Over the last few years, there has been a
significant increase in attention paid to fractional
differential equations, given their wide array of applications in the fields of
physics and engineering. The recent development of using fractional telegraph
equations as models in some fields (e.g., the thermal diffusion in fractal
media) has heightened the importance of examining the method of solutions for
such equations (both approximate and analytic). The present work is designed to
serve as a valuable contribution to work in this field. The key objective of
this work is to propose a general framework that can be used to guide quadratic
spline functions in order to create a numerical method for obtaining an approximation solution using the linear
space-fractional telegraph equation. Additionally, the Von Neumann
method was employed to measure the stability of the analytical scheme, which
showed that the proposed method is conditionally stable. What’s more, the
proposal contains a numerical example that illustrates how the proposed method
can be implemented practically, whilst the error estimates and numerical
stability results are discussed in depth. The findings indicate that the
proposed model is highly effective, convenient and accurate for solving the
relevant problems and is suitable for use with approximate solutions acquired
through the two-dimensional differential transform method that has been
developed for linear partial differential equations with space- and
time-fractional derivatives.
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