Generalized shifted Chebyshev polynomials: Solving a general class of nonlinear variable order fractional PDE

We introduce a new general class of nonlinear variable order fractional partial differential equations (NVOFPDE). The NVOFPDE contains, as special cases, several partial differential equations, such as the nonlinear variable order (VO) fractional equations usually denoted as Klein-Gordon, diffusion-wave and convection-diffusion-wave. To find the numerical solution of the NVOFPDE, we formulate a novel class of basis functions called generalized shifted Chebyshev polynomials (GSCP) that includes the shifted Chebyshev polynomials as a particular case. The solution of the NVOFPDE is expanded following the GSCP and the corresponding operational matrices of VO fractional derivatives (VO-FD), in the Caputo type, are obtained. An optimization method based on the GSCP and the Lagrange multipliers converts the problem into a system of nonlinear algebraic equations. The convergence analysis is guaranteed through a theorem concerning the GSCP and several numerical examples confirm the precision of the method.