Numerical Solution of Partial Differential Equations Using Lucas and Fibonacci Polynomials

Abstract

In this work, we suggested a numerical technique in which Lucas polynomials are combined with Fibonacci polynomials for solutions of partial differential equations (PDEs). These PDEs having either integer or fractional order time derivative. For time discretization of integer case θ- weighted scheme (0 ≤ θ ≤ 1) is used, whereas, for fractional case, the same discretization scheme is combined with a simple quadrature formula. Lucas polynomials are utilized for ap proximation of unknown functions while their derivatives are expanded in terms of Fibonacci series combined with differentiation matrix. Finally, with the help of the collocation points, the given PDE reduces to a system of linear algebraic equations, which are then solved via LU de composition method. Stability and convergence analysis of the proposed scheme are discussed theoretically and verified computationally. The proposed scheme is validated via application to several classes of (1 + 1) and (1 + 2) di mensional PDEs. Accuracy of the computed solutions are measured using L∞, L2, root mean square Lrms and relative Lrel error norms. Validation and efficiency of the scheme in space and time are analyzed through variation of grid points M and time step τ. For the case of a non exact solution, obtained results are confirmed through the spectral radius of the scheme. The documented results, in the form of tables and figures, reveal very good agreement with true solutions, as well as high accuracy in comparison to earlier proposed techniques available in the literature.