Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

Abstract

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

Approximate solutions for a certain class of fractional optimal control problems using Laguerre collocation method

“…5. Solve the resulting non-linear system of algebraic equations using Newton iteration method to obtain c 0 , c 1 , c 2 , c 3 , then the function x(t) which extremes FOCPs (13) has the form (16). Therefore, the control u(t) will obtain as follows The solution obtained using the presented method is in excellent agreement with the already exact solution and show that this approach can be solved the problem effectively.…”

“…i=0 , α > −1 are defined on the unbounded interval (0, ∞) and can be determined with the aid of the following recurrence formula ( [3], [16…”

“…Therefore, the control u(t) will obtain as follows The solution obtained using the presented method is in excellent agreement with the already exact solution and show that this approach can be solved the problem effectively. It is evident that the overall errors can be made smaller by adding new terms from the series (16). Comparisons are made between approximate solutions and exact solutions to illustrate the validity and the great potential of the proposed technique and the advantage this method from the other methods [10].…”

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

“…[25][26][27][28][29] In References [30,31], the fractional-order wave equation was solved using the finite difference and Crank-Nicholson methods, respectively. In Reference [32,33], the space fractional-order wave equation is solved using Laguerre and Legendre polynomials, respectively. The papers introduce approaches for numerically solving the time fractional-order wave equation, 34,35 with fractional derivatives stated in the Caputo sense.…”