigma ^{1-\theta }k_{2})^{2}{\textbf{m}}_{1}-\left( k_{1}k_{2} \varsigma ^{2-2\theta }-\dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }\right) {\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} \left( -\varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( (1-\theta )k_{2}\varsigma ^{-2\theta +1}+\dfrac{dk_{2}}{ d\varsigma }\varsigma ^{2-2\theta }\right) \right. \right. \\{} & {} +\varsigma ^{-\theta +1}k_{2}\left( k_{1}\varsigma ^{-2\theta +1}(1-\theta )+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) +\varsigma ^{-\theta +1}\dfrac{d}{d\varsigma }(k_{2}(1-\theta )\varsigma ^{1-2\theta }\\{} & {} \left. \left. +\dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) \right) {{\textbf{m}}_{1}} +\left( -\varsigma ^{2-2\theta }k_{2}{\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( (1-\theta )k_{2}\varsigma ^{-2\theta +1}+\dfrac{dk_{2}}{ d\varsigma }\varsigma ^{2-2\theta }\right) \right. \right. \\{} & {} +\varsigma ^{-\theta +1}k_{2}\left( k_{1}\varsigma ^{-2\theta +1}(1-\theta )+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) -\varsigma ^{-\theta +1}\dfrac{d}{d\varsigma }(k_{1}(1-\theta )\varsigma ^{1-2\theta } \\{} & {} +\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma })){{\textbf{m}}_{2}} +\left( -\varsigma ^{2-2\theta }k_{1}\left( \dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) +\varsigma ^{2-2\theta }k_{2}\left( \varsigma ^{2-2\theta } \dfrac{dk_{1}}{d\varsigma }\right) \right. \\{} & {} -\varsigma ^{1-\theta }k_{1}\left( (1-\theta )k_{2}\varsigma ^{-2\theta +1}+\dfrac{ dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) +\varsigma ^{-\theta +1}k_{2} \left( k_{1}\varsigma ^{-2\theta +1}(1 \right. \\{} & {} \left. -\theta )+\dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }\right) + \varsigma ^{-\theta +1}k_{1}\left( k_{2}(1-\theta )\varsigma ^{1-2\theta }+\dfrac{ dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }\right) \\{} & {} \left. -\varsigma ^{-\theta +1}k_{2}\left( k_{1}(1-\theta )\varsigma ^{1-2\theta }+\varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }\right) \right) \textbf{t}, \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} \varsigma ^{-\theta +1}k_{2}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{ d\varsigma }\right) \textbf{t}+\left( \varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) \right. \\{} & {} \left. +\varsigma ^{-\theta +1}\dfrac{d}{d\varsigma }\left( k_{1}k_{2}\varsigma ^{2-2\theta }+\dfrac{d\rho }{d\varsigma }\varsigma ^{1-\theta }\right) -\varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{ d\rho }{d\varsigma }\right) \right) {{\textbf{m}}_{1}} \\{} & {} +\left( \varsigma ^{2-2\theta }k_{2}{\textbf{I}}_{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) -\varsigma ^{-\theta +1}\dfrac{d}{ d\varsigma }(\varsigma ^{1-\theta }k_{1})^{2}-\varsigma ^{1-\theta }k_{2} {\textbf{I}}_{\theta }^{x}\left( k_{1}\varsigma ^{2-2\theta }\dfrac{d\rho }{ d\varsigma }\right) \right) {{\textbf{m}}_{2},} \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -k_{2}\varsigma ^{2-2\theta }\left( \varsigma ^{1-\theta }\dfrac{d\rho }{ d\varsigma }\right) \textbf{t}+\left( k_{1}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) \right. \\{} & {} \left. -\varsigma ^{-\theta +1}k_{1}{\textbf{I}}_{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) +\varsigma ^{-\theta +1}\dfrac{d}{ d\varsigma }(\varsigma ^{1-\theta }k_{2})^{2}\right) {{\textbf{m}}_{1}} +\left( k_{2}\varsigma ^{2-2\theta }{\textbf{I}}_{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) \right. \\{} & {} \left. -\varsigma ^{-\theta +1}k_{2}{\textbf{I}}_{\theta }^{x}\left( k_{2}\varsigma ^{2-2\theta }\dfrac{d\rho }{d\varsigma }\right) -\varsigma ^{-\theta +1}\dfrac{d}{ d\varsigma }\left( k_{1}k_{2}\varsigma ^{2-2\theta }-\dfrac{d\rho }{d\varsigma } \varsigma ^{1-\theta }\right) \right) {{\textbf{m}}_{2}.} \end{aligned}$$
