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Numerical Algorithm for the Solutions of Fractional Order Systems of Dirichlet Function Types with Comparative Analysis

Arqub Omar Abu

Fundamenta Informaticae

2019

Vol. 166, nr 2

111--137

The aim of this article is to introduce the reproducing kernel algorithm for obtaining the numerical solutions of fractional order systems of Dirichlet function types. The algorithm provide appropriate representation of the solutions in infinite series formula with accurately computable structures. By interrupting the n-term of exact solutions, numerical solutions of linear and nonlinear time-fractional equations of homogeneous and nonhomogeneous function type are studied from mathematical viewpoint. Convergence analysis, error estimations, and error bounds for the present algorithm are also discussed. The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods.

Approximate Solutions of DASs with Nonclassical Boundary Conditions using Novel Reproducing Kernel Algorithm

Arqub O. A.

Fundamenta Informaticae

2016

Vol. 146, nr 3

231--254

This paper presents novel reproducing kernel algorithm for obtaining the numerical solutions of differential algebraic systems with nonclassical boundary conditions for ordinary differential equations. The representation of the exact and the numerical solutions is given in the W [0; 1] and H [0; 1] inner product spaces. The computation of the required grid points is relying on the Rt (s) and rt (s) reproducing kernel functions. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. Numerical solutions of such nonclassical systems are acquired by interrupting the η-term of the exact solutions. In this approach, numerical examples were analyzed to illustrate the design procedure and confirm the performance of the proposed algorithm in the form of tabulate data and numerical comparisons. Finally, the utilized results show the significant improvement of the algorithm while saving the convergence accuracy and time.