A spectral approach to analyze the nonlinear oscillatory fractional-order differential equations. (May 2021)

• Record Type: Journal Article
• Authors: Hamid, Muhammad; Usman, Muhammad; Haq, Rizwan Ul; Tian, Zhenfu
• Is Part Of: Chaos, solitons and fractals. Volume 146(2021)
• Publication Date: 2021-05
• Subjects: Caputo's derivative -- Chelyshkov polynomials -- Oscillatory problems -- Operational matrices -- Picard iterative method -- Numerical solutions; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2021.110921
• Languages: English
• ISSNs: 0960-0779

Abstract

Highlights: The computational scheme is based on operational matrices of derivative and coupled with Picard iterative method. The approximation of the solution is being made through Chelyshkov polynomials. The residual vector is presented in the form of iterative formula based upon Picard iteration scheme tackled by means of collocation method. The efficiency of the hybrid method is tested for some simple fractional order problems and through comparison with existing numerical methods. The error and convergence of the proposed scheme is presented while comparison with the numerical scheme (RK-4) and then validated with existing published work. The comparative study is being made in the form of tables and set of graphs while the physical behavior of problems is stated in detail. Abstract: The study of complex nonlinear mathematical models of fractional-order needs more attention in recent decades due to its enormous contribution to science and technology. Herein, a combined algorithm is proposed using the Chelyshkov polynomial method (CPM) and Picard iterative (PI) scheme. The proposed Picard Chelyshkov polynomial method (PCPM) is used to attain nonlinear oscillatory problems of arbitrary orders that do not have the exact solutions in the literature. The PCPM covert the highly nonlinear fractional-order oscillatory Problems into a linear algebraic equations system. However, the Picard scheme is to tackle the nonlinearity factor that appears in the differential equations. The proposed method's performance is examined through some test problems of fractional order while authenticated via some numerical methods. A comprehensive comparative study discussed with published work to show that the presented PCPM. To summarize, a more efficient and accurate tool is found to inspect the solution of fractional-order highly nonlinear models.