Hybrid technique of conformal mapping and Chebyshev collocation method for solving time-space fractional order wave equation

This work presents a numerical approach for solving the time-space fractional-order wave equation. The time-fractional derivative is described in the conformal sense, whereas the space-fractional derivative is given in the Caputo sense. The investigated technique is based on the third-kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third-kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non-standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.