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The Helmholtz equation as an elliptic partial differential equation possesses many applications in the time-harmonic wave propagation phenomena, such as the acoustic cavity and radiation wave. In this paper, we establish a numerical method based on the orthonormal shifted discrete Chebyshev polynomials for finding complex solution of this equation. The presented method transforms the Helmholtz equation into an algebraic system of equations that can be easily solved. Four practical examples are examined to show the accuracy of the proposed technique. [ABSTRACT FROM AUTHOR]

In this paper, the distributed-order time fractional diffusion equation is introduced and studied. The Caputo fractional derivative is utilized to define this distributed-order fractional derivative. A hybrid approach based on the fractional Euler functions and 2D Chebyshev cardinal functions is proposed to derive a numerical solution for the problem under consideration. It should be noted that the Chebyshev cardinal functions process many useful properties, such as orthogonality, cardinality and spectral accuracy. To construct the hybrid method, fractional derivative operational matrix of the fractional Euler functions and partial derivatives operational matrices of the 2D Chebyshev cardinal functions are obtained. Using the obtained operational matrices and the Gauss–Legendre quadrature formula as well as the collocation approach, an algebraic system of equations is derived instead of the main problem that can be solved easily. The accuracy of the approach is tested numerically by solving three examples. The reported results confirm that the established hybrid scheme is highly accurate in providing acceptable results. [ABSTRACT FROM AUTHOR]