A Pell-Lucas approximation for solving temperature distribution and oscillator models

In this article, we investigate numerical solutions of two model characterized by fractional-order nonlinear differential equations with boundary conditions. One of these models is model of the temperature distribution in lumped system of combined convection-radiation in a slab made of materials with variable thermal conductivity. Other one is the nonlinear oscillator model. The fractional term is defined in the Caputo sense. In the method based on the Pell-Lucas polynomials (PLPs), these polynomials are first expressed in matrix form. Using this matrix form, the method transforms the models into a system of matrix equations by obtaining the matrix representations of all terms in the two models and using collocation points. The solution of this system through MATLAB allows us to obtain the approximate solution. The error analysis for the presented technique is constituted. Applications of the model for different cases of fractional order derivatives in model are made through MATLAB to verify the efficiency of the presented technique. Moreover, the comparison of the results of our method with other methods in the literature shows that our method is more effective.