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In this research study, a bilinear Chebyshev pseudo-spectral method is considered to solve the two-dimensional nonlinear acoustic wave equation and extended to solve the threedimensional nonlinear wave equations. First, the wave equation is spatially approximated by a bilinear Chebyshev pseudo-spectral method and a second-order Sylvester-type matrix differential equation (MDE) is obtained. Using an eigenvalue decomposition method, this equation is transformed to a set of separated second-order ODEs in the new coordinates. Finally, using an implicit Runge-Kutta-Nyström (IRKN) or stabilized Numerov method, the arising second-order ODEs will be solved. Transforming into a set of separated second-order ODEs causes one to reduce the computational complexity. Specifically, in the higher dimensional models, the cost of computation is considerably reduced. The convergence analysis is also investigated theoretically. Furthermore, the conservation of energy is numerically investigated. Finally, Some numerical examples are included to confirm the accuracy and efficiency and the fast computing behavior of the approach. These examples represent that our approach has superiority over the Kronecker approach. [ABSTRACT FROM AUTHOR]

This paper presents a numerical scheme based on the Galerkin finite element method and cubic B-spline base function with quadratic weight function to approximate the numerical solution of the time-fractional Burger's equation, where the fractional derivative is considered in the Caputo sense. The proposed method is applied to two examples by using the L 2 and L ∞ error norms. The obtained results are compared with a previous existing method to test the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

This paper develops the numerical method based on the Sinc function and B-Spline scaling functions for applying the time-fractional convection-diffusion problem. Here, our approximation is combination of collocation method and transferred functions. For this purpose, we approximate the space dimension of the convection-diffusion problem by using B-spline scaling functions, and for the temporal direction, which has a fractional derivative, we use transferred DE-Sinc functions. With the help of numerical examples, we measure the efficiency and accuracy of the proposed method. In each example, the numerical results obtained with the methods of different references have been compared. [ABSTRACT FROM AUTHOR]

This research study deals with the numerical solutions of linear and nonlinear time‐fractional subdiffusion equations of distributed order. The main aim of our approach is based on the hybrid of block‐pulse functions and shifted Legendre polynomials. We produce a novel and exact operational vector for the fractional Riemann–Liouville integral and use it via the Gauss–Legendre quadrature formula and collocation method. Consequently, we reduce the proposed equations to systems of equations. The convergence and error bounds for the new method are investigated. Six problems are tested to confirm the accuracy of the proposed approach. Comparisons between the obtained numerical results and other existing methods are provided. Numerical experiments illustrate the reliability, applicability, and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]