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Recursion functions of \(\Gamma \left( {\textbf{t}}\right) ,\Gamma \left( {\textbf{m}}_{1}\right) ,\Gamma \left( {\textbf{m}}_{2}\right) \) of timelike \(T{\textbf{m}} _{1}\)-magnetic curves are presentedF–W conformable derivative for \({\varvec{R}}\Gamma \left( {\textbf{t}} \right) \), \({\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right) \), \( {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right) \) recursion functions are obtained$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {{\textbf{I}} }_{\theta }^{x}\left( -\dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta }+k_{2}\varsigma ^{1-\theta }\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta } \right. \right. \\{} & {} \left. \left. +(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) {{\textbf{t}}}+\varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma }{\textbf{m}}_{1}-\left( \dfrac{dk_{1} }{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-2\theta }(1-\theta )k_{1}\right) {\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) =d_{2} {{\textbf{t}}}+\varsigma ^{2-2\theta }k_{1}k_{2}{\textbf{m}} _{1}-(\varsigma ^{1-\theta }k_{1})^{2}{\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) =d_{3} {{\textbf{t}}}+\mu \varsigma ^{1-\theta }k_{2}{\textbf{m}}_{1}-\mu k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}. \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} \left( \left( - \dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }k_{2} \left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right. \right. \\{} & {} +\varsigma ^{2-2\theta }\dfrac{d\mu }{d\varsigma }k_{1}-\varsigma ^{1-\theta }k_{2}\left( \varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) -\varsigma ^{2-2\theta }k_{1}\varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma } \\{} & {} \left. +\varsigma ^{2-2\theta }k_{2}\left( \varsigma ^{2-2\theta }\dfrac{dk_{1}}{ d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) {{\textbf{t}}} +\left( \varsigma ^{1-\theta }k_{1}{{\textbf{I}}}_{\theta }^{x}\left( -\dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta } \right. \right. \\{} & {} \left. +\varsigma ^{1-\theta }k_{2}\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) +\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }\left( \varsigma ^{1-\theta }\dfrac{d\mu }{d\varsigma }\right) \\{} & {} \left. -\varsigma ^{2-2\theta }k_{1}{{\textbf{I}}}_{\theta }^{x}\left( -\dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }k_{2}\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) \right) {\textbf{m}}_{1} \\{} & {} +\left( \varsigma ^{1-\theta }k_{2}{{\textbf{I}}}_{\theta }^{x}\left( -\dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-\theta }k_{2}\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) \right. \\{} & {} -\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }\left( \varsigma ^{2-2\theta }\dfrac{dk_{1}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) -\varsigma ^{2-2\theta }k_{2}{{\textbf{I}}}_{\theta }^{x}\left( - \dfrac{d\mu }{d\varsigma }\varsigma ^{2-2\theta } \right. \\{} & {} \left. \left. +\varsigma ^{1-\theta }k_{2}\left( \dfrac{dk_{1}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{1}\right) \right) \right) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} \left( k_{1}d_{2}\varsigma ^{1-\theta }+\varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\varsigma ^{2-2\theta }k_{1}k_{2})-\varsigma ^{2-2\theta }d_{2}k_{1}\right) {\textbf{m}}_{1} \\{} & {} +\left( -\varsigma ^{2-2\theta }d_{2}k_{2}+k_{2}d_{2}\varsigma ^{1-\theta }-\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }(\varsigma ^{1-\theta }k_{1})^{2}\right) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} \left( k_{1}d_{3}\varsigma ^{1-\theta }+\varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\mu \varsigma ^{1-\theta }k_{2})-\varsigma ^{2-2\theta }d_{3}k_{1}\right) {\textbf{m}}_{1} \\{} & {} \left( k_{2}d_{3}\varsigma ^{1-\theta }-\varsigma ^{2-2\theta }d_{3}k_{2}-\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }(\mu k_{1}\varsigma ^{1-\theta })\right) {\textbf{m}}_{2}. \end{aligned}$$
