A computational algorithm for simulating fractional order relaxation–oscillation equation

Abstract

In the present work, a collocation approach is developed to find an approximate solution of fractional relaxation-oscillation differential equation describing the processes of relaxation and oscillation in many physical systems. The method is relied on generalized Chebyshev polynomials as basis functions, collocation points, and the matrix operations. The proposed scheme converts the underlying fractional initial value problems into a matrix equation, which corresponds to a set of linear algebraic equations consist of polynomial coefficients. An error estimation based on residual function is performed to show the accuracy of the results. Hence, an improvement of the approximate solutions are obtained based upon this error estimation. To illustrate the utility and applicability of the technique, three numerical examples are given and the comparisons between the numerical results of the proposed method with existing results are carried out. The experiments show the accuracy and efficiency the present work.

Simulating accurate and effective solutions of some nonlinear nonlocal two-point BVPs: Clique and QLM-clique matrix methods

“…These methods have been successfully applied to a number of significant model problems with various (orthogonal) basis functions. Among these types of bases, we mention Morgan-Voyce [13] , Vieta-Lucas [14] , Bessel [15] , [16] , [17] , Fibonacci [18] , Jacobi [19] , Chebyshev [20] , [21] , [22] , [23] , [24] , and Vieta-Fibonacci [25] , to name a few.…”

Robust QLM-SCFTK matrix approach applied to a biological population model of fractional order considering the carrying capacity

“…Spectral collocation techniques based on (orthogonal) functions have become very useful in providing highly accurate solutions to ODEs and has an exponential order of convergence. In addition to Chebyshev polynomials [13,18], different (orthogonal) polynomial functions have been utilized inside the spectral collocation approaches in the literature. Among others, we mention Dickson [25], Bessel [14,36,48], Legendre [37], Benoulli [2], Vieta-Fibonacci [1], and Jacobi [47], to name a few.…”