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A space-time spectral collocation algorithm for the variable order fractional wave equation
A H Bhrawy1, E H Doha2, J F Alzaidy3
1Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt.
Springerplus
|August 19, 2016
Summary
This study introduces a numerical method for the variable order fractional wave equation, crucial for fields like acoustics and fluid dynamics. The approach effectively solves complex wave equations using shifted Jacobi polynomials and collocation schemes.
Area of Science:
- Numerical analysis
- Applied mathematics
- Fractional calculus
Background:
- The variable order wave equation is fundamental in acoustics, electromagnetics, and fluid dynamics.
- Solving space-time variable order fractional wave equations with variable coefficients presents significant challenges.
Purpose of the Study:
- To propose an effective numerical method for solving the space-time variable order fractional wave equation with variable coefficients in a bounded domain.
- To reduce the complex fractional wave equation to a system of easily solvable algebraic equations.
Main Methods:
- Utilizing shifted Jacobi polynomials as basis functions.
- Employing the Caputo definition for the variable-order fractional derivative.
- Combining shifted Jacobi-Gauss-Lobatto collocation for spatial discretization and shifted Jacobi-Gauss-Radau collocation for temporal discretization.
Main Results:
- The proposed numerical method transforms the fractional wave equation into a system of algebraic equations.
- Numerical examples demonstrate the effectiveness and accuracy of the developed method.
Conclusions:
- The proposed numerical technique provides an efficient and accurate solution for the space-time variable order fractional wave equation.
- This method offers a valuable tool for researchers in acoustics, electromagnetics, and fluid dynamics.
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