Meshfree Barycentric Interpolation Collocation Method For Solving Integral-Differential Equations Of Fractional Order

Fractional integral-differential equation can accurately describe the viscoelastice materials, signal processing and some other problems. Most of these problems have no exact solution. So the numerical solutions of fractional integral-differential equations plays a great role. In recent years, there are many methods about the numerical solution of fractional integral-differential equation, such as finite difference method, finite element method, wavelet method, homotopy perturbation method and so on. However, these methods exist the defects of the large number of calculation, unstable numerical results, and lower accuracy and so on. The numerical solution of fractional integral-differential equations can be solved by using the meshfree barycentric interpolation collocation method, which can make up for these deficiencies. In this paper, several classes of Riemann-Liouville fractional integral-differential equations are solved separately by using the barycentric interpolation collocation method, and the numerical results are analyzed with reasonable error and accuracy.Firstly, using barycentric interpolation collocation method, the discrete calculation formula of various equations are derived, including fractional order Fredholm integral equation, fractional order Volterra integral equation, fractional order Fredholm-Volterra integral equation, fractional integral-differential equation and fractional order Volterra integral equations. Then through contraction mapping principle, the existence and uniqueness of the numerical solutions are proved, respectively. And the error estimates are also carried out. Finally, numerical results are used to verify the reliability and effectiveness of the proposed method. In the numerical examples, compared with equidistant nodes, the second class of Chebyshev nodes and the general interpolation method, it can be found that barycentric interpolation collocation method has higher accuracy than the general numerical interpolation method. The validity and reliability of this method are verified, and the conditions of affecting accuracy and the range of application of the two nodes are also obtained.