Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

Automated summary

This paper introduces a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation. The method utilizes shifted Chebyshev polynomials to collocate the problem in the time dimension, and introduces certain functions to homogenize the resulting system of ordinary differential equations, producing an approximate solution to the problem. Through extensive experimentation, the effectiveness and accuracy of the proposed technique are demonstrated, highlighting its efficiency and precision in approximating solutions. The convergence of the method is also investigated, providing theoretical insights into its performance. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.