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Recursion functions of \(\Gamma \left( {\textbf{t}}\right) ,\Gamma \left( {\textbf{m}}_{1}\right) ,\Gamma \left( {\textbf{m}}_{2}\right) \) of timelike \(T{\textbf{m}} _{2}\)-magnetic curves are presentedF–W conformable derivative for \({\varvec{R}}\Gamma \left( {\textbf{t}} \right) \), \({\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right) \), \( {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right) \) recursion functions are given$$\begin{aligned} {\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} -{{\mathbb {N}}}\left( {\textbf{t}} \times D_{\theta }\Gamma \left( {\textbf{t}}\right) \right) \\= & {} {\textbf{I}} _{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( \dfrac{dk_{2}}{ d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-2\theta }(1-\theta )k_{2}\right) \right. \\{} & {} \left. +k_{2}\varsigma ^{2-2\theta }\dfrac{d\gamma }{d\varsigma }\right) \textbf{t}+ \left( \dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) {{\textbf{m}}_{1}-}\dfrac{d\gamma }{d\varsigma } \varsigma ^{1-\theta }{{\textbf{m}}_{2},} \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{1}\right) \right) =d_{6} \textbf{t}+\gamma k_{2}\varsigma ^{1-\theta }{\textbf{m}}_{1}+\gamma k_{1}\varsigma ^{1-\theta }{\textbf{m}}_{2}, \\ {\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} -{{\mathbb {N}}}\left( \textbf{ t}\times D_{\theta }\Gamma \left( {\textbf{m}}_{2}\right) \right) =d_{7} \textbf{t}+(\varsigma ^{1-\theta }k_{2})^{2}{{\textbf{m}}_{1}+} \varsigma ^{2-2\theta }k_{1}k_{2}{{\textbf{m}}_{2}.} \end{aligned}$$$$\begin{aligned} {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{t}}\right)= & {} \left( \left( -\varsigma ^{1-\theta }k_{1}(\varsigma ^{2-2\theta } \dfrac{dk_{2}}{d\varsigma }+(-\theta +1)\varsigma ^{1-2\theta }k_{2})+k_{2}\varsigma ^{2-2\theta }\dfrac{d\gamma }{d\varsigma }\right) \right. \\{} & {} +k_{1}\varsigma ^{1-\theta }\left( \dfrac{dk_{2}}{d\varsigma } \varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) -k_{2}\varsigma ^{2-2\theta }\dfrac{d\gamma }{d\varsigma } \\{} & {} -\varsigma ^{2-2\theta }k_{1}\left( \varsigma ^{2-2\theta } \dfrac{dk_{2}}{d\varsigma }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) +\varsigma ^{3-3\theta }k_{2}\dfrac{d\gamma }{d\varsigma }){\textbf{t}}\\{} & {} +\left( \varsigma ^{1-\theta }k_{1}{\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma } +(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) +k_{2}\varsigma ^{2-2\theta }\dfrac{ d\gamma }{d\varsigma }\right) \right. \\{} & {} +\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }\left( \dfrac{dk_{2} }{d\varsigma }\varsigma ^{2-2\theta }+(1-\theta )\varsigma ^{1-2\theta }k_{2}\right) -\varsigma ^{2-2\theta }k_{1}{\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{ d\varsigma } \right. \right. \\{} & {} \left. \left. \left. +\varsigma ^{1-2\theta }(1-\theta )k_{2}\right) +k_{2}\varsigma ^{2-2\theta }\dfrac{ d\gamma }{d\varsigma }\right) \right) {{\textbf{m}}_{1}}+\left( \varsigma ^{1-\theta }k_{2} {\textbf{I}}_{\theta }^{x}\left( -\varsigma ^{1-\theta }k_{1}\left( \varsigma ^{2-2\theta }\dfrac{dk_{2}}{d\varsigma } \right. \right. \right. \\{} & {} \left. \left. +\varsigma ^{1-2\theta }(1-\theta )k_{2}\right) +k_{2}\varsigma ^{2-2\theta }\dfrac{ d\gamma }{d\varsigma }\right) -\varsigma ^{1-\theta }\dfrac{d}{d\varsigma }\left( \dfrac{ d\gamma }{d\varsigma }\varsigma ^{1-\theta }\right) \\{} & {} \left. \left. -\varsigma ^{2-2\theta }k_{2}{\textbf{I}}_{\theta }^{x}(-\varsigma ^{1-\theta }k_{1}\left( \dfrac{dk_{2}}{d\varsigma }\varsigma ^{2-2\theta }+\varsigma ^{1-2\theta }(1-\theta )k_{2}\right) +\varsigma ^{2-2\theta }\dfrac{ d\gamma }{d\varsigma }k_{2}\right) \right) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{1}\right)= & {} \left( k_{1}d_{6}\varsigma ^{1-\theta } + \varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\gamma k_{2}\varsigma ^{1-\theta })\mathbf {-}\varsigma ^{2-2\theta }k_{1}d_{6}\right) {\textbf{m}}_{1} \\{} & {} +\left( k_{2}d_{6}\varsigma ^{1-\theta }+\varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\gamma k_{1}\varsigma ^{1-\theta })-k_{2}d_{6}\varsigma ^{2-2\theta }\right) {\textbf{m}}_{2}, \\ {\widetilde{D}}_{\theta }{\varvec{R}}\Gamma \left( {\textbf{m}}_{2}\right)= & {} \left( k_{1}d_{7}\varsigma ^{1-\theta } + \varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\varsigma ^{1-\theta }k_{2})^{2}\mathbf {-}\varsigma ^{2-2\theta }d_{7}k_{1}\right) {\textbf{m}}_{1} \\{} & {} +\left( k_{2}d_{7}\varsigma ^{1-\theta } + \varsigma ^{1-\theta }\dfrac{d}{ d\varsigma }(\varsigma ^{2-2\theta }k_{1}k_{2})-d_{7}k_{2}\varsigma ^{2-2\theta }\right) {\textbf{m}}_{2}. \end{aligned}$$