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<p>In this paper, the time-fractional telegraph equation (TFTE) for a transmission line is derived using the fractional forms of Kirchhoff’s law and Ohm’s law. The dissipative behaviour arising in an electric circuit is analysed using fractional calculus. Here, an equation for an electrical circuit containing the parameters – conductance (<i>G</i>), inductance (<i>L</i>), resistance (<i>R</i>) and capacitance (<i>C</i>) – is proposed within the framework of the newly developed Caputo–Fabrizio (CF) derivative. To balance the dimensions, we introduce a new physical parameter, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, which represents the inner fractional structure of the circuit. First, we analyse the sinusoidal solutions for the basic parameters of the model. In the second stage, we analyse the effects of varying both damping coefficients while keeping the transmission velocity fixed, observing the pulse shape at different times and fractal orders. Finally, we examine how the voltage pulse changes with the fractal order <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\( \alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> at different damping coefficients, transmission velocities and time instances. The fractal order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\( \alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> is found to be the most influential parameter of the telegraph equation in characterising inhomogeneity.</p>

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Construction and different analytical solutions of time-fractional telegraph equation with non-singular kernel

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Swapan Biswas,
  • Shantanu Das,
  • Uttam Ghosh

摘要

In this paper, the time-fractional telegraph equation (TFTE) for a transmission line is derived using the fractional forms of Kirchhoff’s law and Ohm’s law. The dissipative behaviour arising in an electric circuit is analysed using fractional calculus. Here, an equation for an electrical circuit containing the parameters – conductance (G), inductance (L), resistance (R) and capacitance (C) – is proposed within the framework of the newly developed Caputo–Fabrizio (CF) derivative. To balance the dimensions, we introduce a new physical parameter, \( \sigma \) σ , which represents the inner fractional structure of the circuit. First, we analyse the sinusoidal solutions for the basic parameters of the model. In the second stage, we analyse the effects of varying both damping coefficients while keeping the transmission velocity fixed, observing the pulse shape at different times and fractal orders. Finally, we examine how the voltage pulse changes with the fractal order \( \alpha \) α at different damping coefficients, transmission velocities and time instances. The fractal order \( \alpha \) α is found to be the most influential parameter of the telegraph equation in characterising inhomogeneity.