Finite and Infinite Dimensional Reproducing Kernel Hilbert Space Approach for Bagley–Torvik Equation

In this paper, two different numerical approaches are presented in finite dimensional and infinite dimensional reproducing kernel Hilbert spaces for the fractional order Bagley–Torvik equation with boundary conditions. The reproducing kernel functions are obtained in finite dimensional Hilbert space \(\varPi _{\rho }^{n}[0,A]\) ${\Pi }_{\rho }^n[0,A]$ using Legendre polynomials, while they are obtained by a known classical method in infinite dimensional Sobolev-Hilbert space \(W_{2}^{3}[0,A]\) $W_2^3[0,A]$ . A comprehensive theoretical analysis is given for both approaches, which have different forms of reproducing kernel methods. Numerical results are calculated over wide intervals with both proposed approaches. In order to compare the efficiency of these proposed methods, the numerical results obtained for the considered six examples are presented through tabulated data and graphical representations.